So welcome all to the session on inverse functions now I have not named it as inverse trig functions because I'll be also touching upon inverse of a function in general That is actually covered under the functions chapter So relations and functions altogether is a different chapter and under that we cover the inverse of a function But I think it's the right time that when we talk about inverse of trigonometric function We should know what's inverse of a function also Okay, so let me take this as an opportunity because sooner or later. I have to do that chapter as well with you So I'll start it today itself inverse of a function So what is the inverse of a function? Let us understand that first Inverse of a function has this concept been taken up in school Yeah, Anjali has it been taken up in school No, okay. I think Nafal has done it. So I think NPS Aajaji Nagar is yet to do it. Okay, so let me explain this concept So all of you please listen to this very carefully. It's a very important concept not only for competitive exams But also for your board exams. Okay, a lot of questions are going to come on finding inverse of a given function Okay, so as a layman, how do you understand inverse of a function? So let's say I have a function f of x Okay, I have a function f of x And I have another function I'm naming it as this Okay, now this is to be read as a symbol. It is not to be read as a operation. I'm not doing one by f Okay, I'm just writing it as a name. Okay, just like a symbol. Okay So think that I was in a shot of alphabet search instead of f I chose this symbol for my function. So this is another function now these two functions are related in this way So let's say functions are mathematical machines.  Processed it and gave you x back When there are two such machines which are doing such an activity We call these two machines as inverses of each other So they would be called as inverses of each other Okay, a very layman definition a very very layman definition Fine to give you an example for the same Let's say there's a machine f e to the power x Okay, and there's another machine L and x Now I claim that these two machines are inverses of each other. Why for the simple reason Let's say in this machine if I put in To this machine will give me e square isn't it? Okay, sorry, let me write in yellow because yellow is the original machine So sorry for that. So let's say in this machine if I put to I'll get to e square from this machine So when you put e square in this f inverse machine Okay, this machine is going to give you ln of e square ln of e square is what ln of e square is actually a 2 So you had put two here You got e square if you put e square here you got a two So these two machines would be inverses of each other. So these two machines would be inverses of each other This is fine. In short what I'm trying to say is that if your f works from a to b Okay, then f inverse will work from b to a Where b must be the range of the function Where b must be the range of the function So range of f becomes the domain of f inverse and Domain of f becomes the range of f inverse and vice versa Okay, so we already know this is domain. So I did not write it, but let me write it for clarity Okay, so domain of f. Let me write it over here domain of f becomes the range of f inverse and Range of f becomes the domain of f inverse Okay, but this is a very very superficial explanation of what is inverse of a function mean I'll go into deep Inside the concept and discuss a lot of Nitigrities about the concept meanwhile any doubt regarding what do you understand from an inverse of a function? Please let me let me know any case or in case of any doubt or any concern. Please highlight over here. Is it fine? No problem. Good afternoon, Anjali. Good afternoon to everybody who has joined in and All the best for tomorrow school There may be some initial Technical challenges. So couple of things There may be some drop-off data and all so you may be missing out hearing something from the teacher So ensure that you are you know asking a lot of questions there Secondly get get to know the tool very well because MS team is a new team a new Platform for most of you because we have been conducting classes mostly on zoom and moxa So all the best for tomorrow session Okay any questions now One important question that arises in our mind is are all functions invertible? Can I find inverse of every function on this earth? Yes, sir. Could you repeat my connection this time? Yeah, this is what will happen tomorrow. Yeah repeat from there So domain and range yeah, okay So see what happens f works in some on some input. Okay, and gives you some output, right? So it's inverse will work on the output of f and will give you the input of f back So if f works from a to b where a is the domain and B is the range not the co-domain base the range Then f inverse will work with B as the domain and a as the range So this will be the domain of f inverse and this will be the range of f inverse So the connection here between f and f inverse, which is very much evident is That domain of f is the range of its inverse and Range of f is the domain of its inverse and vice versa. Is this fine? Okay, so this is very important now the next thing which we are about to discuss is Are every functions invertible? Can I find inverse of every function? Right? The answer to that is no So what type of functions are invertible? So remember if f of x is invertible If f of x is invertible it means f of x is a bijective function So only bijective function Also called as bijection also called as a ka as bijection So only bijections are invertible now you must be wondering. What is this new word bijections? I have not heard about it in my function chapter last year. Yes, it's a new word for you so I'll be not talking about you know bijections and In course of expanding bijections, there would be many other new terms that will come your way. So slowly We'll talk about those terms as well. So what's a bijective function? What's a bijective function? What's a bijective function now functions which are Both one one and on two Right. So bijective function is basically the one which is both one one One one and on two Okay, again these two words are new to you. What is one one? What is on two? What is this? You know already ready made term that sir is using today. Don't worry We'll explain will go into details of each one of these today One one means injective functions Injective functions Also called as injections. It's a combination of inject and functions or injections and on two means surjections or surjective functions So these are some technical terms Which you need to know so functions which are both injective and surjective are called bijective Now what is injective? What is surjective? Let me go into details of this Okay, so let me start today's proceeding with explaining you. What's a one one function? Now we are entering into a part of your CBSC curriculum, which talks about types of functions So let me give you a give a header to this types of function types of functions So functions are normally categorized into two complementary sections one is called one one function another is called many one function So one one and many one Now, what are these one one and many one functions as the name itself suggests there is a one to one Correspondence in case of a one one function Okay, definition wise every image Let me define it for you When a function is one one remember every image What are images? images are your elements of the range Right, so every image that means whatever elements are there as a part of your range under this mapping Those every image will have a unique pre-image a unique pre-image Okay, let me give you a dummy arrow diagram for it. So let's say there is a Set a and there is a set B and let's say this is a function Your function maps a b c to one two three four and the mapping goes like this a maps to one B maps to two c maps to three So you can see that one two three are your images under this mapping Right and a b c are their corresponding pre-images Correct. So you can see every image has come from a unique pre-image That means no two pre-images are mapping to the same image Correct. So when no two pre-images are mapping to the same image we call such a function as one one function However, let me tell you it is allowed for a function to have two images I'm sorry two pre-images mapping to the same image Correct, but in a function it is not allowed to have one pre-image mapping to multiple images correct So now in this function in a one-on-one function There's one more restriction that every image should have a unique pre-image Of course vice versa is always true because then only it will be a function Right Okay, so this is a typical arrow diagram that suggests you a one one function Okay, to give you some examples of one one function Let's say I have a function from real numbers to real numbers I can take an example of let's say two x plus three correct This is an example of a one one function That means every output will come from a unique input Well, let's say if I put a five for it five will come only from one It cannot come from any other value Correct. So this is an example of a one one function Is it clear what's a one one function everybody? Is this clear any questions? Okay, no question. Okay. So, how do you define a many one function then in many one function at least at least one image has more than more than One pre-image More than one pre-image That means out of the images there exists at least one image which is coming from more than one pre-image So a typical arrow diagram here would look like this Let me just make a Set a and set B Okay, and this is mapping. So let's say a mapping is like this. So this has got a bc This has got one two three four And let's say the mapping goes like a is mapping to one B is mapping to two C is also mapping to two Okay, so this will be a case of a many one function Many to one as the name itself suggests Now what are the examples for this so I can take an example of let's say from R to R X square Okay, in X square you would realize that if you just make a rough arrow diagram If you just make a rough arrow diagram, there would be two Pre-images which will map to the same image for example two square and minus two square will give you the same answer Isn't it same way three and minus three will give you nine four and minus four will give you sixteen I'm just giving you a rough example. So this is a case where At least one pre-image. Sorry at least one image has more than one pre-images has more than one pre-images This is fine Now many people I have seen here They actually decide on one one or many one By looking at the definition of the function. This is very wrong Now the same function as you can see if I change my Domain to R plus Okay, the very same function X square now you would realize that If you have R plus then this four can only come from two Correct nine can only come from three sixteen can only come from four Are you getting my point? So in this case, you would realize this function is a one-one function So here it was an example of a many one and the very same function started behaving as one one now When I made a small modification to its domain So dear students, this is very important for us to understand that never pass a judgment on the function by looking at Its definition many people X square many one. Oh X to the power four many one oh Sinex many one No, don't pass the judgment unless until you have a concrete idea about its domain So you can curtail your domain to make a many one function as one one also Right. So domain is a very important part in Decision of one one function or many one function. Is this idea clear? Any questions here? Okay, so before we move on to any other thing how to identify a one-one function and all I would I would like to ask you All a question the question is very simple question is in fact, this is a theory that we need to Keep in mind. So let's say there is a set a which has got N elements and there's a set B which has got our elements fine, okay my question here is How many one one mappings are possible from set a to set B? How many one one mappings are Possible or one one functions are possible from set a to set B. I would like you to put the answer in the chat box By the way, everybody knows total number of functions possible How many total functions are possible from set a to set B? Shanken total number of functions. Okay, let me start from back total number of relations possible So, let me go back a little bit. How many total number of relations are possible? 2 to the power nr correct Absolutely, Siddhartha. How many functions are possible? R to the power n R to the power n remember I told you it's number of elements in the code domain raise to the power of number of elements in the domain Fine, this you should never forget very important Okay, so if a1 has got Da-da-da-da-da till n elements b1 has got b1 b2 da-da-da-da till b are how many one one mappings are possible How many one one mappings are possible? Take a call. Take a call. Let it be wrong. No worries. Take a call Okay, Anjali says dash. I'll not tell what is her answer Anybody else? Wow. Trippan, Sudha very good. Okay, Ashish nice A, A where is A ma? It's only R and N here Ashish Sir, the last number sir Last number? A answer A answer A, N. Oh, the last number you're saying that will determine how much how many one-one functions are there, okay? Okay, so let me ask you this question in a slightly simpler format If your R is less than N, how many one-one functions are possible? Now, this is not a very difficult question. You can actually take a dummy example Okay, so what I'm going to do is I'm going to take a scenario where R is less than N Okay, so A and B Okay, I'll take Three elements here A, B, C and I'll take two elements here. So R is less than N. Correct. So R is 2 and N is 3. Correct. How many one-one functions are possible? Now see let me start a mapping So A maps to 1. Let's say Okay, now B cannot map to 1 else it will not be a one-one mapping So B has to map to 2. Poor fellow. Okay. Now, I cannot leave C behind Because if I leave C behind it will not be called a function only. Correct So C has to either map to 1 or has to map to 2. So the moment you map C to 1 or 2 My dear, what do you realize? You will not be able to make any one-one function. So Shankin is absolutely correct So Zilch No one-one function will be formed if R is less than N Okay, now what about when R is equal to N? Let's say when R is equal to N. So let me take a case R is equal to N So let me make A and B as two dummy sets Let me take three elements here A, B, C. Let me take 1, 2, 3 So A, let's start mapping for A. A has got three options Okay, so A can map to any one of them. Let's say A goes and maps to 2. So A has got three options B has got now B cannot go to 2. So B has got 1 and 3 only. So two options for B Okay, so let's say B goes and maps to 1. So C will have only one option Whichever is left it has to go with it. Correct So the total number of functions that could be formed in this case is 3 factorial Correct If you scale this up and say both RNN are equal Then your answer will either be N factorial or R factorial because both are same Okay, don't take this slightly because this may be a question in your school exams also fine Let us take the last case now when your R is greater than N Okay, so let me make a let me make a dummy diagram for this also when R is greater than N So let's say there's a set A and there's a set B Okay, and let's say A has got A, B, C and This has got 1, 2, 3, 4, 5. Okay Now how many options A has? How many options A has? A will have five options It can you know go and map to any one of the five. So let's say A goes and maps to three Okay, how many option will B have now? Of course B cannot now B cannot choose three now It has four other options to be chosen from so B has four options. So let's say B goes and hits two Okay, so C will have one two three four options. So C will have four options Okay, I'm so sorry. Uh, it'll have three options my bad Okay, so your total number of cases or total number of one one functions that can form is basically five into four into three Which is actually five factorial by By two factorial right two is five minus three factorial, right? Okay. This reminds me of the formula five p three five p three So in this case you will have r p n number of one one functions Is this fine? dear students So whenever such a case arises you take a miniature scenario and just try to scale it up Getting my point now you may combine the last two scenario because When r and n are equal even r pr is r factorial. So you may combine this formula to be Zero when r is less than n And r p n when r is greater than equal to n Please remember this this will help you save time in your exam also Any questions here anybody any questions here? Okay This is clear. How many one one functions? Fine Now if a function is defined for me, how do we identify a one one function? So the next concept that we are going to discuss with you is identification of one one function identification of one one function Now we normally identify a one one function in three ways Any one of the three ways could be used The first method is your graphical method graphical method Okay Now, how do you identify a function from its graphical method? What do you think would be a method? Let's say if you know a graph of a function from the graph, how will you identify whether it's one one or not a one one? Any idea NAPL people don't respond. I want nps r and r to respond NAPL is please don't respond only nps r and r people to respond if graph of a function is known Okay, let's say I make a graph like this Let's say I make a graph like this Okay Is this graph or is this graph a graph of a one one function? Yes or no If no, why not? No, okay. So shawmik is saying no. Why not shawmik? Sir, because at One one function if we draw a horizontal line at anywhere in the graph it should Intercept the perv only at a single point Absolutely. So sahana also has an explanation If it is only one coordinate for the same x then it will be one one function, right sahana that is very well stated So, how do you you know implement that statement of yours in this case? So as shawmik rightly pointed If I draw a horizontal line Okay horizontal line means line parallel to the x-axis Okay, any horizontal line Now you should always draw it to scrutinize the function. Don't draw horizontal line here Okay, we need to scrutinize the function. You don't have to save the function Right. There is no friendship between you and the function. You have to scrutinize it If such a line cuts the function at more than one point Then it will not be a one one function. The reason being it is clearly saying that a particular output Let's say an output y Okay, it is coming from three separate inputs x1 x2 and x3 That means if you make a arrow diagram from it Okay, set a is your domain set b is your you know go domain Then you would realize that there would be three such inputs in this case in this case Which will map to the same output. So this is definitely not a one one function Connect So what we say is that if a function has to be one one, it should satisfy Horizontal line test now don't get confused with vertical line test vertical line test is there to see whether a relation is function or not Right every function must satisfy vertical line test But those functions which also satisfy horizontal line test those would be called as one one function or injective functions Okay, so here the idea is I write it down. This is not a one one function So the idea is A one one function will always pass the horizontal line test And what is the horizontal line test? Let me write it here one one function satisfies Horizontal line test and what is this test this test is This test is Any horizontal line will cut the function Only at one point Only at one point Okay So as I told you don't try to save the function don't draw a line here and say oh, sir It is cutting only at one point here. No, you have to scrutinize the function That's why I wrote any horizontal line will cut the function only at one point Now this nature comes from the fact that one one functions are known to be monotonic The reason being one one functions are known to be One one functions are known to be monotonic Are monotonic What is monotonic Monotonic means it is either strictly increasing or strictly decreasing it cannot show you both the characteristics Okay, so I'll write it down here in bracket. It is either strictly increasing or Strictly decreasing Or strictly decreasing They will not be stagnated Or they will not show you rise and fall both simultaneously either it will what is strictly increasing Let me just you know show you through some diagrams Strictly increasing means the function will either be like growing like this Right or it will be growing like this Or be growing like this. This is called strictly increasing cases So in any of these case if you draw a horizontal line They are going to cut the function only at one point getting my point Strictly decreasing means what strictly decreasing means it is only decreasing in this domain. So either it is falling like this Okay, let me draw it in yellow falling like this or falling like this or falling like this Okay, so any horizontal line that you draw in the function will end up cutting it only at one point Okay, it is not like The parabolic function that we are seeing in x square. So if it is like this The moment you draw a horizontal line, it will cut at two points. So this is not a case of a 1 1 function Is this okay understood Now, what is this monotonic and all word use use here? This is something which is Coming up for you in application of derivatives chapter So don't worry. We'll do one complete chapter only on this monotonic nature of functions or monotonicity what we call it Any question anybody clear Yes, Vibhav has a question Yes, Vibhav all linear functions are 1 1 functions All linear functions y is equal to any linear polynomial is a 1 1 function. Is that what you wanted to ask Vibhav Now it would be wrong to say any quadratic function is not 1 1 It would be wrong to say that because as I have already told you it is in the hands of the domain It is in the hands of the domain So even I can make quadratic functions 1 1 if I want to I can make cubic functions if I 1 1 if I want to So never judge a function on its you know degree that oh, it's a cubic. It must take turn It's a quadratic. It must have you know a u shape. No, it is not It is not you know the shape it is basically where are you looking in which interval of x are you looking at the function Okay, so the idea is if you don't allow the function to take a turn somewhere It will be 1 1 The moment you allow it to take a turn somewhere it will lose its 1 1 nature Okay, so the word here is turn it should not have any turning point Is the idea clear so it should be monotonic in nature monotonic means either strictly increasing or strictly decreasing in its domain good enough Okay, so what is the next test next test is the Uniqueness test this is what I call as the uniqueness test uniqueness test Now this test is basically a test based on contradiction Okay, so what do we do? Let's say I have a function defined from domain to co-domain Okay, I'm not writing a and b. I'm just you know writing straight away as domain and co-domain Okay, and the function is defined as something whatever Okay, so this is given to you and they ask you Test whether this function is a 1 1 function or not And let's say you are not in a position to sketch its graph because probably the graph is too complicated to sketch So what can you do in such a test? So we'll say let x1 and x2 be two inputs from the domain of the function such that f of x1 is equal to f of x2 Okay, so this is your initial assumption This is your initial assumption while solving the question. So you're assuming that Let there be two separate inputs which give me the same output Let there be two separate inputs, which gives me the same output Correct, and let's say by using this assumption Dot dot dot you realize x1 is equal to x2 is the only possibility x1 equal to x2 is the only possibility Then what do you comment on this function? Then this function will be a 1 1 function in that case Now this word only possibility is very important Why I'm saying only possibility because this will always be a possibility See try to understand when you are comparing f of x1 to f to f of x2 when you're equating these two It is very obvious that x1 equal to x2 will satisfy this equation Even a blind person can tell that boss if x1 and x2 are equal This relation to ship will be true. So this will always appear as your solution That doesn't mean you will blindly say that it is 1 1 function But if that possibility is the only possibility That means whatever other possibilities you are getting if you are able to negate those possibilities Or say okay this that cannot happen Then you can say that the function is a 1 1 function I'll take a simple example to illustrate this Okay, very simple example Let's say Let's say I define a function From r to r as x square Our same old parabola Okay, but now I'm Dealing with two examples containing x square one having r to r and the other having r plus 2r Okay, same function in both of them Okay, the only difference is in their respective domains. So this is all real numbers. This is only positive real numbers, right? Now here if you see let me apply my uniqueness test over here So let's say There are two inputs x1 and x2 coming from the domain of the function such that f of x1 is equal to f of x2 Okay, that is you are saying x1 square is equal to x2 square correct So f of x1 is x1 square f of x2 is x2 square Okay, so this is my initial assumption fine Now, dear students a very sincere request to all of you do not cancel out square square Please Okay, now you are matured enough not to do that Okay If you do that you are losing on informations Okay, cancelling out powers means loss of information Fine, so do not do that So what is the right way to solve it bring one of them to the other side? So you write it like this Okay, now this is factorizable as x1 minus x2 and x1 plus x2 So this will give me two possibilities one is x1 minus x2 is equal to zero that is number one possibility And number two possibility is x1 plus x2 is equal to zero okay In other words, you are getting one of them as x1 equal to x2 and the other one is x1 is equal to negative x2 correct So as I told you x1 equal to x2 will definitely come out from your from your You know this relation right as I told you it will always come out from this But is this the only possibility? No There is one more possibility arising Can you negate it? Can you say this is not possible? Can you invalidate this condition? Unfortunately, no There could be two real numbers which are negative of each other and can give you the same squares For example, four can be obtained from two square also and minus two square also Right, so this is telling you that Pause that this condition is possible when x1 is equal to x2 that is number one But it is also possible when x1 is negative of x2 Right So there could be two separate inputs which can give you the same output So in this case, we have to say this function is not one one in other words. It is many one Right Are you getting it? So this is not the only possibility. This is also possible along with it Many students have seen making a mistake. Oh, I got x1 equal to x2. So one one. No, that will always come out Don't worry about it. You should be worrying about negating the other, you know possibilities Okay, or even the other possibility should give you x1 equal to x2 But unfortunately it is not happening in this case so I have to conclude here that x square under this definition from r to r is a many one function Okay, you can further you can further emphasize it with a diagram Now my dear students, let me tell you one more thing That when you are trying to prove this in exams school exams Please do not just draw a graph and tell your teacher see it is cutting at more than one point. So it is not one one Your teacher will give you zero for it Okay, graph and all is good for your competitive exams not to be used for school exams If at all you're using it it is to be like a supplement. It is it is to be like a support for your answer Do not just solve a question by using graphs. You can always use graphs to support your answer. That is fine Okay, now let me show you the other case Now here I'll start with the same proceeding. Let x1 and x2 come from r plus That is the domain of the function such that such that f of x1 is equal to f of x2 That means you're saying x1 square is equal to x2 square Okay, or same thing same approach. I mean there's no difference in the approach So you'll end up getting x1 minus x2 x1 plus x2 equal to zero Okay, that again gives you two possibilities either. This is zero Okay, or x1 plus x2 is equal to zero So this gives you x1 equal to x2 as one case another case is x1 is equal to negative x2 no difference in it But now I can say with surety that this is not possible Why it is not possible Because two positive real numbers cannot be negative of each other Have you ever seen two any two positive real numbers which are negative of each other? It's not possible right so in the light of the domain you are able to negate the other scenario That means this is the only possibility my dear friends This is my only possibility and if this is the only possibility then you can say Yes, this function is a one-one function This the function is a one-one function and graphically also you can support it So when you draw x2 you will only draw the part which is on r plus side You will only draw the part which is on the r plus side Getting my point So if you draw any horizontal line, it will only cut at one point It cannot cut it at more than one point and hence it is a one-one function So don't go by the definition go by its domain along with the definition Is the idea clear how it works So this is something which is very important and this is What your school will also ask you important for your school Any question any doubt? Please stop me Okay, let's have a quick question on this Let's have a quick question on this I think nafflers would be knowing it. So Apply uniqueness test to prove that x cube is a one-one function Okay, apply uniqueness test to prove that x cube is a one-one function Now your your school may not use the word uniqueness says they will say prove that it is a one-one function So you have to only apply the uniqueness test So prove that f of x is a one-one function Or it's an injection Everybody please attempt this I'm giving you one minute See zero. Yeah, I could have included zero also that that won't make a difference But since I said r plus that's why I didn't include zero. Okay with zero also it will be one-one Anjali Yes, exponential functions are one-one Siddhartha Exponential functions are one-one logarithmic functions are one-one Okay, okay done Yes, I've done one once you're done done trippan is done. Okay Patham is done. Ayush is done Aditya see is done Do me. Okay Atharva is done Aditya Manjunath is done. Okay, so most of you have done it. So let me discuss this with you now If I apply the uniqueness test, I'll say let there be two inputs x1 and x2 belonging to r Okay, such that such that f of x1 is equal to f of x2 Okay, that is You're saying x1 cube is equal to x2 cube Now again god for god's sake do not cancel out the powers The right way to solve this is bring it to one side factorize it Factorize it like this. Okay aq minus bq formula. Everybody knows in banglore So This will give me two possibilities This will give me two possibilities x1 minus x2 is equal to zero So as I told you x1 minus x2 will pukka come. Okay, so don't make a judgment over here, right now other than this You also get x1 square plus x1 x2 plus x2 square. Is this fine? Okay, now, how do you show that? number one It's not possible or if it is possible that it can only be possible when x1 and x2 are equal to each other and equal to zero Each now there are several ways to do it You can you can make a you can make a perfect square from it like this x1 plus half x2 the whole square Plus three by four x2 square and you can say that since everything is a perfect square The only thing that will make it zero is when x1 And x2 both are equal to zero each Which anyways gives you x1 equal to x2 as your you know final condition correct Some people would like to multiply with the two another approach is multiply with the two Okay, and write it as I write it as uh x1 plus x2 Uh, the whole square plus x1 square x2 square. So as you can see here everything is a perfect square So if they have to be uh, zero individually each one of them has to be zero Which will ultimately lead to x1 equal to x2 equal to zero So what do you see is x1 equal to x2 is the only thing that is coming out from here Okay, even if you take this possibility or this possibility ultimately The end result is x1 equal to x2 one more school of uh, you know approaches Um, I don't know whether you remember your complex numbers. This can actually be factorized as x1 minus x2 times x1 minus omega square x2 What is omega here omega is the complex cube root of unity minus half plus i root 3 by 2 hope you remember this I told you that this can be factorized Okay, so if this is possible that means you're saying a real number could be written as a complex number times real number Right, so if this is real this is real and this is complex This scenario can never arise until unless x1 and x2 both are zero each Okay, neither will this situation arise Tell your x1 and x2 are equal to zero each Okay, so whatever approach you take ultimately you end up getting x1 equal to x2 only Okay, so by all these we can say that yes this function conclusion is yes xq function under the definition of uh, real to real Is is a one one function. So this function is a one one function Any questions here anybody any question any concern any doubt Try one more. Can you all see this question? So this is a function defined from r to r as It's a rational function x square plus 4x plus 7 upon x square plus x plus 1 is f of x 1 1 done very good anybody else So I can see uh Okay Is this so difficult now sketching this would be slightly difficult because it's a it's not a polynomial or something. It's a rational function Okay, so you have to take care of all those asymptotes kind of a scenario. So what we'll do is I'll first make my life easy by You know writing this term like this. So numerator. I'll break it up as x square plus x plus 1 plus 3x plus 6 Something like this. Now, I'll start my proceeding. Let me apply the uniqueness test. Let's x. Let's say x 1 and x 2 belongs to real numbers Okay, such that Such that f of x 1 is equal to f of x 2 Okay, now the benefit of doing here is that the moment you Put x f of x 1 and f of x 2 and equate You can safely cancel out 1 and 3 Okay, so it slightly reduces the complexity of the scenario now cross multiply So x 1 plus 2 times x 2 square plus x 2 plus 1 is equal to x 2 plus 2 times x 1 square plus x 1 plus 1 Fine Now let me bring everything to one side Uh, so x 1 x 2 square and from this side, I will get x 1 square x 2 Okay, I will also have x 1 minus x 2 and I will have 2 x 2 square minus 2 x 1 square And I will have one more term 2 x 2 minus x 1 Okay, so this will be zero Now you can see you can easily pull out x 1 minus x 2 from all the terms So example from here I can pull x 1 minus x 2 and along with the minus x 1 x 2 From these two terms, I can write it as minus minus x 1 minus x 2 And this term here would be minus 2 x 1 minus x 2 x 1 plus x 2 So if I pull out x 1 minus x 2 which definitely would come as my answer Remember I told you x 1 minus x 2 will have to come as your one of your solutions Okay, the other solution would be minus 2 x 1 plus x 2 Okay minus x 1 x 2 And a minus 1 Okay, so the two possibilities arising either x 1 minus x 2 is zero that is number one possibility and the other possibility is Uh, the other factor is zero which I'm writing as without a negative sign So 2 x 1 2 x 2 plus 1 equal to zero now Is there anything that you can do to rule this out? Or are you able to find an example of x 1 not equal to x 2 which satisfies this condition? If you're able to do the latter that means you can claim it is not a 1 1 function This is very interesting my dear all of you please listen to this This proving something 1 1 is very uh, you know easy. So if you want to prove it's not 1 1 I think a simple example will suffice So if you're able to come an example So if you're able to cite an example where x 1 and x 2 are not equal and this condition is fulfilled That means you have said that there is another possibility other than x 1 equal to x 2 for which f of x 1 and f of x 2 can be equal And in that case you are you know, I you're saying that this function is many one In that case you're saying that the function is many one. Okay, so to prove many one It is sufficient to cite an example like this Else if you want to prove this cannot happen you have to give me a generic proof that this cannot happen Okay, so shankin has given a proof that if you take x 1 as 1 and x 2 as minus 1 Okay, let me do that. So this will become a minus 1. This will become a 2 This will become a A minus 2 plus 1 No, yeah, it's becoming zero. Absolutely shankin is very very correct So shankin has given an example where if x 1 is 1 and x 2 is minus 1 Then this condition is going to hold true.  This is also 4 This is also 4. So yes So many people say said could we do this in the beginning of the question itself? Yes, why not? But you have to have that vision you have to have that, you know Par You can say, you know, you you have to think ahead of time to prove it Okay, so at this stage Or at the initial stage if you are able to cite an example Where x 1 and x 2 are not equal and meets this condition. That means this condition can also hold good So this is not the only possibility. This is not the only possibility Are you able to understand the nitty gritty of things over here and hence this function is many One function Okay Now they can be so many examples I can give x 1 equal to let's say minus half and x 2 equal to 0 Right, so if I put x 2 is 0 then this and this will go for a toss and this will become minus 1 minus 1 plus 1 will be 0 So another example This could be another example. Is it understood? Okay, so this function is not a 1 1 function. It's a many 1 function any question any concerns, please ask me Then we'll get a set of solutions from this equation, sir Of x 1 and x 2 Yes, so here if you can get a this is basically a two variable single equation So if you're able to find To any two real numbers any two numbers belonging to the domain of the function which are different from each other And makes the second condition true Then you have to abandon the fact that it is a one-one function. You have to accept it as a many one function Right, Tipan Yes, sir, but then we'll get a set of solutions from this one equation, sir. Yeah Like supposing we have some solvable equation and we get some set of solutions Then will that be the number of times the Graph will turn Of the original function Oh, no, actually we cannot say that for example Let's say if you have x 1 is equal to minus x 2 coming in case of a quadratic equation, right? Okay, so two and minus two will give you the same input Correct four and minus four will give you the same sorry output four and minus four will give you the same output Six and minus so that doesn't mean it is going to turn that many times. It is only turning once Correct. So this is a relationship This is a relationship between The two inputs so that the output becomes the same Getting my point. It doesn't signify how many times it is Okay Now coming to the last test the third test Coming to the last test That is called the calculus test This test is basically a advanced version of the first test where I told you a monotonic function is basically a 1 1 function A monotonic function is basically a 1 1 function. So if a function is always increasing Either like this or like this or like this Right, what will happen At any point if you draw a tangent at any point of this function you draw a tangent, okay Doesn't matter whether whether it is this case this case Or whether it is this case You would realize that for all these situations your derivative of the function will always be greater than zero For all x belonging to the domain of the function Correct and if the function is strictly decreasing Let me write it as strictly increasing if a function is strictly decreasing Let's say it decreases like this or it decreases like this or whether it decreases like this Okay, you would realize that in any point of the function if you draw the tangent Any point of the function if you draw the tangent Your slope of the tangent will always be negative for all x belonging to the domain of the function So listen to what this test says This test says that if the function derivative Is either positive or Negative but not both That means this or is exclusive or Right, so either your derivative of the function is positive or This or is exclusive or exclusive or means both cannot happen together Okay, or the derivative of the function is negative For all x belonging to the domain of the function Then your function will be a one one function So a one one function will satisfy this characteristic Okay Now in some cases you will have to include zero also But this is to be used with a lot of discretion. So I'm putting it with dotted this thing. Why? Is because you will be tricked in such functions like x cube graph x cube graph Okay, you know that at zero The slope is zero right the slope of the tangent at x cube is zero, but it is a one one function Right, so you say it is not in the derivative is zero. So it should not be a one one Will you say that? No, that would be wrong to say So in the case of x cube you have to include the slope of the tangent being zero also But that cannot be included blindly Because there may be a function which will be something like this. Let's say it went and then it stayed flat Okay, if it stayed flat means any point here if you choose Any point here if you choose the tangent would be zero zero zero So this is not a one one function, but this is a one one function Okay, so inclusion of zero has to be You know done with a lot of discretion a lot of thought upon manner Are you getting this point? Now this test, please do not use it now when your school reopens and let's say your teacher teaches you this chapter Do not try to solve questions by use of calculus method Because this is something which you probably will learn in your Application of derivatives chapter. So she might not honor this but get a confirmation from her Is this fine any question here? So three tests we saw horizontal line test Uniqueness test And the last one the calculus test Fine to show an example of how calculus test works So let us say if I talk about Under this definition is your x square function A one one function. So if you differentiate this you will end up getting two x And since your x belongs to r plus F dash x will always be positive Right if this is always positive or always negative not both It should not show you both the characteristic because if it shows both the characters Characteristics that means it is rising somewhere and it is falling somewhere. The moment rising and falling both are happening It'll cut it'll be cut by a horizontal line at more than one point. So this will not be a one one function, right? So in this case, you will say since the slope is always positive in the domain of the function Yes, it is a one one function Right, but if you had only r then two x could be both positive and negative depending upon whether your x is positive or negative So in that case, it would be a many one function Idea is clear Now many one. I don't have to speak much about it because many one is basically The complementary of one one. It is the complementary of one one So what is not one one is many one Okay, so by default if a function is not one one, it is many one Okay, so I don't have to talk about different tests for it Right if it fails one one test, it is a one. It is a many one function It's like corona virus. There is no two tests whether you have it or you don't have it right only one test So if it is negative, you don't have it if it is positive, you have it simple as that Okay, so all the three tests that is horizontal line test uniqueness test and Calculus based test will be equally applicable to this as well. Any questions here? Okay, just a quick question just to wrap up this topic Again, the question is let's say there is a function from a to b Where a has got let's say n elements and b has let's say r elements Okay, how many many one functions are possible? How many many one functions are possible from a to b? fast anybody can unmute the Himself or herself and respond to this Go by the fact that many one is a complementary of one one Sorry It is how much If r is less than n Then can I say It will be r to the power n only because when r is less than n the number of one one functions are zero That means all the functions that are possible will be one one. Sorry will be many one So indirectly what i'm doing i'm doing total function total function minus one one function right One one function was zero right just refer to your previous notes previous page Okay, when r is greater than equal to n it is r to the power n minus r p n Is it fine any questions here? So no questions. So let me move on to another You know Division that the functions are categorized into The other two complementary divisions are onto and into Onto and into okay, they may sound like Pintu and chintu So it's onto and into so onto functions are called surjective Okay Of course, what will be into non-surjective so we don't have a separate word for it Now what's an onto function a plain and simple definition that you will remember for the rest of your life In any function if the range is equal to its core domain. It's an onto function Simple as that That means every element of set b must have a image or must have at least one image Okay, so definition wise Every element Of set b. So let's say the function is from set a to set b every element of of b has at least At least one preimage At least one preimage is what makes a function onto function This is a more easier way to remember range is equal to the core domain Okay, and that's precisely why you were taught to find range in your class 11th Right, so domain range exercise that you had done was done for this day So that you can easily make out whether the function is onto or not Okay, so in order to figure out whether a function is onto you have to do two steps one find the domain But most of the time domain would be given to you. So even that step need not be done But find the range And check whether the range and core domain. This is your core domain. This is your core domain Check whether the range is equal to the core domain If it is a subs proper subset of the core domain, then it will be into function If it is equal to it, that means an improper subset Then it would be a onto function Is this clear or to give you an example of a onto function is let's say Uh Simple x plus two Okay, if you find the the range of x plus two it will take all real numbers Okay, is this fine Okay What is into same complimentary of onto Complementary of onto is into so this is complement sets So now understand here a function cannot be both of them simultaneously Just like a function cannot be one one and many one simultaneously They cannot be onto and into simultaneously because they're complimentary of each other But there can be a cross connection. It can be like one one onto It can be like one one into It can be like many one onto It can be many one into so like that category categorization can be done Okay So what is an into function to define it in a plain and simple word? at least at least One element of set B of set B does not Does not have a pre-image So if you're able to show that there is at least one element in the set B which is unmapped That means that is alone Okay, then you are able to show that the function is an into function Let me give you a typical arrow diagram for it. Let's say a b c d And there is one two three four. Okay. So let's say a is mapping to one b is mapping to one c is mapping to two d is mapping to three Then you can see that four is unmapped So if you're able to show the examiner that there is one element from your core domain set, which is not participating in the mapping Which is not a part of your range Which is not able to become an image Then you're able to show that that function is into function or a non-surjective function Is the idea clear? Is the idea clear everybody? Okay, so here comes a question for you In fact, it's a part of the theory which I would like to discuss if let's say set a has n elements and set b has r elements and There is a mapping from set a to set b Okay Tell me How many onto functions How many onto functions are possible from set a to set b? Think carefully and then answer You may take case wise also you may take case wise also Okay, sorry shankin. I missed out on your question. Uh, should there be a local extrema? Yes, I think you're talking about Many one right Shankin, okay, so I'm getting different answers. Okay. Let me take a scenario over here I think there's not a single scenario here I'll break this up into three scenarios when r is greater than n When r is equal to n And when r is less than n Okay, let me first take r greater than n Let me take r greater than n so r greater than n Let me just take a dummy case So let's say We have Three elements in set a let's say a1 a2 a3 And five elements in set b Okay Now I have to make onto functions onto function means what? onto function means None of the elements of set b must be left out That means every element of set b must be your range must be a part of your range So let's say a1 goes and maps to b1 okay Now even if I take worse case a2 Will mark will map to somebody else correct a3 has to map to somebody else is the worst case or this is the best case you can say But despite doing that you will see that these two You know elements are left unmapped These two elements are left unmapped In other words you can actually think the scenario as there are three different balls There are three different balls And there are five persons Okay, how many ways can you distribute these three different balls such that each person Each person Gets at least one ball Now you'll say it is not possible Because even if you give you know one one ball to b1 b2 b3 b4 and b5 will be without balls correct Right, so you cannot ensure that each person gets at least one ball So the answer form for this case will be zero Answer for this case will be zero Is that fine Any questions here? Okay, let's take the other scenario when r is equal to n. Oh, sorry my mistake r is equal to n So r is equal to n. So let me make a scenario where r and n both are equal to each other and let's say I take 5 5 in both of them So let's say a to b So this has So there are five different balls and there are five different people Correct, of course people are different only So how many ways can you ensure that each person Goes with at least one ball. So you cannot give two balls to one person because if you do that one person will go empty hand So the only possibility is you have to give one each to them correct So how many ways can this be done? It's very simple. So a1. Let's say has got or a1 has got five opportunities so a1 Can map to any one of b1 b2 b3 till b5 a2 has also Four opportunities a3 will have three a4 will have two And a5 will have one so by fundamental principle of counting the total number of Onto functions you can make will be five factorial So if you scale this up if you scale this up Your answer would be either be n factorial or r factorial because both are same Now one thing that you would realize over here that when the number of elements in set a and set b are equal Then the number of onto functions Formed is the same as the number of one one functions formed Isn't it even for one one function the answer was r factorial or n factorial whatever you want to call it So when the number of elements in set a and set b are equal You will always get A one one onto functions from it So one one onto function number will always also be n factorial Now this is what we used to call it as a bijective function correct So this answers one more thing indirectly that If you want to make a bijective function Then your total number of elements in set a and set b must be equal Right So the number of bijective functions you can form from set a to set b having same number of elements Let's say r or n Is actually n factorial So i'll write it down over here It also gives you the number of bijections Okay, so bijections cannot be formed Bijections cannot be formed till the number of elements in set a and set b are equal Is this fine? Any questions here with r equal to n scenario? Okay Now when n is greater than r When n is greater than r this scenario is slightly tricky Okay, so i'm putting a question mark because i'm going to take this up in the next board Okay, this situation is actually tricky when your n is greater than r I can see answer coming from aiush, but aiush that is not the answer actually When n is greater than r okay Then this scenario becomes a scenario of distribution of Distribution of End distinct objects Into r different groups Into r different groups r different groups Without any blank group Without any blank group Okay, if you recall this is basically a concept which we had done in permutation combination chapter some of you We're not there with us last year But those who were with us Please remember this is a concept just go and check my notes Okay, and you can find those notes on the Uh On whatsapp group also so distribution of n distinct objects Into r different groups When blank groups are not allowed Without any blank groups without any blank groups Okay, so how do you carry out this scenario? Now before I start many people would think in it in this way They will think that if blank groups are not allowed we can first give one one object to all the groups correct One one object to all the groups So they'll select r objects from n objects in ncr way And distribute it into r groups in r factorial correct And the remaining n minus r objects Okay They can be distributed in any way you want. So basically it is n minus r will have r choices So many people think the answer is this Okay, but this is not correct Why it is not correct. This is something that you will find it out for homework and let me know On my personal chat This will lead to over counting This will lead to over counting Now, how does this result also come? Let me explain you. So many people think in this way see Let's say These are my r different groups. I can call it as r boxes. So let's say box one box two Box r Fine And let's say there are different balls ball b1. Okay. I'm making it of different Shapes and sizes ball b3 and so on till ball bn Okay Now if you want to distribute distribute means give away these balls into these boxes Without leaving any box empty, then how does one do this?  So how many are left? So if r is gone, you are left with n minus r Okay, you are left with n minus r, right? Left with n minus r balls Now each ball here has got Each ball here has got r choices correct Each ball here has got our choice. Okay, so this is basically ulta R to the power n minus r. Yeah, so each ball has got our choices So r into r into r up till n minus r times You will end up getting r to the power n minus r. Okay. Sorry. I interchanged the position But this answer is not correct And you have to find it out. Why it is not correct Okay, hint is it will lead to over counting Fine, so please let me know over the whatsapp group why this answer Is not going to give you The right answer Okay, so what do we do in in that case? How do we accomplish this? So we have to apply something which we call as principle of inclusion and exclusion I want to know One people here that you are heading towards a slightly difficult concept So pay attention if possible put away your pens just listen to what I have to say Okay So how to do this? This is not a correct way to do it. So I'm just putting a cross mark over here This is not a right formula Okay So how to do this so everybody please listen to me everybody please listen to me Let me put up Okay, so let me again make a scenario that we have our different boxes Dot dot dot box one box two till box are Okay, and there are n different balls Okay of different shapes and sizes Okay, so let's say b1 b2 Da da da da da till bn Okay now My first question to all of you is if there is no restriction That means you can put any number of balls in any box you want So boxes can can be empty also. So if there's no restriction If there is no restriction In how many ways can you place these balls into these boxes? R to the power n very good because every ball will have our options So ball one can go in any one of the r boxes ball two can also go in any one of the r boxes Ball three also can go in any of the r boxes till ball number n So r into r into r into r into r into r and number of times so r to the power Let me call this figure as nu Okay, that is the number of elements in the universal set that means this is the Our distribution without any restriction Now pay attention everybody Let's say even is the event When box number one is empty Even is the event when box number one is empty Now try to understand here. I'm not saying that other boxes may not be empty Or other boxes will not be empty I'm just saying box number one is empty Others may or may not be empty that I don't care But I'm here where what I'm saying here here my box number one has to be empty Okay, so if there's no ball allowed in box number one Then How many ways can you perform this distribution? I have to the power n minus one Ah r minus one to the power n R minus one to the power n right because every ball will have r minus one choices, right? Because box one cannot enter sorry ball one cannot enter box one So ball one will have r minus one choices Correct even ball two will have r minus one choices Even ball three will have r minus one choices So all these balls will have r minus one choices. So this will be your event. Okay So if I continue making a list till er being the event that box r is empty Box number r is empty Will you all agree that that will also be r minus one to the power n? So I'm just putting dot dot dot. Okay. I'm not writing anything. Is that fine Any questions here? Now if you're wondering, why am I doing this? I'll connect the dots little later on So just understand the scenarios of now Now, let's say even intersection e2 is an event When box number one and box number two are empty, correct? Tell me in how many ways can I now perform my distribution When I don't want any ball to enter in one and two Now it doesn't mean others cannot be empty Others may or may not be empty, but even an into pukka pukka will be empty So how many ways can I perform this? Excellent you guys are awesome So if I just make a generic event saying that even intersection ei intersection ej is an event where box i and box j are empty You would all admit to the fact that The number of phase in which I can perform this event Is r minus two to the power n Okay and similarly I'm not just I'm not writing anymore here Number of phase in which you are ensuring i th box j th box and k th box are empty will be r minus three to the power n Right and so on And so on Now let me now connect the dot over here. What am I trying to find here? I'm trying to find out I'm trying to find out the number of phase in which I don't allow my box one to be empty And my box two to be empty And box three to be empty and so on till my box number r to be empty. So this is this is what I'm looking for I'm looking for the number of elements In even complement intersection e2 complement intersection e3 complement all the way till you are compliment Right see because my situation is I don't want an empty box without a blank group blank group means empty box right So I don't want box one to be empty And I don't want box two to be empty and I don't want box three to be empty Till I don't want box r to be empty correct Now there is a famous law called de morgan's law de morgan's law says that just recall de morgan's law What is de morgan's law? What is de morgan's law n of a complement intersection b complement is It is basically what n of a union b whole complement right n of a union b whole complement is what n universal set minus n a union b right This is de morgan's law So I'm applying the same over here and I'm saying it is n universal set minus n of e1 union e2 union e3 Till union er correct Now again recall your cardinal number properties which you had done in class 11 I hope you all remember the property n of a union b union c is what? Na plus nb plus nc Let me write it as n summation na minus na intersection b nb intersection c and c intersection a let me write it as Na intersection b. So I'm just using in a short form notation Okay, I don't want to write that down So you can extend this particular cardinal property. By the way, this is this itself is principle of inclusion and exclusion This itself is principle of inclusion and exclusion Okay, why it is called principle of inclusion and exclusion? Because you include then you subtract subtract means exclude then again you include then again There would be a subtract if it all there are more sets, right? So you include one at a time Remove two at a time then include three at a time then remove four at a time like that Okay, so according to principle of inclusion and exclusion you can write this as Summation of ne y e 1 or e i you can write minus summation n Even intersection e 2 plus summation Ne 1 intersection e 2 intersection e 3 Let me move slightly to the right And this will go on and on till you till your number of sets are there. Okay, so I'm not writing Doesn't mean infinity. It will go up till it can go. Okay. That's what I'm trying to say Okay, now time to fill in this gap. So now we are at the last step This was r to the power n. You only told me this is r to the power n Here r to the power.  So if you add all of them Remember, there are r of these so it'll become It will become r times r minus 1 to the power n correct Now just to maintain uh A pattern I'll write rc 1 RC 1 is also r only no problem, right minus Even intersection e 2 if you see you only told it will be r minus 2 to the power n And how many such pairs will be possible? How many pairs you can make from r? RC 2 So rc 2 r minus 2 to the power n. So this trend will continue my dear And this will become your Answer So this is the formula of number of onto functions From set a to set b where set a has got n elements and set b has got r elements Right So the expression is slightly complicated and many times I have seen Uh, the students forget this formula But I'll tell you a method by which you can do away with this formula also So this formula is just optional. It is for those people who like remembering formula When I was preparing for my je I never remembered this formula, but still I could solve all problems. I'll tell you how I just use my basic understanding of pnc So please note this down as of all as of now, please note this down Remember if possible If possible, okay I'll give you some demonstrations on this. I'll give you some illustrations on this. Don't worry Do you want me to move the screen left right? It is actually not important for you to copy this Just understand what is the core principle by which this formula was derived So let me give you an example That will make things more clear Now, let us say Let us say you want to make number of onto functions from set a to set b Where set a has got five elements and set b has got three elements Okay In other words, you can think as if there are five balls And there are only three takers for it. So let's say a1 a2 a3 a4 a5 are five balls And b1 b2 b3 are three people who wants to take these balls such that each person gets at least one ball Okay Now if I use the formula From the formula What is the number of onto functions? Number of onto functions from the formula is R to the power n minus rc1 r minus 1 to the power n minus rc2 r minus 2 to the power n Plus rc3 r minus 3 to the power n Correct Now it'll automatically start becoming zero at one point of time. So I don't have to you know bother about it So our hair is this guy And hair is this guy Correct. So your answer would be 3 to the power 5 minus rc1 is what rc1 is 3c1 This will be r minus 1 which is 2 2 to the power 5 Then you will have 3c2 R minus 2 will be 1 to the power 5 And I think after this term it will become a zero. So you stop over here So your answer would be 243 minus This will be 3 into 32 which is 96 minus 3 which is 93 So 243 minus 93 that means 150 ways Okay Now this is by use of formula Okay, now I'll use My common sense to solve this Without using formula Okay Now assume that there are three boxes And there are five balls and you have to place these five balls in these three boxes such that no box is empty Okay, so there are five balls waiting outside b1 b2 b3 b4 b5 b4 b5 Okay, so how will you distribute it? So one distribution could be 2 2 1 All right If you're distributing two balls to the first box you can choose two from five in five c2 Correct Yes or no If you're choosing two balls to the if you're giving two balls to the second box You can choose the remaining from the remaining three in three c2 ways Okay, and the leftover ball will go in the third box. So that is one c1. You need not write it also Now there could be a shuffling of the balls also. There could be shuffling over here. You can have one two two also Two one two also like that. So how many shuffling can happen? You can shuffle it is in as many ways as you can make a three digit number from 2 21 How many three digit number you can make from 2 21? Three factorial by two factorial, right? Okay, so in this case your answer will come out to be 10 into three Into three which is 90 Okay, now this is one of the case Second case is 3 1 1 that is also a case Okay, so here also you are ensuring that Every box gets a ball Or every person gets a ball So in this case your distribution will be five c3 Then two c1 and of course one c1 And how many ways can you shuffle it you can make it as one three one or one one three So again three ways which you can actually write it as three factorial by two factorial again So your answer will be 10 into two into three which is 60 So cases means you have to add these two and you end up getting the same answer 150 ways So here by use of formula Here by use without use of formula. I got the same result Okay, so go with this method. You will never forget your stuff. You'll never forget your Basic understanding another way to look at it is many people say sir. Can we apply division into groups formula? division into groups In this case you are doing division into groups With the order of the group being important Okay, so let me recall this also. So all your pnc as I get getting tested over here see If you divide five into Three groups of two to one Okay, how many ways can you do it? You will say five factorial by two factorial two factorial one factorial But since these two groups contain same number of items you have to further divide by two factorial And since order is important you have to multiply with three factorial Now don't ask me how because this is already explained in the chapter pnc If you want you can go today Check out that video and watch that video n number of times again. Okay So this will give you if i'm not wrong 120 by 8 into 6 120 by 8 is 15 15 into 6 is 90 Okay, so this is what we had got here also 90. Okay, another is we distribute it as five Object as 3 1 1 So same formula here also five factorial by three factorial one factorial one factorial And since these two are having same number of items you have to divide by two factorial And since order is important into three factorial So three factorial three factorial gone. So 120 by 2 is 60 So 90 plus 60 Again, you end up getting 150 It doesn't matter which way you take up and result is the same Any question here anybody Okay, trip and let we'll discuss it. Okay, so Just give me this argument. You just type this out and send it to me on my whatsapp. Okay Is this fine everybody So now you know how to find out the number of onto functions also Okay, let's take a question on onto function very easy one. I'll start with a very easy one Okay question is if this function is a surjection Find the Set a over here. Excuse me, sir Yes, can you please go to the previous slide? Oh, yeah, sure. Thank you sir. Thank you Done one Show me. Are you sure one is inclusive? Okay, so basically this is an exercise which you would have done endless number of time in class 11 finding the range of a function Whose domain is an infinite interval So here the approach used to be remember used to make y is equal to the function And then used to make x the subject of the formula and apply the constraints of the domain on it So when you make x the subject of the formula Let's write it like this So x square y plus y is equal to x square So you can write x square 1 minus y is equal to y So x square is equal to y by 1 minus y okay now Domain of this function is all real numbers. So when x is all real numbers We already know that x square has to be positive Which means y by 1 minus y should be positive Okay, you can go for a wavy curve for this case. So this will be zero This will be one and now please be very carefully minus plus minus Okay, because here it is 1 minus y not y minus 1, okay So greater than equal to zero is this inside zone. So your y should lie between zero to one Zero inclusive, but one cannot be included Okay, so this is primarily the range of the function and the range should be equal to for onto function For onto function a range should be equal to the codomain codomain is your a in this case So you can say that your set a can be zero to one one exclusive Yes, why not we can write x square is x square plus 1 minus 1. Why not? It won't affect it won't affect the answer All right, okay All right now into I don't have to spend much time with you. Okay. I'll just talk about into also Into is basically complimentary of into is a complimentary of onto. So what I'll do here is Let's do a quick exercise on You know finding the number of into functions finding the number of into functions Okay, I would say it is very easy sir because now I know number of onto functions So I can easily find out the number of into by subtracting the total number of functions minus the number of onto functions remember number of onto function when r was R was greater than n When r was greater than n How many onto functions used to be there zero right? When the number of people are more than the number of balls Okay, r is the number of people n is the number of balls You cannot provide every person with at least one ball Okay, when r is equal to n your answer would be r to the power n minus n factorial or r factorial whatever you want to call it And when r is less than n it is basically And now try to recall the formula the formula was r to the power n r c 1 r minus 1 to the power n Minus r c 2 r minus 2 to the power n So if you subtract this from r to the power n you will realize that you can you will only see r c 1 r minus 1 to the power n Minus r c 2 r minus 2 to the power n and so on Okay, so one of them if you remember you will be able to find the other one also easily Is this okay? okay So before we move on to bijections there was this one question that I would like you all to attempt Okay, so read this question carefully There's a function from r to r There's a function from r to r defined as a piecewise function x mod x minus 4 when x is rational And x mod x minus root 3 when x is irrational Identify the type of function that means you have to comment whether it is 1 1 onto 1 1 into or many 1 onto many 1 into so four categories So which of the four categories is it fall into so identify the function as 1 1 onto Or 1 1 into or many 1 onto Or many 1 into Which of the four categories is the function falling into Think carefully and then answer you can type your response on your chat box To the previous slide aditya you want me to go to the previous slide? Okay Oh, is it I think it's going to rain soon at my place as well Done taken a snapshot Yes, sir. Thank you sir. Sure Okay, so Ayush has given his response What about others Okay, skanda Noted down your answer Now here don't try to solve this question by graphing it because you know, you can't make graphs for such conditions where you have some directly Directly type functions involved. Directly function is where you're You know, um Domain is rational irrational type of situation. So it's very difficult to plot such graphs. Okay So graphing will not help you Now, how do I solve this question?  Let's try to prove that it is, you know, not a one-one function Okay, can it ever have See this function We all know x mod x graph x mod x graph is like this So it's a downward parabola down and upward parabola Okay Now in itself this function of course minus four will just shift this down So it doesn't change its one-one nature. So in itself this function is one-one Correct So no matter whatever, you know number you put it is going to give you different values for different inputs So this is a continuous graph, but even if you take any, you know, different different rational numbers It is going to give you different different answers In the same way this guy also is one one in itself It is one one in itself Okay So overall it gives a feeling that oh it has to be a one-one function, but wait Can I find such a value? For which both of them Gives me the same input. Oh, sorry the same output. So let's say if I put a two here Let's say two mod two minus four And let's say if I put fourth root of three Okay, so all of you please Try to figure out what are these Outputs so f of two will give you two being a rational number will follow the first definition So two mod two minus four that will give you a zero And fourth root of three is an irrational number. So it'll follow the second definition So it'll give me three to the power one fourth into three to the power one fourth minus three to the power half That will also give you a zero What do you realize over here? So you realize that there are two inputs which will give you the same output zero In this case One example is sufficient for us to show it is a many one. So these two options are ruled out Now whether it's many one onto or many one into how do we figure this out? So now I have to take some real number Which cannot be obtained from I have to take some real number which cannot be obtained from any x What will it be if I'm if I'm unsuccessful in doing that that means it is a onto So can you try out thinking of an example? Where The output is such which is a real number, but doesn't come from any input try minus root three Let's say the output is let's say There is some input for which I get the answer is minus root three Okay, let's see. What is this input? Now if you follow the first definition You end up getting x mod x minus four as minus root three. That means x mod x is equal to four minus root three Okay, now this is either minus x square or x square I don't know any real number or any rational number whose square gives you an irrational number So there is no such There is no such x belonging to rational number for which the square of it or negative of the square of it gives you four minus root three Right, so obviously minus root three cannot be obtained from any x coming from the first definition So what is left? It should come from any x coming from the second definition. So let's say if I do this You would realize that x becomes zero area, which is not which is actually a rational number But this definition was supposed to be for not rational. See it was crossed over here, right? Right, so ultimately you cannot you cannot put zero into your second function So you cannot get minus root three So what do I mean? I mean That this real number cannot come from any x neither from rational number nor from irrational number So this answer is it is a into function into function That means your function is one one and into Sorry many one and into Is this clear anybody has any doubt in this any concerns in this please ask Into part see shashank what i'm trying to say here is Recall the definition of onto and into when is a function onto When the entire co-domain, this is your co-domain, right Co-domain is all real numbers, right when your entire co-domain is your range also So if i'm able to show that there exists at least one real number which is not mapped Or which cannot be obtained from any x or which doesn't have any pre-image Then it becomes into so my effort is towards that direction. So what I do is I choose a number minus root three Right, and I want to see whether minus root three can come from any Rational number or any irrational number Now no rational number give will give you x mod x is equal to four minus root three because this is clearly irrational So x mod x will always be rational if x is rational So a rational number can never give you so this is ruled out Now can an irrational number give me x mod x minus root three as minus root three So when I tried that when I tried equating I got x as zero, but x is not irrational x is rational So here it should have been irrational here. It should have been irrational, right? So even zero is not permitted correct So minus root three cannot be obtained from any x And hence this function will not be a onto function. So it will be an into function Now is it clear? Okay, so now you all take a break Everybody, please take a break Okay, take a break Uh, we'll resume at We'll resume at 6 23 p.m. Fine Good enough time Eat something have water Come back in 15 minutes Bijective functions so bijective functions. We have already discussed. What's the bijective function? So bijective functions are ones which are both Injective as well as surjective So injective plus surjective makes it bijective Okay, now functions which are invertible have to be bijective Why? So that's number one question why functions which are invertible Have to be bijective Why a function has to be one one necessarily and onto necessarily for it to be An invertible function Not it is basically coming from the very basic definition of a function and its inverse. So when f acts on a to b Okay, it's inverse acts from b to a right Correct. So this is something which is known to us from the very basic definition of a function and its inverse So let's say my function is not one one My function is not one one Then see what will happen Let me take a dummy example of a function, which is not a one one function. Okay So let's say we have a b c And let's say we have one two And the mapping is like this. So a is mapping to one b and c both are mapping to two two Okay Now for such a function if you start making its inverse arrow diagram So inverse will operate from b to a Okay, so let me write f inverse on top And it will work on one and two to give you a b c. So one will give you a Remember one came from a So when you put a into the function, it gave you one, right? So when you put a into the function, it gave you one So putting one in the inverse should give you a back. So this is how the mapping happens correct But putting b and c both gave you two. So putting two will mean giving you both b and c now This is violating the very basic definition of a function The very basic definition of a function is every preimage must have a unique image So this is not a function at all For it about it being a inverse inverse function. It is not even worth calling a function Okay, so your function f cannot be a one one function. I'm sorry cannot be a many one function Right, so it has to be one one. So it has to be injective Okay, now let me show you what will happen if it is not onto If it is not on to see what Problem will you face? So again, let me give you an example of a function which is Not a onto function. So let's say a b c and you have one two three four. Okay, so let's say it is one one But it is not on to you can see four is left out. That's why it is not on to okay So when you plot its inverse I don't diagram for its inverse So it goes from b to a Okay, so one two three four and this will be a b c So putting a gave you one. So putting one in its inverse will give you a Putting b gave you two. So putting two in the inverse will give you b Putting c gave you three. So putting three in the inverse will give you c But what about this? This is again unmapped which is again violating the basic definition of a function So it is not a function forget it To be a inverse of a function. So it is not a function itself That's why Function has to be onto or has to be surjective in order to be invertible. So these two criteria are important So in any function where Your function is a one one and onto Then only you can find the inverse for it Else you will not be able to find the inverse for such functions Is this fine any questions here? Okay Now the next thing that comes in your mind How do we find the inverse of a function that is a very important exercise that is going to come for you in your school as well Okay, so how to find the inverse finding the inverse of a function? Okay Now all of you please pay attention to this. This is slightly tricky Now you're finding y is equal to this and what are you given? You have been given a function Which is a one one and an onto function and it is defined like this Okay, so assuming that it's a bijection Assuming that it's a bijection How do you find out the inverse of the function? Now Do you all agree that saying this? Saying this Is equivalent to saying x is equal to f y x is equal to f of y Correct both are same thing mind you Both are same things Right, so now I have to find this correct So is this simple what is there in this? Just stop the position of x and y and you're done Of course that is obviously correct So in your any given function just stop the position of x and y Right and make y the subject of the formula And make y the subject of the formula and you are done Okay, so these two steps you have to do one is this step and second is this step because you Normally your teacher will ask you to represent it as y is equal to something Right, so this is what you have to finally represent So the moment you swap it you get this equation. However, that is also the answer But that will look very weird. So you have to write y in terms of x I'll give you an example I'll give you an example Acha by the way, when you write the inverse, please mention its domain And range if if not, uh, uh, you know, you you don't know the domain but at least mention that It's a domain will be the range of the original function Okay So let's say I define a function Let me take an example I define a function like 3x minus 5 Right, we all know that it is a bijection. We all know it's a bijection because It will pass the horizontal line test And its range is equal to its codomin codomin is all real numbers. Correct. So it is one one and on to both Okay, so how do you find the inverse of the function? So first write This has y equal to this and then replace your Or you can say swap your y with x You can say swap y and x The moment you do that Basically, this should be your answer. This is your answer Okay But normally your teachers will say You they want to see y is equal to something they want it you want the answer They want the answer to see like look like this. Okay. So what you can do is You can write this second step as 3 y is equal to x plus 5 So y is equal to x plus 5 by 3 Okay, so this becomes the inverse of your given function. So this is f inverse x Okay, so if this is your f of x If this is your f of x, then this is your f inverse x. Is that clear? Any questions here? Okay, now some teachers will tell you to do the opposite way first They will say make x the subject of the formula. So there's another way. This is method number one Method number two is They will say make x the subject of the formula. So from here we can say x is equal to y plus 5 by 3 Okay, and then they will say swap the position of x and y Okay, so ultimately the end result will be the same So it is up to you. So whether you want to first make x the subject of the formula Sorry, this is the first step And then swap the position of x and y Or you want to do the other way round first swap the position of x and y and then make y the subject of the formula It's your call. Is this fine Any questions regarding this? now How do one verify that they are inverses of each other? So I did something but I did not verify whether they are inverses of each other. It is very easy to verify If you put f inverse The entire function if you put In the given function so f inverse you put in the given function f of x you would realize you will get x Okay, now one important thing I would like to add over here. This x here comes from set b See this function was defined from a to b, right? So your inverse would be defined from Remember your inverse would be defined from set b to set a Now we have not done composite function officially But soon we'll do it you would realize that the input that you can feed to this function inside Should be the dome in the domain of your f inverse function. So this x will come from your set b Okay, similarly f inverse f of x will also give you x But this time this x will come from This x will come from domain of f Correct and domain of f will be your a this is something very important. So many books will write write this in this way They will say f of f inverse is i b and this they will write it as f inverse o f will be i a They just mean to say that it is an identity function i stands for Identity function. What is identity function? Something which returns you the same input as output remember identity relation you have done Correct. So only a comma a kind of elements are there Okay, so if you put some let's say five it will return five to you So five comma five will be there in this particular kind of this is a relation by the way All all functions are relations at the end of the day. Okay, but we give it a name i b because Your x that you get will will come from the Set b And this is called i a because this x that you will get will come from set a Is the idea clear about how to find inverse and how to check whether Uh, one is the inverse of the other. So as of now, I've not done the test Let me do this test. So let me do the first activity first So I'm putting x plus five by three in place of three x plus sorry three x minus five, right? Yeah, three x minus five You yourself would see that three three gets cancelled Five five will get cancelled and you'll end up getting an x. So this is satisfying this particular characteristic. So this is correct Okay, similarly In the expression of the inverse if you put so this is the expression of the inverse if you put the original function So let's say three x minus five. You're putting in place of x And do this operation Again, you realize five five gone And then three three will be gone and you'll be ending up getting an x. So this is satisfying the other one. So this is also correct Okay, so what we have found out is a correct result. So both these functions are inverses of each other Okay, so if you know, it's domain and range the right way to complete your answer is Your f inverse is this For all x belonging to the domain of the function domain of the function will be the range here So you can say for all r Okay, or x belonging to all real numbers Okay, for all you can remove and just write a comma also Any question here on finding the Domain, uh, sorry on finding the inverse Could you go down? Could you go down? So this is the verification verification Now in your exams, do you have to verify? No Not for at least showing the examiner, but you can do it for yourself If you want to if you're not very confident about your approach you can So any of the two verifications will work because both will be inverses of each other Should I go to the next page now? Yes, sir. Okay So Let us look into the properties of inverse Properties of inverse of a function. This is very important. So keep this in mind The first property is inverse of a function Is unique and it itself is a bijection This is very obvious because if it is not a bijection then Uh, see f is the inverse of f inverse, isn't it? So f and f inverse both are inverses of each other So since f is the inverse of f inverse that means f inverse is invertible If f inverse is invertible, it means it has to be a bijection Okay, this point is very important unique unique means you cannot have two inverses for the same function You cannot have two inverses for the same function Okay, so they can only be one answer for the inverse This is important. I'm telling you you don't write two answers for inverse of a function However, the situation will arise where you will feel that there are two outcomes coming So which one of them will be your answer? That is something that you have to decide there I will do some problems of that type also with you Okay, so inverse of a function is unique and itself is a bijection Okay second thing is If there is a composition of if there is a composition of two functions, let's say f and g Okay, then inverse of it Follows the law of reversal This is called the law of reversal law of reversal So f o g I think I have not done a composition of functions with you yet Sir can two functions have the same inverse? For two functions have the same inverse That means The inverse should give you two opposite inverse. So how it can happen? See let's say f and g gives you the same inverse Okay, let's say h then opposite should also be true No, so inverse of h should be f also inverse of f should be g So I think again the uniqueness says our uniqueness criteria is violating Right So I think the answer in fact what I think I'm sure the answer. There is no, okay Now what is law of reversal? I'll just take a simple example here Just to illustrate that how it works Let us say I have two functions 2x plus 3 and g of x Which is e to the power x. So let's say these are the two functions f of x, which is 2x plus 3 And there's another function g of x, which is e to the power x. Okay Now again, I'm repeating this. I have not done officially the inverse of a function with you Sorry composition of a function with you, but As of now, this is a very simple task if you want to find the composite function f o g All you need to do is put this in place of x So the answer will be 2 e to the power x plus 3 Okay, correct Now find the inverse of this Find the inverse of this function So which approach you want to follow do you want to swap the position of x and y and then make y the subject of the Formula or do you want to make x the subject of the formula first and then swap x and y the call is yours Okay, so let me do one thing. Let me swap the position of x and y over here Okay, and then let me make y the subject of the formula. So it'll be x minus 3 by 2 ln of that Okay So this function has its inverse.  It's just a verification. I'm doing. It's not a proof. It's a verification. I'm doing Okay, just to make you clear about it so When I do this F inverse We all know is what can you just figure out f inverse from here anybody can speak it out? I think it's a mental game. You don't have to pick up a pen for this What do you think will be the inverse of this? Minus x minus 3 by 2 absolutely What will be the g inverse now? This is something which is very famous. We all know it's ln of x Now if you do g inverse o f inverse, that means if you feed this function in place of x over here You will see that you end up getting what we had got on the Left side. So these two are same Okay, so this is the law of reversal and it can be applied to any number of functions It can be applied to any number of functions. So let's say if you have f of g of h of x And you are inverting it it will follow the law of reversal. So it'll be h inverse o g inverse o f inverse Is this fine Now for a homework, please give me a generic proof for this Please give me a generic proof for this for homework why it works Okay, why this law of reversal is followed Any questions here up up till property number two Okay, third property is something which is very important the graph of The graph of the function And its inverse Are mirror images Are mirror images of each other Each other About the line y equal to x Right Now, why is it so if you look at these two functions This is of course this And this is actually x equal to f y f of y So you can see that you have actually swapped the position of x and y to obtain this Okay, so this is obtained by swapping Y with x Okay, when you're swapping y with x Okay, so think as if you are swapping x to y and y to x it can only happen if you are Keeping a mirror which is at 45 degrees like this Okay, of course Mirror at both the sides So only way your y can become x and x can become y is when there's a mirror like this place, isn't it? So we say that the images The function f of x and f inverse x are mirror images about y equal to x line Okay, you can try it out in the previous example At given y is equal to 3x minus 5 and its inverse was y is equal to x plus 5 by 3 right i will just quickly draw their Graphs on g o g bra and just show you Where is my g o g bra? So y is equal to 3x plus 5. Sorry 3x minus 5 And y is equal to x plus 5 by 3 x plus 5 by 3 Okay, now i'll also draw y equal to x for your reference now see This is y is equal to x This is y is equal to x plus 5 by 3 And this is y is equal to 3x plus 3x minus 5 Okay, there exactly mirror images about this line as a mirror So if you treat this line as a mirror Of course mirrored on both the sides Okay, you would realize that one is the reflection of the other Okay, what happens is for the simple reason If you take a point over here, let's say I take this point. Um, let me take an integral point. Yeah, let's say I take 1 comma minus 2 And if you want to switch the position of 1 and minus 2 that means if you want to make it minus 2 comma 1 That means you want to come over here Okay, it can only happen when you are treating the mirror image of this point about y equal to x line So this will be equidistant from this mirror. Okay, so when you bring about this transformation When you bring about this transformation, you'll have to use y equal to x as a mirror for this transformation Isn't it? This is how the functions work, right? So a if you feed it'll give you b In inverse if you feed b, it'll give you a right Okay So when you're bringing about such transformations, you will have to reflect it about y equal to x line Any questions here? Now a very important outcome from this result is And I mean it's actually a debatable outcome But I've seen many books including Cengage and all the right such things So our outcome of it is Or you can say a derived result of this is If you are solving an equation where a function has been equated to its inverse Okay, then many books says that the solution of this Is same as the solution of this is same as equating the function to x In other words in other words f inverse and f of x meet at the same positions where y equal to x meets the function Now it is because many people They take a very generic view out of it Let's say my this is my function Okay So this is the inverse of the function Right And you know that these two are mirror images about these two are mirror images about y equal to x line So let me make it in blue y equal to x line Correct Okay, just consider it to be y equal to x only okay Now the thing that if you want to know where does a function and its inverse meet It is as good as knowing where does the function and y equal to x meet. This is what they're trying to claim Right But I feel that this property Is not completely true It may work for many cases, but it will not work for all the cases So there are few cases that I have figured out where it will not work For example Let us say if you want to solve Let's say your function is minus x cube Okay, so what is the inverse of this function? Anybody can tell me the inverse of this function minus x cube class class Yes, anybody minus Cube root of x correct. Okay. Now. Let us try to solve them. Let's try to equate them Okay, by the way cube root of minus this will come as a minus minus x to the power one third correct minus minus minus minus Gone correct, isn't it? So that'll give you x to the power nine is equal to x Okay So one solution could be x equal to zero another solution could be one and minus one Okay, so these are the three solutions coming from it Right So equating these two will give you three solutions But if you equate your f of x to x only You would realize that in this case In this case The only solution coming is zero That means I don't obtain these two solutions Okay So people or books who say that solving this is as good as solving this because they both meet on y equal to x line That is not a right way to Find out the number of solutions. I'll show you from the graph also See I'll hide this I'll first draw the graph of y is equal to minus x cube Okay, and I'll draw the graph of y is equal to Minus x whole to the power one by three Okay, what do you see of course there are mirror images about y equal to x But the points where your function that is your green line and your Orange line if I'm not colorblind Their meeting are these three Minus one zero and one correct And your function is meeting y equal to x only at this point. So only zero is coming out from it So other two solutions are missing. Yes or no Other two solutions are missing in this case Fine, so please do not go by this kind of logic that Solving this is as good as solving this the answer the conclusion is May not be true for all cases May not be true for all cases And some good authors have also made this mistake Okay Now We'll talk about how to find How to find inverse of a piecewise function how to find inverse of a piecewise function Of a piecewise function Piecewise defined function This is very important especially for j Especially for j. This is very very important concept Let me take an example Let me take an example for finding the inverse of a A piecewise function Let me see if I have a question on that. Okay. Let's say this example Okay for a single function the process was very easy, right? You swap the position of x and y and then made y the subject of the formula Or you made x the subject of the formula and then start the position of x and y So how will I find the inverse in cases where your function is defined in a piecewise manner Now since this is the first question I'll give you one more follow-up question Let me do it for you. Let me do it for you Okay See the approach is not very different. We'll again start with the fact that y is equal to this Okay Then step number one is you swap the position of x and y so step number one You swap y and x Okay, so when you swap y and x x will come over here and here y will come over here inside so y when y is less than one And y square When y is between one and four And a true to y When y is greater than four Correct. No doubt about step number one Okay Now step number two is make y the subject of the formula. So let me break it into pieces So first you're saying x is equal to y and y less than one Now when you make y the subject of the formula Okay, it becomes y is equal to x And this y is x itself. So it'll become x less than one Okay, so the first part of the conversion will give you The inverse as x when x is less than one So there will be no change and it is obvious. Why why why will not be any change in the inverse? Because that itself is y equal to x line. So a line When it is itself y is equal to x What will be the reflection of it about y equal to x the very same line What is the reflection of the mirror on the mirror the mirror itself, isn't it? Okay Now next part next part Next part is where you are making y is equal to Root plus minus root x Now, which one will you take will you take plus root x or minus root x because they cannot be two answers Your inverse is unique Which one will you take? Class this is the question for you less root x. Why? Because why lies from one to four? Correct So in this case your negative has to be dropped off Okay, of course, we don't write positive. We just write root x. Okay Now here also you have to make the same replacement. That is your root x will lie between one to four A sober way to write it is x lying between one to sixteen Okay, so your function will behave as root x When your x lies between one to sixteen Correct now coming to the last part Last part when you say x is equal to eight root y y will be equal to if i'm not mistaken x square by 64 correct And x square by 64 Will be greater than four Yes or no Yes or no So if x square is greater than 64 that means x square is greater than 256 That means x should be greater than 16 Yes or no Right now If you combine these three pieces and make a one function out of it Your final answer would be F inverse x is x when x lies From minus infinity to one It will behave as root x when x lies between One to sixteen And will behave as x square by 64 when x is greater than 16 Because this is going to be your answer Is the approach clear Any questions regarding this is the approach clear? Any questions here Now you may try to make the graph out of it. See I'll I'll I'll I'll plot the graph for both of them And I'll show you how this function actually works Okay, so all of you, uh, please help me to copy this on GeoGibra So I'll be plotting it now for you So your y is equal to if x is less than one it follows x right Okay If x is between One to four It follows Can somebody tell you what is that one to sixteen One to sixteen No, no, no original function original function y square Original function is what original function is x square. No Yeah, this one I'm typing Yeah, so x square x square Okay and when it is And when it is greater than four It is following a true text. So x is It is wrong x is greater than four It is following eight root x Okay So this is your graph So of course it is invertible because it's a one-one function any horizontal line you'll draw it'll make it Um cut at one point and it is on two also Okay Now if I want to write down the inverse of this function, what was the inverse that we had got can somebody dictate me if x is less than one it follows x correct Yes, sir Okay, and what are the next one root x when it is Less than equal to equal to one Greater than equal to one less than equal to 16 16 and the following root x right x to the power half comma comma And when x is greater than 16 it is following the graph of x square by beautiful Do you see that Exact mirror image about y equal to x line exact mirror image about y equal to x line So I'll I'll draw that also to show complete the scenario Okay This is how you basically make the make the Inverse of or find the inverse of Let me take a snapshot of this and put it on your notes. Okay, so this is how you Find the inverse of a piecewise function There is it. Is this okay? Any questions here? Okay, I'll give you one more exercise just for you to try out because obviously you would like to do one on your own So let's say I have a function I have a function Which is defined as x plus four When x lies between one and two And is defined as seven minus x when x lies between five and six Okay find the Inverse of this function That's one first part of it and second is sketch both of them So sketch f of x and its inverse Everybody please try this out Just follow that process You'll be able to find the answer without any problem Just type done once you're done So dhiyan has got it. Good dhiyan How about others? Are you sure it's done? Okay, let's discuss this Not very difficult. So first thing that you would do is you write y as this Okay, so I'll just copy my function Remember step number one step number one is You are swapping the position of x and y Step number two, you're making y the subject of the formula. So let me take one by one So let me take the first piece first piece of the function So in first piece of the function if you make Seven minus y Thank you So in the first piece if you make Make the change you'll end up getting y as x minus four So this interval will also be for x minus four Just add a four to all the sides will become five to six So this will be your function Definition of inverse when you are between five to six correct Let's take the other one Let's take the other one so x is equal to seven minus y so y is equal to seven minus x And this definition will be true when seven minus x lies between five and six Okay Do one thing subtract seven from all the sides will become minus two two minus one Our drop a negative sign and reverse their positions Okay, so Collating both these two informations I can conclude that my inverse would be x minus four when x lies between five and six And it is seven minus x When it lies between one and two Okay Now let us plot their graph and very interesting thing will be seen over here. Uh, please remember the function Okay, and it's inverse. I'll ask you. Okay Can I go to jujibra now or anybody who's copying from here Coping done class Coping partisan. Okay Yeah So now Somebody please dictate me the function. So first tell me the interval and then the function definition. Okay So i'm hiding this y is y is equal to If yeah, tell me the first interval x between one and two And it is defined as x plus four Then Five and six Five and six Defined as Seven minus six Seven minus six So if you can see I've actually zoomed in a lot. So I'll just zoom out from the given Scenario Okay, so you can see there are two Line segments you can see here Two minor line segment in blue. You see this these two lines. Okay. So this is the graph of the original function Now what was the graph of its inverse? It's looking five and six Five Six, how is it defined? X minus four and then It's between one and two seven minus seven minus six Okay, now see the green ones how beautiful they're coming as crosses Now this example is another example where you would realize that Where you would realize that the function and it adds and its inverse Don't even meet at all on y equal to x line. Isn't it? So this is a classic example where your function and its inverse are not even meeting Are not even meeting on y equal to x line. They're meeting on their own positions Okay, so this further, you know Makes that situation Makes that property In fact, I will not call it as a property, but this is what many Authors will state so that further invalidates the fact I'll plot it over here and keep so Just a concluding word over here before we go to some next set of problems. So this is an example where f of x And f inverse x Do not Intersect on y equal to x line And there can be several such cases Right There could be a cases where they don't even meet at all. For example e to the power x and lnx case Now here is a case where they're meeting but they're not meeting on y equal to x line. Is this fine? Or really i'm terribly colorblind Okay, so we'll conclude this idea of inverse with few few problems which are going to be important not only for your Je but also for your schools. Okay, so let me give you a set of problems We'll not solve all of them Here I've Selected few problems for you Not all of them. I'll do today I'll just ask you to do the following Do the first one Do the first one Do the third one And do the seventh one Okay, rest you can do it for homework And then we can actually start with our inverse trig functions Our first one if you're done, just write first done Number one done So that I can discuss it So assuming it is invertible. You don't have to prove it is invertible assuming it is invertible Assuming it's our bijection Find the inverse of this function Done schaumic is done with the first one Trippan also done We have Vibhav also done. Okay, so this is a question which is a question in your school exams also. Okay So we'll discuss it So y is equal to e to the power x minus e to the power minus x By e to the power x plus e to the power minus x plus 2 by the way if you recall it is actually Tan hyperbolic x dan dan plus 2 actually. Okay This is just for your extra knowledge Can you hear me? Can you hear me everyone? I'm audible. Hello. Can you hear me? Yes Sorry, there was a power cut and now it has come back Okay So yes, so this is just for your extra information that is actually dan x plus 2. Okay Now we have to make now again There are several approaches whether you want to swap x and y first and then make y the subject of the formula Or you want to start you want to make X the subject of the formula and then swap x and y so let's say first of all, I would write Write it like this Okay, and now I would be making x the subject of the formula. So if you make x the subject of the formula You can use component or end of dendo so use Component or end of dendo Okay, so when use component or end of dendo first you can just add it on top So it'll make y minus 1 And then you can subtract it from here. So it'll be 1 minus y minus 1 So it'll be 1 minus y plus 2. So that will be your e to the power x Yes or no So e to the power x is y minus 1 by 3 minus y So x is ln of y minus 1 by 3 minus y Now you can just swap the position of x and y from here. So this is what you will see Okay, so this becomes your inverse of the given function. This becomes your inverse of the given function Now one thing that we can check here is that Uh, the range of this function will be the domain of this function Okay, that I will verify for you over here So let us find the range of this function So what is the range of f of x? So for finding the range of f of x You can take the limit of Extending to infinity of e to the power x minus e to the power minus x by e to the power x plus e to the power minus x plus 2 Now all of you have done your limits, right? Can you evaluate this? So what do you do in this case first you take e to the power x common right? Something like this e to the power x e to the power x gone and when x tends to infinity when x tends to infinity Remember these two terms will go for a toss. They'll become a zero. So answer will be 1 plus 2, which is 3 So this is the maximum value Okay, and for minimum value you have to take the limit when x tends to minus infinity So When you apply the limit when x tends to minus infinity in that case, what do you do? In what in that case, what do you do? put y as minus x isn't it So when x tends to minus infinity y tends to infinity So it will be like e to the power minus y minus e to the power y By again e to the power minus y plus e to the power y correct And do the same act that we had done in the previous um in the previous uh uh limit So we will take here e to the power y common So we'll take e to the power y common So it'll become e to the power minus 2 y minus 1 By e to the power minus 2 y plus 1. Okay, plus 2 will remain as it is This is gone. And now these terms will become zero. Don't ask why they become zero because That's a quantity whose modulus is less than 1 1 by e is a quantity whose modulus Is less than 1. So if you're raising it to an infinite power, it will become zero So your answer will be minus 1 plus 2, which is 1 So your range of the function is from 1 to 3 Now check the domain of the function over here Check the domain of f inverse So for domain of f inverse you will say since this is a argument of log this guy must be Greater than zero right Okay, now by the way, let me tell you one more thing 1 and 3 cannot be achieved because infinities cannot be achieved So they will be around brackets correct Now here if you solve this by wavy curve This will be 1 this will be 3 and this will be negative positive negative Check the sign scheme over here fine Greater than zero means it will be between 1 and 3 So this will be your domain of f inverse. So it further re establishes that a function and its inverse The domain and range position will be swapped So domain of one will be the range of the other and vice versa Got the point Any questions here anybody So this is a question which will come for your definitely come for your school exams as well Please try the third one And once you're done, please type done for the third one If you want I can take the question separately because we have almost Use the entire space over here. So let me just put the question in the next sheet So I'm taking your attendance right now. Hope everybody is present third one is super easy. See your function is x minus gif of x Now, what is the domain of the function that you're looking at domain is from 2 to 3, you know from 2 to 3 your gif is going to be 2 correct So this is your function So anybody can find the inverse of it. What is it x plus 2 done? And remember this will be valid when your x is between 0 and 1 so range will become the Domain of its inverse. So, please remember that Fine, uh, try seventh one now So this was your third one You try seventh one now everybody So assuming that it's a bijection Look at the domain and the range in fact, uh, yeah domain and range of f of x We have to find the inverse of this function. So in the interest of time, I'll also, uh, start contributing over here So first of all, we'll call this as y Now we'll make x the subject of the formula. So for that first, you have to take log to the base of two on both the sides So log y to the base two is x square minus 2x. So you can say it results into a quadratic right Let us use the quadratic equation formula the sheathalach area formula over here, which is 2 plus minus under root b square minus 4 ac which is plus log y to the base two All divided by 2a So I think the factor of 2 can be cancelled from everywhere. So it'll become 1 plus log y to the base two now Here there is a problem coming up We have both plus and minus but we cannot write both of them because we know inverse of a function is unique I cannot write two answers for it Now which one do you think is accepted? Which of them is rejected? And why look at the domain? domain says minus infinity to one so your x value should be less than one Right, so we'll take minus not the plus So In light of your domain, you choose one of the answers. This is very important. That's why I told The property that inverse is unique is very important. You cannot write plus minus both and state that this is your inverse Now you can swap the position of x and y so sorry Now when you swap the position of x and y, this is what you end up getting as your inverse This is your answer. Okay Any questions here anybody? So last half an hour. I'm going to now start your inverse trigonometry But this was a useful and it was a much required exercise that we need to have Probably while doing functions chapter, we can, you know, skip this part Is that fine? And tomorrow, uh, whatever is starting in your school, please message me privately anybody that this chapter is has started in school Fine So that I am also prepared with respect to the next chapter Yeah So now we are going to start with inverse trigonometric functions So nafflers have already done this and I'm sure They had a very good experience with this chapter, right skanda shankin Anurag Say something How did you find this chapter? I studied easier chapters Skanda, how was this chapter? How did you like overall tell tell your feedback to others? I didn't like it that much. So Didn't like it that much Anurag Anurag is quite Okay They don't believe that it's very easy chapter. Okay. So yeah, so what is this? Basically, this is a concept where you are learning the inverses of trigonometric functions But wait a minute Let me begin with the first trigonometric function sin x defined under this Okay now under this function Under this definition if you plot the graph of sin x if you plot the graph of sin x Okay, you would say sir wait a second When I plot this function, I realize it's not a one-one function because In the domain of the function if you plot this graph it is going to take so many turns In other words, if I draw a horizontal line like this It is going to cut the graph at millions and trillions and zillions of positions, right? So this is not a one-one function correct So because of this it is not a one-one function Okay, so one one And okay now it's range is from minus one to one Okay, but here the core domain is all your numbers Right, so it is not onto also That means chapter over we don't have to study inverses trigonometric function Because it is not one one and onto So how can you find the inverse of it? Wish life were This easy for us But it is actually not So right now it is not one one And onto but you can definitely make it one one and onto remember This is what I was speaking in the inverse trigonometry chapter. So don't judge a function by its definition Okay, if you curtail your domain and co-domain in such a way That you don't allow it to take a turn And secondly you limit it between your range Then you will make this function a invertible function You can make this function invertible function, right? So agreed in the present definition. We cannot find inverse But what if I define my Define my sine x function between two such points Where I'm not allowing it to take a turn back Now I would ask from you Suggest me an interval Where number one the function doesn't take a turn anywhere Number two it can cover all the values from minus one to one. So basically I'm asking you To treat this as your range So what should be your domain says that your function doesn't take a turn and it covers all values from minus one to one What will you say? Minus pi by two to pi by two Okay Now there can be so many answers for it. Why not pi by two to three pi by two? Why not three pi by two to five pi by two and so on? See all of them are accepted But normally there is a standardization of the interval that we have done as A principal value branch So normally we restrict our function to minus pi by two to pi by two in this interval Okay So all the other parts will be removed off So only in these two vertical lines Your function will not take a turn back And if your function doesn't take a turn back in this interval your function will be one one Now why not any any other interval? Why was you know this standardization made? See if you don't make this standardization What will happen? Everybody will start making their own intervals And then there would be an utter chaos when you are finding its inverse all the rules will change Dhyana is a question sir. Could you please repeat from where you rule out? It being on to internet good. Okay fine. See when I started talking about inverse trick function I just you know draw the graph of sine x for its domain And we realized that it was not one one and onto correct Then I told the class that even if it is not one one and onto we can make it one one and onto how By restricting our domain and of course restricting our core domain So core domain. I made it the range. So this became the range. So I made it onto Okay, and how did I restrict my domain? I restricted my domain in such a way that I don't allow it to take a turn back And secondly it covers all the values in its range Okay, so why not zero to pi by two zero to pi by two also. It's one one, right But it will not cover all the values in the range So I want to go from minus one to one So it can only happen when I start from minus pi by two to pi by two or pi by two to three pi by two Or three pi by two to five pi by two or five pi by two to seven pi by two But normally this branch is chosen as the principal value branch principal value branch It's like your principal root What is root of four What is root of four two why not minus two minus two is also correct But why don't we write minus two because Two was considered as the principal root So there has to be standardization in mathematics also just like you have it in iupsc names Okay So if you Are studying trigonometry you will have if Nothing is mentioned. This is the interval under which you are treating your sine x function to be one one and on two All right Unless until stated this is the interval where you will be considering your function to be one one and on two Now there is there are other rules also which I would like to tell you in the beginning also beginning itself Normally the interval that we choose not only we ensure that the function is one one there The function is The entire range of the function is taken care by that input range Input set But we also ensure that there is no discontinuity in the function in that particular interval But this discontinuity criteria is not very important Preferably we choose that interval where there is no discontinuity But if there is a discontinuity Coming up at the cost of making it one one then we'll have to buy that discontinuity Okay, so this is something that you should keep in mind later on we'll use this concept So when you're writing the inverse of this function You will write it like this f inverse f inverse will be defined as Domain is minus one to one and range is from minus pi by two to pi by two. This is very important unless until stated Your range and domain for sign inverse function. Of course, there is nothing you can do with the domain Okay, you can only curtail it between minus one to one. So range cannot go beyond this That's very very important Now i'm just predicting Hypothetically If j has to make this particular situation worse Or let's say j advance and all what they will say that Let's assume that there is a sign inverse function, which is defined from minus one to one To pi by two to three pi by two Okay Then remember all the rules that you have learned. I'm just talking to the naffolas All the rules and properties that you have learned they will all go undergo a change Okay So whatever we are studying in this chapter is under the fact that i'm assuming that this is your domain of the inverse And this is the range of your inverse Okay This is fine any questions here Now how would the graph Be related graph. You already know if this is the graph of sine x under which it is invertible So let me just draw here. This is minus pi by two. I should actually draw it a little farther Let's say this is minus pi by two This is pi by two And this is one This is minus one Please note pi by two is a bigger value than one and minus one, right? So don't draw it at the same levels I've seen people making that mistake So your graph would be like this Okay, so this is a graph of sine x function Now if you want to take its inverse What is the criteria for taking the inverse? A drawing the inverse graph You need to take the mirror image about y equal to X line. So let's say this is your y equal to x line Okay So if you draw the mirror image about this, let me draw it separately Has it been done in uh raja jeena at this chapter? No, sir Okay, so its graph will be like this So whatever was on the x axis will now jump and go on the y axis And whatever was on the y axis will now come on the x axis So all of you please Carefully observe the shape the shape will now become like this Okay So this is the shape of it'll stop here. It'll not go any further Okay, so the graph is only defined from minus one to one and range is only from minus pi by two to pi by two The function for uh sine inverse on geo zebra is arc sine So I'll just type it for you Y is equal to arc sine x Okay, so this is the function as you can see this is the graph Is this fine This graph from now on word should be in your mind. It is as good as knowing the graph of sine x itself Now a couple of things that I would like to comment about this graph Is it monotonically increasing? Yes Yes, it is always increasing Now this is something that you will learn in your application of derivatives So if a function is monotonically increasing Okay It's inverse will also be monotonically increasing Is that fine any questions here and if a function is monotonically decreasing its inverse will also be monotonically decreasing So number one couple of things that you have to uh note down here the graph shows that Sine inverse x is a monotonically or you can say strictly increasing function Now what is the strictly increasing function many people ask me said How mathematically should we understand strictly increasing function? It means that in sine inverse If you put let's say if you put two inputs X1 and x2 such that x1 is greater than x2 Then sine inverse of x1 will also be greater than sine inverse of x2. This is a layman definition of an increasing function Okay And one more thing I would like to highlight over here inverses will always give you an answer as an angle in radiance inverses will always give you an angle in radiance Is that fine? So if I ask you what is sine inverse half What will the answer be Pi by six What is sine inverse root three by two Pi by three, okay What is sine inverse minus half Minus pi by six. Okay, so always your answer would be angles mentioned in radiance Okay Second thing that you would observe about this graph this graph is symmetric about origin This graph is symmetric about origin Which means Sine inverse x is an odd function Now what is odd and even is something which is again a subject matter of functions chapter. I'll talk about it when the right time comes Okay, so this is an analysis that I'm doing as of now for you fine So sine inverse x Is an odd function. Is this fine any questions here anybody? Can you all hear me? Hello Yes, sir. Okay. I think the power just Bent off again Okay Now let's move on Without much ado to cos inverse Let's move on to cos inverse So now when you're studying cos inverse Again, I'll save your time over here In order to make it onto we have to keep it as a range of the function now cos inverse graph Cos inverse x graph Sorry cos x graph Is again a sinusoidal curve like this Okay Is again a sinusoidal curve like this So in which interval do you suggest me to break this graph so that it is one one function And it covers the entire range from minus one to one Zero to pi that's excellent. Okay, you could have written