So, dear all in the class, we had started the discussion about finding the domain and and a range of a range of real functions. Okay. So, this is what we had started talking about domain and a range of real functions. So, in the class, we had taken up the domain and the range of rational functions to begin with. Under rational functions, we covered polynomials. Under polynomials, we had covered the domain and the range. By the way, when I say domain and the range, I hope this is clear that I'm talking about the exhaustive domain, okay, and the range resulting out of that exhaustive domain. There is no curtailment, there is no restriction to the input that, you know, somebody is putting on the function. Okay. So, what is the exhaustive set of inputs that we can feed to the function is what we are calling as the exhaustive domain. Okay. So, continuing with the polynomials, we covered constant functions. We covered linear functions, quadratic, cubic, bi-quadratic. So, this was under our polynomials discussion. Then we started with the rational functions of the type 1 by linear. If you remember, we did 1 by AX plus B, and there we also found out the domain and the range. We also discussed linear by linear, right? Somebody please confirm, linear by linear, AX plus B upon CX plus D. You may use chat box also to communicate. You can unmute yourself also and communicate, right, Anirudh, am I audible? Yes, sir, we can hear you. Sorry, muted your song. Oh, yeah, sorry. Is this fine? Yeah. So, we basically discussed linear by linear also, correct? After that, we started talking about the wavy curve sign scheme, and some of you had some questions related to that. So, why don't we do one thing? We'll start our today's class by solving few more questions on the wavy curve sign scheme. What do you think? Will that be a good idea? Okay. So, let me just pick up some questions which was given to you on your assignment. Then when are we doing the one? Sorry? Sir, will we do the 1 by quadratic also today? We'll do that. Yes. Don't worry about it. Okay. So, we'll take one of these questions. Maybe we can pick up this question, the one which looks the slightly ugly part, this one. Okay. Let's take this one. So, let me upload the question here. Not this one. This one. Yeah. Yeah. So, this question is basically for what values of X do you think this inequality will be satisfied? Okay. Those who have already solved this question, it would be a quick check of your answer. Those who haven't, please listen to me very, very carefully because as I told you, the wavey curve sign scheme is going to play a very, very vital role in your future concepts also. Like, we'll be talking about the wavey curve sign scheme for finding the range or finding the interval where the function is increasing, decreasing. That is the concept of class 12. So, your seniors are doing that presently. Hence, it is very important that you know this idea very well in class 11 itself. Okay. So, what do you do in this case? We first make a number line. Step number one. Make a number line. Okay. And on this number line, please do not show a zero till it is needed. This is what I discussed with you. Step number one. Make a number line and do not show a zero till it is needed. Okay. So, what are we going to show on this number line? We are going to show the zeros of each of these factors. Okay. The factors which I am circling on your screen with yellow. So, zeros of each of these factors I am going to show on this number line. So, what is the zero of X minus one to the power four? One. Isn't it? So, we'll show a one. Maybe let's start with one position here. What is the zero of the next one? X minus two to the power 12. Two. What is the zero of X minus three to the power 17? Three. Right? So, that number which makes that particular factor zero. That is basically what is called the zero of that factor. And that is plotted, that is shown on that, this on the number line. What about X minus five to the power 2012? Five. Yes. So, five will be the zero of that particular factor. Now, after having put these numbers on this number line, what are these numbers do? These numbers divide the number line into intervals. Right? So, you can see one, two, three, four, five intervals have been created because of these four numbers. Right? So, what I will do is I'll first start with the rightmost interval. The one which I'm showing right now with a yellow tick. Okay? So, that is my rightmost interval. Correct? So, what I'm going to do, I'm now going to figure out what will be the sign of this rational function. This rational function. This rational function, the one which I'm showing with a blue box. Right? So, what is the sign of this rational function in that interval? Right? So, if let's say I call this whole thing as let's say Y, I want to see the sign of Y in this interval. Are you able to find out what I'm trying to say? So, basically you are looking for the sign of the rational function in all the intervals created by these four numbers. So, what we do is we do a sample check. We take any number which is greater than five. Any number you can take. Okay? Up to your convenience. Let's say I take a six. If I put a six here, I get six minus one to the power four, six minus two to the power 12, six minus three to the power 17, six minus five to the power 2012. But you don't have to calculate these values because you will unnecessarily waste time. What I see is each one of them are positive, positive, positive, positive, positive. So, overall this entire expression will be what? Positive only? Yes or no? In the class, some of you were asking this question, sir, does it always remain positive on the right most interval? Need not be. In some cases, you may also get a negative value, right? So, it's always advisable that you check by putting a sample value which is lying in this interval. Okay? And it could be any value. Even if you put seven or eight or a hundred or a 5,000 or a 1 million, the result is still going to be the same. Okay? So, Samrit is already ready with the answer. Okay? Samrit, just hold on. Okay? You may have got this. So, let me write down the sign here. So, this sign is a plus. Okay? Now, move to the next interval which is to the left of it. Okay? You start moving to the left interval. So, while moving to the left interval, you cross this number five, right? This number comes from this factor, correct? This factor is subjected to a power which is an even number. So, 2012 is an even number. So, what did I tell you? I told you a simple rule to follow. The rule is if the power is even, then retain the sign. If the power is odd, switch the sign. Correct? So, here it is even, so you will retain the sign. So, this will also remain positive. Correct? Any questions? Next. Now, you are crossing three. So, three comes from this factor. This factor has a power of 17. 17 is an odd number. So, if it is odd, you will switch the sign. So, if you switch plus, it will become minus. That means reverse the sign. Switch the sign means reverse the sign. Okay? Next. The number you are crossing now is two. Two comes from this factor. Again, this factor has got power of 12. 12 is even. So, again, retain the sign. So, even means retain the sign. Correct? The next one, one comes from x minus 1 to the power 4 factor. Right? So, one is a zero of that particular factor and four is an even number. So, we will again retain the sign. So, this will be negative. Is that fine? Any question? All right. Let us answer this question. Where is this particular expression greater than equal to zero? See, look at the question. The question is demanding you where it is greater than equal to zero. Yes or no? So, wherever you have written positive sign, that interval you have to state. Okay? So, your positive sign is in the interval 3 to 5 and 5 to infinity. Okay? Immediately put a union sign in between. Now, will I include 3? No. Not because 3 is in the denominator. Even if it was in the numerator, I would not have included it because here I am talking about a pure inequality. Okay? So, in pure inequality, you have to not include any of these numbers. So, you have to start putting round brackets about everything. Okay? So, your x should belong to this interval for your expression on the left-hand side to be always greater than zero. So, this is your answer to the question. Is it fine? Sir, I have a question. Yes. Sir, can we just write x belongs to 3 comma infinity? But if you write 3 comma infinity, you have included 5 also there, right? Oh, yes. So, you have to remove the 5. So, either you write like this or alternately, you can also write x belongs to 3 to infinity excluding a singleton set 5. This is also acceptable. Both are acceptable answers. Okay? Excuse me, sir. Hi, Arjun. Good evening. So, what we were doing was we were just quickly recapping the wavy curve sign scheme. I have not started teaching anything new. So, I just talked about wavy curve sign scheme, which we had ended our last session. Okay, Adhika? Good. Any other doubt? Arjun, is it fine? Okay. So, if you want, I can take one more problem so that everybody who had joined in late, you are all benefited. We'll take one more problem just a second. Which one should we take? Which one should we take? Let me take the one where we have, okay, let's take, let's take the last one, please. Could you do the 10th one? 10th one, sure. Okay. Let's take the last one. Okay. So, this has our next question. Solve this inequality by using wavy curve. Okay. Now, this is also a rational function, guys, angles. You must be thinking that, oh, this is a polynomial, yes, polynomial, in fact, is a rational function. Okay. Now, before we start making our number line, it is advisable that you start factorizing it completely. So, I think some of the terms, in fact, most of the terms are quadratic expressions in themselves. So, let us try to factorize each one of them. So, x square minus 16 is what? x minus 4, x plus 4, x square plus 5, x plus 4 can be factorized as x plus 1, x plus 4 again. Now, remember, x plus 4 is already there, so you put a square over here, correct? I hope the factorization, you are able to understand. x square minus 9, I can factorize it as x minus 3, x plus 3, right? And x square minus 7, x plus 12 is x minus 3, so let me write it on top. This guy is x minus 3, x minus 4, am I right? So x minus 4 is already there, x minus 3 is also there, so you can write the powers on them as 1 more. Is that okay? Any question? So, indirectly, we are trying to solve this inequality. Is everybody fine with the question? Okay. So, now, let us begin the process. So first step is make a number line. So let me make a number line, okay? On the number line, I will now show the zeros of each one of these factors. So for this, it is plus 4, so plus 4, let me write it on the rightmost. This fellow, it will be minus 4, let me write it here. For this, it is minus 1, minus 1, maybe I can write it over here. Then for this, it will be 3, 3, let me write it over here. And for this, it will be minus 3, minus 3, let me write it over here. Is it fine? So all the zeros of the factors which are seen in this inequality on the left-hand side, even before that, I told you this in the class that please ensure you have a zero on the right side to begin with. If the zero is not there on the right-hand side, then you cannot start with your wavy curve. You can do something with that zero, with that non-zero number and try to bring a zero at that position, right? So wavy curve sign scheme, please do some checks and balances before you start. Number one, you should have a rational function and a zero on either side of the inequality. Then only you can use wavy curve, right? So you should have a rational function and you should have a zero. Then only you can begin with the process, else you cannot use wavy curve for solving any inequality. Is it fine? Okay. See, many people ask me this question, sir, what is this wavy curve representing? How do these signs actually are related? See, these signs are basically telling you the position of this function, right? So let's say I take this as a function, okay? Y equal to this. And this is your x-axis actually. So wavy curve tells you how is this function positioned with respect to x-axis? If it is above the x-axis, you put a positive in that interval. If it is below the x-axis, you put a negative in that interval. So at the back end, that means the heart of this concept is basically you are trying to show without actually plotting the graph, how is this function positioned with respect to the x-axis? So if the function graph, of course, you're not drawing the graph here because drawing the graph is not that easy, it will be time consuming task. So had you drawn the graph, you would realize that the part where you have written positive in the interval, in that part, the graph is above the x-axis where you have written negative, the graph is below the x-axis, right? And in this case, minus four, minus three, minus one, three and four are the positions where it will be cutting the x-axis, okay? But it may not be the case in all the rational functions. So those factors which are in the denominator, listen to my words very, very carefully, for those factors which are in the denominator, the zeros of those factors at those positions, the function is actually becoming undefined. So those are not the points where the function is cutting the x-axis. However, this information is just optional for you. It has no bearings on solving the question. So just an extra knowledge which I'm giving you. All right, let's begin the process now. So let us pick up a number greater than four, okay? So don't take chances. I don't start guessing it to be a positive sign. Of course, it'll come positive in this case also. But take a value which is greater than four and just see what is the sign of this expression, the one which I'm showing right now. What is the sign of this expression for the value chosen by you? So for example, if I choose a five, this will be positive. This will also be positive. This will also be positive. This will also be positive. This will also be positive. So everything will be positive. So put a positive sign, okay? Now come to this left interval. So you are crossing four. Four comes from this factor. This has got an even power. So retain the sign. Then you are crossing three. Three comes from this factor. This has again got an even number. Again, retain the sign. Then minus one, minus one comes from this factor which has got an odd power. If nothing is there, it is an odd power. So switch the sign. Again, minus three comes from this which has again got an odd power. Again, basically switch the sign, clear? Then minus four comes from this factor. This has even power. Even power means retain the sign. Is this fine with everyone? I hope the sign was clear. Any question, any concern? You can unmute yourself always. Yes, somebody was saying something. So if it's even power, it's retaining an odd power. It's changing that. It's switching. Yes, yes, yes. Absolutely. Okay. So let us try to address the requirement of the question. The question requirement is, for what interval is it showing less than or equal to zero? Okay. So let us first address less than a part. So less than means negative, correct? So it is negative here, correct? So write down that interval. So that interval will be minus three to minus one. Okay. Hold on. Don't put anything right now without confirming. What should we put next to minus three? A round bracket or square bracket? Now here, less than equal to, and all the terms are in the numerator only. There is no denominator. In fact, denominator is one if you want to say a denominator. So every term, which is basically on the numerator, and it is less than equal to has to be included. So this will be included because at minus three, this becomes a zero. So it is fine for me to get a zero because inequality says less than equal to zero. Okay. Similarly, minus one can also be included, but hold on for a second. Aren't there more values for which the function can become a zero? Like, isn't minus four a value, right? Three a value, four a value. So you have to say union of sets which contain minus four, three and four. Is it fine? So your X must belong to this set for your inequality to be satisfied. Okay. So as I was talking about, minus four, three, four. Why did I include minus four, three, four? See, at minus four, the value of the function becomes a zero, no? So that should ideally come into your answer. But if you just write this, it will not come in your answer. So you have to separately take care of those single, single values for which the function is becoming zero. Yes or no? Yes, sir. So minus four is a value for which the function was becoming zero. So I had to include it in my answer. Three is also a value for which the function was becoming zero. So that also have to include in my answer. Four is also a value for which the function was becoming zero. So that also has to be included in my answer. Clear? Sir, before starting, putting the sign on every curve, when we take a bigger value, sir, won't the value become greater than zero because the sign? No, no, no. Don't look at the end result, Advika. End result is something which you will take care after putting the sign. So what are we doing? We are ignoring this information as of now. And we're just trying to give sign to this number. Later on we'll say for what interval was it negative, for what interval was it positive, depending upon what question was asked. Okay? Initially, when we put the sign, we don't look at this part. We just look at this part that, yeah, it should be zero here. That's it. Okay? Beyond that, you don't have to worry about it. So we take a greater number than four. Do we have to insert it in all of the X's? Or do we just have to insert in the? X value. So the number that you're taking, you have to put in the X value everywhere? Everywhere. Everywhere, yes. That is your X value. So what if it doesn't match? So, sir, what if in one bracket, says positive and one is negative see if I take a five or a six or a seven any number you take this will be positive only you can check if you want to right then was that your question not all like in some specific questions overall understood what you're trying to say you're saying that what if this is positive and this is negative no yeah yeah overall you have to see the sign of this expression oh it's a majority not majority you let's say plus plus minus yeah I got it see product of three numbers where two are positive and one is negative overall will become negative so you have to yeah you have to see that depending upon what signs you're getting what is the collective sign of the number that you would be getting yes sir so but what if there are a lot of factors and there are many thirty forty are there thirty forty do you think such kind of question will be asked to you what if it comes thirty forty see thirty forty you are smart enough to figure out how many events are there how many odds are there correct right so every even odd in every negative which is present even number of times will be one even one positive number right yeah you can figure out I mean of course the question setter aim is not to bombard you with a very gigantic expression because see this is not a concept in itself it is helping you to solve a problem it is a tool are you getting my point so it is going to be one of the assisting tools in solving a different problem may be somewhat some problem in calculus or some problem in functions okay so the the question setter may not ask you a direct question on this idea okay may not ask you I'm not saying we'll never ask you okay I think Arul also has a question can you give an example where the right most sign can be a negative yes why not Arul okay let me give an example for that maybe I can make a function on my own here let's say I have a function like x minus 1 to the power 3 4 minus x to the power 6 7 upon let's say x plus 2 to the power 2 and x plus 1 to the power 4 okay let us say I want to solve when is this rational function less than equal to 0 okay so just hold on everyone I'll be solving this problem on behalf of you all but if you want to attempt it you are more than welcome so first we'll make a number line then we will put the zeros of these factors on the number line so as you can see for the first one for this fellow the zero is one okay so put a one somewhere let's say I put a one here for this it will be four so put a four here for this it will be minus 2 put a minus 2 for one second yeah minus 2 and for this it will be minus 1 clear okay so have you all put the numbers on the number line okay now start with the right most interval take any value which is lying in this interval you may take a 6 maybe so if you put a 6 here this is for Taren as well if you put a 6 here this is 6 minus 5 cube so we don't have to find the value just put the sign on top of it let's say positive okay this will be 4 minus 6 4 minus 6 is what 4 minus 6 is minus 2 minus 2 to the power odd number will be negative okay this is 6 plus 2 to the power 2 again positive 6 plus 1 to the power 4 positive right now you have a quantity where on the numerator there is a positive into negative which is negative and down it is positive into positive positive so negative by positive is what negative correct no so this is how you end up figuring out that the right most interval is a negative sign clear no now clear Taren also you are clear how I am figuring out the sign of the entire number okay it is not the majority my dear it is collectively ka number ka sign ban kya tha getting my point okay now if I'm going to sir what if we have like a x minus 1 the whole cube in numerator and then x minus 1 whole square and denominator serve with the same zero is that repeat repeat once again sir like if you have the like x minus 1 in the numerator x minus 1 whole cube and then in the numerator also you have x minus 1 whole square so how will you solve that type of inequality where both the values zeros are the same zero are the same so is it not equivalent to saying that this is you remove this guy sorry remove this guy and write a power of one down correct yes over okay okay and but in that case if you realize if let's say the power on the numerator was higher then you will have x minus 1 on the top correct but still one cannot be included in your answer despite the inequality being a non-hard inequality see I'll tell you the reason why let us say I give you x cube by x okay and I ask you this inequality okay pay attention pay attention everyone let me ask this inequality greater than equal to zero now see you say sir x is there on the numerator also and x is there in the denominator also of course the powers are different so what you'll do you'll remove the powers and you make it as x square right so you know make the problem like this correct now when you plot the zero of this on the number line the zero of x square is zero only okay take any value greater than zero let's say I take a one one square is positive and since the power here is even there will be retention of the sign but here please pay attention you would be thinking this is greater than equal to zero for all real numbers right right but here I can't put all real numbers because if I take a zero the original expression which was x cube by x this will not be defined for zero sorry this is x correct because your denominator is becoming zero for zero so zero will not be defined so you have to remove the zero from your value of x got the point yes or no okay are we has a question what if x square was less than zero less than zero means there is no value satisfying it's a null set clear are we so your answer your set will be a null set understood so very good question that you ask mehul is this clear what I'm trying to do here so you have to always look at the original function since x is present in the denominator I cannot include zero in my answer okay even if after simplification that x is no more to be seen that is okay but still we cannot use because there's a difference between x cube by x and x square x cube by x you can't put x as zero simple as that in x square you can put x as zero so even though it is simplifying to x square still zero is not allowed have I made myself clear this is a very interesting question on this which came in one of the competitive exams I'll show you the question just give me a minute just give me a minute okay let's complete this problem since we have started with it right so four comes from this factor this factor and this factor has an odd power so they will be switching off sign then one one comes from this factor not this okay this for this is responsible for minus one the top one is responsible for one okay this is the one which is responsible for one again this has got an odd power so switching of sign again minus one comes from this factor which has got even power even power means retention of sign and then minus two comes from this factor which is again even power so again retention of sign is it clear any questions okay let me just complete and then maybe I'll take few more questions yes so let's complete this so the question setter is asking when is this less than equal to zero right so less than equal to zero means wherever you have written negative sign you first write down those intervals so minus infinity to minus two then minus two to minus one correct somebody retracted his answer then minus one to one okay and then four to infinity I hope I have covered up all the intervals now let us try to see what kind of a brackets will go around them I'll quickly write a union before I forget the to write the union yeah now let's start putting brackets so minus infinity can never be included infinity can never be included so always put round brackets around infinity or minus infinity now minus two can I include minus two no because minus two comes from the factor which is sitting in the denominator I can't put a minus two in X because then the denominator will become zero there by making the entire expression undefined so minus two cannot be included so since minus two cannot be included here it cannot be included here also okay so put round round there then minus one again also cannot be included so put a round here round here one can be included because one is coming from this factor right and when I put X as one it makes everything zero and zero is perfectly acceptable by the inequality right so inequality says it can be zero also so one will be square bracket four also will be square bracket okay now one quick check you have to make are you covering all the numbers or all the zeros of the factors on the numerator that means is one and four included in my answer because zero is allowed yes one and four are already included in my answer so this becomes your interval of X so X belongs to this interval Varun has a question self of the wavy curve science scheme is it okay if we start assigning the science with an interval to the left of the number line see you can do it not an issue okay you can start doing it any questions alright so with this we are going to proceed with our next concept so what are the next concept we are now going to talk about domain and range let me write in yellow domain and range off this type of function so what are the type here the type is one upon quadratic I promise you to give a question which came in one of the comparative exams wait wait wait sorry before starting with this I'll let me go back to the previous slide okay a very simple question but many people get this wrong the question is the question is if a function is X minus 4 by 4 minus X okay find its domain so find the domain of the function and find the range of the function give a response on the chat box and give it at one go okay don't give a domain and then put an enter and then range then put an enter give it in one shot in one shot a varun because before you type range somebody else would have typed something I mean okay don't speak out the answer first of all write down the answer in the chat very good very good nice okay and then I'm not seeing right or wrong to answer let's let's see okay as in as in always remember domain and range are sets domain is not a number or set of not then there's not they're not numbers written like series you have to always treat them as a set yes domain range they are sets at the end of the day okay guys so first of all the confusion and the mistake which people do is they'll say oh the X minus 4 is negative of 4 minus X so the function is minus 1 right so is this a constant function and if this is a constant function my domain should be all real numbers do you think this is correct no this is not correct now what is the mistake the mistake is you whenever you're simplifying it it's fine but when you're simplifying it simplify it under the assumption that X cannot be 4 then only it is minus 1 because if X is 4 you can't cancel 0 by 0 and say it is 1 no this is not allowed in mathematics correct so this is a mistake which people make they think that if you cancel it off it becomes a minus 1 minus 1 is a constant function constant function domain is all real numbers no idea this is not going to work this is going to give you the wrong results okay so most of you have done this correctly so it is all real numbers but for 4 you can't put a 4 clear because 4 is going to make the denominator become 0 that is not allowed correct see same idea which we had discussed in the beginning any Px by Qx rational function you have to write down r minus those values of X for which the denominator is becoming 0 so ideally by that formula the answer is this correct but yes if r is not if your domain is not 4 let's say it's any real number then of course it stands cancel and gives you 1 minus 1 only so your range is a singleton set minus 1 okay what do you arrange range is a singleton set which only contains minus 1 is it clear see many of you have got this wrong so the range adding machine see if X is not equal to 4 because you're not inputting 4 inside the function then it can be cancelled no and you can get this minus 1 so your output is always a minus 1 correct so your range is only 1 value which is minus 1 always so if you put a 100 you'll get a minus 1 60 you'll get a minus 1 0.3465 you'll get a minus 1 right are you getting what I'm trying to say any question yeah so you put any value of X and then 4 and then you'll get minus 1 minus 1 yes correct is this okay all right so what we are going to start today yes but for any question we're not allowed to put in a random value and then check so will the result always be the same range for or is this just a special case no I mean random value and check is just for your you know satisfaction but random value checking will not give you the range you may miss out some values how many inputs you will put all real numbers may infinitely many quantities come how many values will you put in check right so it is just a I mean I was just showing it for this question that will always give you a minus 1 okay okay Aditi no issues all right so how do I deal with this kind of question maybe to start our discussion today I'll be taking a simple example to make it clear okay let's take an example to understand how to deal with domain and range of this kind of a question to make things very simple for all of you I'll be taking a function like this 1 by x square minus 1 okay I've purposely kept the quadratic to be simple so that you know everybody understands it so first step how do I find the domain let me put the heading here I am now finding the domain of the function so remember the domain approach I'll write down the formula over here that we had discussed in the class so formula for domain was what formula for domain was domain of this kind of a function where these two are polynomials what was the formula we had discussed the domain formula is r minus x such that q of x is equal to zero that means you remove those values for which your denominator becomes zero clear so now tell me in this case what values of x will make the denominator become zero okay very good so here what I'll say domain is r minus such x is such that x square minus one is equal to zero correct so what are the values for which x square minus one becomes zero you are rightly said it becomes zero when x square is equal to one which means x is plus minus one so you will say r minus a set containing one minus one clear everyone any question with respect to the so we always subtract the real like the number which makes it makes the denominators you know it is not actually subtracting my dear it's an operation on sets you know operation of difference of sets it's like excluding it you're saying it will be numbers in r excluding these two values correct any doubt related to the domain part any question related to the domain anyone okay can you move out for finding the range okay pay attention as I told you in the class you have to find the domain before you start finding the range there is no existence of output without input I hope you remember this famous statement of mine which I keep giving in the class there is no existence of output without knowing the input so these are your inputs my dear right and I already know what inputs I am making to the function right now we are geared up to find the outputs coming because of these inputs okay now please pay attention the process remains the same more or less of course here and there a few minor changes will happen so you have to listen carefully to what I am saying so first write y equal to this because your output is plotted on the y axis hence we write it as y this is just for convenience you can keep using f of x also but that would be difficult you know expression to manage so write a y y is simple single alphabet you can easily manage it now what you do is you make x square here the subject of the formula ideally you were making x the subject of the formula in this case you can actually make x square the subject of the formula it will not matter matter to us even if you make x the subject of the formula but x square will do the job for us we don't have to go to x okay so what I'll do now is I'll take the denominator and multiply it to the left side expand it so x square y is equal to 1 plus y so x square will be what 1 plus y divided by y is it clear any question okay now a small question I would like to ask from you since your x belongs to all real numbers except 1 and minus 1 who will tell me x square belongs to this set or x square belongs to which interval write your answer on okay Arul are you sure I know that are you sure sir I think so I think so okay okay okay see guys once again I have a very basic question to ask you those who are saying all real numbers except one let's say can I say x square can be minus 100 think again real number square can it be minus 100 or for that matter any negative number that means your answer is not right no no negative number don't go to natural numbers now because same real number natural number is a you know it's it's a restriction of your answer if you square a real number you can get a non-natural numbers also know let's say I take x as 1 by 2 right 1 by 2 is a real number and it is not 1 or minus 1 so square of it will be 1 by 4 Aditi you are you are very close to the answer right it will be a number which is between 0 to infinity and not 1 am I right no no no don't go to whole numbers again you're doing the same mistake Arul let's say I take an example of x as 1 by 2 which I already took square of it will be 1 by 4 it's not a whole number why you're saying all whole numbers so it will be a quantity which is between 0 to infinity 0 inclusive but cannot be 1 why it cannot be 1 because to get x square as a 1 your x should be plus or minus 1 which you are not allowing see 1 and minus 1 we're not allowed in the inputs and now Smaran saying positive also will not be correct and positive rational numbers will also not be correct you are putting lot of restriction on your answer it is just a non-negative real number except 1 over right does it make sense see is everybody happy with this part then only I'll move forward sir can you hear me I can yeah so I have a doubt please ask what basically what is the root of a negative number non-real quantity it's a non-real number we are not going to talk about that non-real quantity will come out when you take a root of a negative number does it answer that question of yours may hul any any question understood understood no yeah are you all convinced with this first of all say yes then only I'll move forward but why can't x square be negative square of any real number cannot be negative sir but they're taking x square as a whole no so if it's minus 1 then y is equal to you take any okay Anirudh you take any real number for me except 1 and minus 1 any real number which you think whose square will give you negative let's say minus 10 minus 10 square will you about 100 no this is greater than 0 even if you take 0 0 square will be equal to 0 right so it can't be negative it can be 0 at the least got this point okay square of any real quantity will always be non negative some people wrongly use positive right it's non negative you should say that means 0 is also allowed okay what is the difference between this answer in the range and the domain sir I have not found out the answer yet okay range I am still finding out did I say this is the answer for the range I'm just in one of my intermediate steps okay have patience I'll I'll complete this question okay now see now why am I so interested in x square all of a sudden because you can see this expression is where you have equated this y terms to x square see this part that means this fellow that we are talking about this fellow is actually what this fellow is 1 plus y by y so from here I will end up getting some idea in which my in which interval my y lies okay so this will help me this step will help me to get my interval of y okay see how pay attention okay just paying attention everything will be resolved now this says that this has to be greater than equal to 0 but not 1 now remember in the class I told you in one of the types that let's try to put this as 1 and see what values of y make it 1 we will remove those values right you remember that approach everyone so let's say I am trying to see what values of y will make 1 plus y divided by y equal to 1 so that I can remove those values of y from my range but you will get a surprise when you do that so let's say this is my rough work column if I if I start putting 1 plus y by y equal to 1 you'll end up getting see you'll end up getting a shock the shock is 1 is equal to 0 which you know is not possible that means this guy's anyways cannot be 1 so let's not worry about it being 1 let's only worry about it being in this interval that means 1 plus y by y should be greater than equal to 0 yes or no so 0 to infinity means greater than equal to 0 ultimately correct what has this brought you to this has brought you to your inequality that is why inequality concept was taught to you a little while ago so this brings me to again my wavy curve sign scheme that is going to help me to get the answer so what I'll do now I will make a wavy curve sign scheme for this expression so all of you focus on this expression tell me what are the zeros of these factors so what is the zero of this guy minus one what is the zero of this fellow zero correct yes or no same thing I mean earlier we used to deal with in terms which were in x now we are dealing with the term which is in y but idea doesn't change just changing the name of the variable doesn't change the concept behind it okay right so tell me what sign should I put on the right most interval what sign should come here who will tell me write down on the chat box plus or minus plus very good what sign should come what sign should come in the left to it minus yes again right most sorry left most plus because they are all having odd powers odd powers means constant switching will happen clear any question any question anybody okay now what is the questions that are asking us this is with special attention to Advika so Advika we put the sign without looking at this okay we are we are not considering this for the time being we just put sign depending upon the function now we are looking at it now we are looking at it yes so now we try to figure out where is it greater than or equal to zero so greater than zero is minus infinity to minus one don't put any brackets okay put a union okay now put the brackets so minus infinity will never be included infinity will also never be included what about minus one yes minus one can be included because it will make the expression zero and zero is allowed as per the inequality zero cannot be included because zero will make the denominator zero correct so this is your interval of y and now my dear friend mehul we are there with the answer range is minus infinity to minus one union open interval zero to infinity this is your answer to the question clear everyone so the approach is slightly different from whatever we had done in our previous types right so this is how range becomes slightly unpredictable it is not predictable like how you have for the domain domain there is a ready made fixed formula to take care of but range becomes slightly predictable and you have to keep such things in your mind while solving the question sir can you zoom out so that i can my dear i cannot zoom out in this i can just drag this screen left and right just tell me where to go is this just looking through it no i'll just want a domain okay you want to me to start with domain yes yes harsha i'll give you another example don't worry i'll not be happy till you give me an answer because this problem i have solved it no only you should solve a problem and give me the answer see and one more thing normally when i take online sessions i share the pdf of the class this entire notes will go to you in the group so if you're missed out on anything don't panic don't ask me to go to the particular sheet and left right you can easily open the pdf after the class and copy complete your notes wherever you are missed out okay sir then we can why not why not let us say if i put a zero as x value won't my answer be a minus one nickel so what is that logic that since we cannot put minus one i cannot get a minus one there's no such logic like that because function processes that it processes as per the functionality definition and gives you an output right so you you can get any answer which is in the range here why do you subtract the zero from the real numbers in the domain where where where in the domain no i didn't subtract zero why will i subtract a zero like why do you exclude from the real numbers which one said in the domain why do you exclude the zero like one comma minus one from the real number because at one this will become zero no at minus one also this will become zero ah it should not be zero no so whatever is making that fellow zero you have to exclude it from your input that means your input basket cannot contain those numbers plain and simple can you show the inequality yeah yeah sure mia don't worry just tell me where to drag so i'll drag the screen will you give another example yes yes please i'll do that can you explain how you did that x square thingy for the spot where you put that smiley face below that area this one so see Laksh since x square is equal to this saying x square and saying one plus y by y is the same meaning no so if this belongs to this interval even this guy will also belong to the same intro isn't it like like how we would do the range in the previous functions didn't we make x the subject of the formula and whatever was equated to x then we put that in place of x and solve for y the same approach here as the slight extra thing which i'm doing is i'm not dealing with x per se i'm dealing with x square here yes sir yes so those who are copying it i hope you're done with the copying part shall we take another problem may i move to the next slide please yes okay yes sir don't worry as far as i i could have given you but in that case that the concept will become slightly more complicated okay so let me first cover up things which are easy okay definitely we'll come to those kind of problems also don't worry about it okay let's take another question let's say i have a function uh four upon nine minus x square for this function find the domain of the function find the range of the function and give your response together for domain and range on the chat box very good mea smaran you're you're writing the answer in an incorrect way so you're basically writing the answer which is actually not the answer rest everything is the answer see be careful because it'll become a habit for you to write it like that okay okay i understood that see that that's what i said i know you know the answer but if you write it like this it'll become a habit and then you'll purposely make a mistake domain i don't think so anybody has an issue because i can see all the right answers coming for the domain nice aria i didn't get that what is that nine comma four her mayor check your answer once again okay varun aditi range will take a little bit of time but domain is straightforward not even five seconds you'll take to answer domain okay sam so i think varun has only answered with the range part i guess the rest of you are still doing it let me give you two minutes for it do it carefully because this is going to be tested in your school as well okay harsha domain is root five what laksh why okay mea we'll check we'll check i myself don't remember the answer for these so we'll check it out so keep your answer ready with you then i'll go back and check who all have answered correctly sharan give me the answer okay shall be shall we discuss everybody's ready anybody who thinks that i want 30 seconds more 20 seconds more 10 seconds more i can oblige okay good good yes sir you can shall be okay thanks okay arul we'll check your answers the first one is domain so let's talk about domain domain is plain and simple for domain we already know that there's a well defined formula all real numbers except those values of x which make the denominator become zero right so let's go to our rough work side and let's say i take the rough work here okay where do you think nine minus x square becomes a zero that means when do you think x square becomes a nine when x is plus minus three correct so your answer would be all real numbers except plus and minus three sets so this is a pair set pair set means the set containing only two elements okay and don't put a round bracket and a square bracket here it's a set set means curry braces should be there so check your answer everyone i'm sure everyone has got this response hermare understood where you went wrong next for finding out the range we'll take a similar approach what we do for the previous question let's first write y equal to four by nine minus x square let's make x square the subject of the formula so we'll have nine y minus x square y equal to four adhika what was your answer i didn't see your response not range okay domain did you get it correctly okay sam we'll check all right this is what you're getting for x square right now same question to you all same question since x lies in this interval which interval will x square lie right zero to infinity excluding nine am i right anybody has still any doubt related to this transition since x lies in all real numbers except three and minus three the square of it can take all non-negative numbers except nine clear okay now since x square is this expression which i'm showing with a circle on your screen that means i'm trying to say nine y minus four by y belongs to this interval okay now again let's go to our rough work section and try to check when does nine y minus four divided by nine actually become a nine let me write my four properly it is appearing like a nine no let me write a four so let me go to my rough work column and i'll i'll try to check when does nine y minus four by y actually become a nine and you will realize that it actually never does because if it does then minus four become equal to zero which you know is not possible okay so this is the equation way of communicating to you equation aapse baat karthai right so the equation is communicating to you that c was your nine y minus four by y can never become a nine if you try to do it you will get a shock okay yes somebody said a serene between uh yes sir in exam do we have to show and prove that it cannot be not really but uh yeah i mean this is for your own satisfaction okay sometimes yes you may realize that there are some values coming up but maybe not in this case so better not to take a chance on erud right it hardly takes a fraction of a second to verify that okay rather taking a chance now see that means i only only have to ensure that this is between zero to infinity including zero that mean this should be greater than equal to zero in short i am now solving a wavy curve sine scheme question sir it's nine y minus four by nine y right sorry by y right sorry sorry sorry sorry slip of pen okay is it okay any questions i'm happy 31 eyes are watching whatever i am doing so you minus three how does that become nine is it just the x square volume sorry the value of x square not only you're putting like minus nine uh you x square you put three minus three to x square the value will become nine so you're canceling that oh huh that's what if you square this up this will become nine no and i don't want this to be in your x square values that means i don't want nine to be taking uh been taken up by this fellow okay okay so ultimately if nine is not taken then i'm i don't have to worry about it i just have to worry about this fellow being in this interval okay so if i solve my wavy curve the approach is very simple make a number line make the zeros of this factor the zero of this will be zero the zero of the numerator will be four by nine please put the sign take any number greater than four by nine twelve okay so nine into twelve minus four divided by twelve is positive so put a positive and since every every factor has an odd power keep alternating the sign without any second thought you can save your time okay now the question sitter is asking you where is it greater than or equal to zero that means wherever you have written a positive please state that interval let me write it below here so it is going to be minus infinity to zero and four by nine to infinity infinities and minus infinities will always be round basis and zero can never be included here because zero will make denominator become zero which is never going to be entertained four by nine i can put because four by nine makes everything become zero and zero is acceptable acceptable because it is greater than equal to zero see here hence four by nine will be included so put a union here so this greater than equal to zero it was not given in the question was it given given in the question yeah like that greater than equal to the question just said find the range this is something which i am inferring in between so well so how can you say it can be less than zero also right oh for me will they call me why did you put a close interval and zero hold on a second i'm explaining no are you convinced with this step with this step if not then tell me so could you explain that step again see if any real number is squared and that real number is not three or minus three can i say the answer will always be a quantity greater than equal to zero and not nine is this creating an issue in understanding i understood clear no any question okay now since this is greater than equal to zero may well greater than equal to zero will come no yeah correct no why didn't i worry about nine being excluded because greater than equal to zero will also contain nine right why didn't i worry about nine being excluded because i already checked that it cannot take nine understood what i'm trying to say yeah it is possible sometimes that it can be equal it is possible it is possible we'll see in future if some problem comes like that after that is it you know clear to everyone is it clear to everyone so this is the interval of why so your range is going to be minus infinity to zero union four to nine so those who had questions right now is it all answered have i missed out answering anybody's question no sir sir i had asked you one question yes sir tell me sir a quadratic in quadratic functions when you do the wave curve will all of them in all of them will you get a greater than equal to zero can you get even a less than equal to zero okay see it depends upon question like later on we'll do something related to irrational functions there we have to you know consider a few things where you may get greater than zero also so i will not be generalizing things like that but yes to answer your question there may be problems which will be taking up where only greater than zero will be required okay i'll take up those questions not to worry about it sir could you give an example where there's a value for why value for why as in nickel i didn't get your question these are all values of why you know i meant like when we equating it right like let's say we got we got the the x square in terms of y and we're equating that to the value of x which gets the entire function is you get the denominator of the function zero so could you give an example where you get a value of y rather than getting a not possible case oh you're saying if i equate it to this number i should get a value of y rather than impossible yes maybe i'll have to think of that separately it's not coming at the top of my mind maybe while solving few questions who knows i may get a scenario like that okay let's let's stay tuned i mean as of not till now it has not come but there may be cases where it may come so we'll see we'll see when it happens okay meanwhile is it possible sir is it possible they'll give us the range and ask us to find the function no no that is not possible because there can be so many functions having the same range okay let's now take a situation where i have a quadratic where x term is also present i think some of you were demanding for this question let's take this question one by x square minus three x plus two would you like to attempt this question or do you want me to help you with this can we attend it once okay can we attempt and then please please please please go ahead your attempting is more important oh we have to find domain right yes sorry let me complete the question you have to find the domain and range yeah that's correct me yeah okay harsha nirudh har maher all right what i can sense is many of you are stuck at the domain not knowing how to proceed further am i right anybody who's getting okay sure sure me i'll wait i'll wait let's not give up the fight come on everyone let's do this failing is not a problem giving up is a problem okay so let's not be scared of failing let's only be scared of giving up let's not give up okay aditi pranav smaran hayush okay okay let's discuss so for the domain part okay one second i can wait for one second let me at least i domain no i don't think so there's such a pattern there some i don't think so there's a pattern existing so before concluding on a pattern we should at least write out for five six problems to see i don't think so there's a pattern like that all right we'll check we'll check okay mea mea has finally given one answer see domain the concept is same the domain is going to be all real numbers except such x is which makes the denominator become zero now again let's go to the rough work area as a work area so what makes this fellow zero okay so this is easily factorizable i hope all of you know how to factorize quadratic so x becomes one or two so here the answer would be all real numbers except one comma two this is your answer for this question no issues let me box it okay mea we'll check we'll check we'll check the answer next range range is a different ball game so you have to be careful while as you know finding the range let's start with the same process the processes we start calling it as y okay now we'll try to make x the subject of the formula now here making x the subject of the formula is is not as easy as what we had in our previous types okay so for that you have to pay attention take the denominator to the other side okay take the one also to this side let me write it like this minus 3 y x 2 y minus 1 okay now why have I written it like this is because I want you to realize that I have actually written a quadratic expression in x what is this read this as if it is a quadratic in x okay so we all know how to solve quadratic equations by using the quadratic equation formula so here this is your a this gentleman is your a this is your b this is your c so it's like ax square plus bx plus c what is the quadratic equation formula what are the quadratic equation formula minus b plus minus under root b square minus 4 ac by 2 a right so let's do that x is equal to let me write minus b will be 3 y plus minus under root of b square which is minus 3 y square which is 9 y square minus 4 ac am I right whole divided by 2 a any question any concerns in this anybody has up till now up till now any concern anybody has please please please highlight no questions okay now pay attention everyone now pay attention since x belongs to all real numbers except which two values one and two am I right x cannot be one and two right what does it mean it means this fellow should be real and should not be one or should not be two okay by the way I'll just try to simplify this little bit more so that you know you can easily work with this so if you see the expansion of this here will give you you 9 y square minus 8 y square which is y square okay plus 4 y am I correct if I just expand it won't I get y square plus 4 y okay so this should be real number except one and two am I right correct now what are the things which we need to take care for this to be a real number except one and two what is the number one thing you have to take care of who will tell me that shouldn't be equal to one okay first thing is this fellow should not be equal to one and two so that expression under the root shouldn't be negative this should not be one this should not be two okay let's let's take that into our account okay second thing yes somebody was saying the expression within the under root should not be negative means this should be greater than equal to zero why is this restriction placed because if this restriction is not placed what will happen under root of a negative quantity will yield a non-real results but you want your answer to be some real number isn't it and one more thing we are missing who will tell me that why can't be zero right why can't be zero sorry for writing down because I have almost reached the end of the sheet okay so these three conditions in fact I can say these four conditions because in one they are two conditions we have to simultaneously take care of okay any question anybody has here if you have any questions till this point feel free to highlight so not equal to do right which one so the second one on the right it's not equal to not equal to do yeah not okay is it fine any question anyone okay pay attention now let's try to figure out whether these guys can be one or not or can be two or not so let's start let's go to our rough work area once again so what if you put this expression to be equal to one it means you're trying to say three y plus minus under root y square plus four y is equal to two y that means you're trying to say plus minus under root y square plus four y is equal to minus y if you square it you will again get something like this four y equal to zero that means y should be equal to zero but you know this is not allowed by my third expression the third expression the third restriction which you're putting here see here the last restriction why should not be zero correct so if you want to make this zero why should be zero which is not allowed this is not possible not allowed correct so this anyways can't become one this anyways can't become one and same will happen or similar thing will happen if you try to make this as a two as well let's do that quickly also meanwhile can i just write less and just write it as three y plus minus this is equal to four y that means again this is going to be a y i'm just saving my space here square it that again will become four y is equal to zero which is not possible again this is also not possible so let's not worry about the first restriction because this expression can anyway not become a one and two so this restriction is actually no restriction for us okay let's now try to go to the second restriction second restriction is actually a wavy curve so here if i say i factorize it like this i can solve it by wavy curve sign scheme so for wavy curve sign scheme i need a number line i need to show the zeros of these factors is it fine zero of these factors are zero and minus four respectively i hope that is clear okay what sign should i put in the right most interval write it on the chat box quick quick quick quick positive yes yes character then negative then positive right now you're trying to find out where is this greater than equal to zero that means wherever you have written a positive sign which is minus infinity to minus four remember minus four can be included union zero to infinity and zero also can be included okay but the third condition says zero can't be included right so what i have to do is i have to keep all these three restrictions in my mind means i have to take an overlapping scenario of all the three conditions so first one says i have no issues with any value of y any value of y you put i'm fine with the first restriction second restriction says i want want value of y only in this value minus infinity to minus four inclusive of minus four union zero to infinity inclusive of zero but the third condition says i don't want it to become zero okay so tell me which will be the overlapping condition here uh shall i erase this part because this part is eating up my space yeah so what is the overlap of the first and the second and the third scenario won't you say that the overlap is going to be y belonging to minus infinity to minus four union open interval zero to infinity this becomes my dear the range of the function can i see who all got this right because you really need a round of applause because if you have solved this problem now means you have done a great job wow mia do you i mean your answer seems to be very very close to it but is is it the zero included for you if yes then you are absolutely right the rest i am not able to see anybody close to the answer so this really needs an applause good mia nice so if you don't mind could you show us one more something i guess okay i'll send it for assignment okay three times more of that time because see i didn't have to make a progress you have a test also coming up on 18th of june so do you always need to check the first restriction is this zero included or not uh sorry this is the zero included or not in the range no see the final answer this is the final answer the one which i have box zero is not included oh yeah yes uh mia to answer your question do we always have to check the restriction um see uh if you don't if you want don't want to take chances you should check right because maybe in future you may face some some issues because see i have not solved infinitely many problems to tell you that a problem will arise okay but from my logic i would always check yes you may say sir so far it has not happened so in future also it may not happen but that's just a conjecture sir yes sir so i have a doubt yes yes please ask really go back to that three years later okay okay yeah right so uh over here it says uh the answer says negative infinity comma minus four june zero comma infinity but then uh we know it can't be negative say it can't be one or two so like doesn't that come out domain can't be one and two a range can be who's stopping range from becoming one okay so then what do you call that three y plus or minus what do you call that what's the name for that this yeah yeah this is your x value from this x value you're trying to get the y values oh like domain is helping you to get the range hence domain is always needed okay yes got the point okay dear all so i'm very happy that some of you are getting these difficult problems also right at this stage but by the way for school exams they will not ask you range for these kind of functions because they take a lot of time okay so in school i can assure you that mostly the questions will be around domain finding only which you know you are good at okay all right shall we move on to the next type the next type that i'm going to address very quickly is where you have a linear by quadratic blocking my screen once again yeah linear upon quadratic so let me take a quadratic as px square plus qx plus r okay now again i'll just take a problem to explain this idea okay and the process is very similar to whatever we have done so far okay nothing very difficult nothing very out of the box we are going to do here okay let's take an example question let's take this as an example okay so this is a linear term a very simple linear term i have taken because this is just an example to understand the idea and a very simple quadratic also i have taken okay now let us find the domain and the range of this function okay so first find the domain and range would you like to try because the idea is more or less the same would you like to try or do you want me to do it you tell me we will try okay good that's the spirit like see carefully my dear very good samyukta that's right no domain is a common value my dear you don't find domain and range separately domain is for the complete function what you can put inside the complete function because what x you are going to put up you are going to same put the same x down also no these two x's are not different x's oh akshita has responded akshita where were you just joking okay all right i can see a lot of answers coming from the domain let me wait for the range good good good varun good nickhil arjun please check your answer once i don't quite i didn't quite get that why you are removing that interval and why you're putting curly braces there my dear see if you put curly braces around infinity means you are you know where is infinity nobody knows where is infinity okay promote see whatever you feel that you are not going to put inside the domain try putting that value once for example somebody has said all real numbers except zero why can't i put zero in place of x i'm getting zero for that so output is zero is fine zero is a real number what is what is the issue with that i can't put such a number for which the function becomes undefined right so zero is not a threat to the function zero the function can easily eat or feed on 18th june paper is very easy i don't want to impose my expectation what i'm expecting most of you to get above 80 85 out of 100 in maths okay it's a set solution mostly a bit of functions so very easy paper again arjun you're not very sure about the bracket see you try to you're trying to say everything between zero to infinity correct zero inclusive or exclusive so put round or square bracket around zero getting my point it cannot be a half a set huh put that bracket all right let's discuss the domain everyone see domain will be just domain mia if i may okay thank you so domain will be all real numbers except those values of x for which the denominator becomes zero but mind you all my dear students x square plus one when x is real can never become a zero that means in reality the answer for this is a null set so r minus null set is r only yes there's no real values that you have to remove okay so your answer will be r only so domain of the function will be all real numbers okay any question here okay aruhi very good mia also okay included means both are included okay okay mia anybody else so aruhi and mia have given me the answer for range as well shall i discuss the range or somebody needs more time so just one moment so okay arul there's a small mistake you have done arul but that mistake i'm sure many people will make the thing that you are excluding do you really think it should be excluded this is to arul only yeah i mean you're trying to say that your output can never be that number right but i can see it can become that number for a particular value of x yeah this this problem solving has a lot of surprises okay let's discuss it so range so first we are ready with the domain so not an issue next range the approach is going to be the same approach which we had taken for the previous question call it as y take the denominator to the other side get a quadratic in x once again okay get a quadratic in x once again so what do you have got here you have again got a quadratic in x right okay now thanks to the shidh racharya formula sorry x will be minus b now remember b is minus one here so minus b will be plus one plus minus under root of b square minus four ac b square minus four ac by two a okay anybody having any issue with the quadratic equation formula no issues let me further simplify it okay now pay attention everyone okay arul please carry on please carry on recording is anyway has been done okay sanjay adity will see will see the answers now see everybody pay attention you want this guy to be real your x is all real numbers right because we just found out that domain is all real numbers which means that even this fellow which is actually your x this should also be real correct so tell me what all things you must keep in your mind if you want to keep it real what all you know restrictions comes into your mind so first you can't be equal to zero first first restriction what what what what y cannot be equal to zero right yes okay now this is something which i am asking everybody to check out check is this actually happening so what is the problem with y not equal to zero can't y become zero look at this expression if i put my x as zero y also become zero so is there any problem with y becoming zero this is a surprise to many of you know so this restriction is actually overruled why because why can be zero if it wants for x equal to zero y is zero right then then if y is zero then as for this x becomes not different wait wait now many of you would be wondering sir how did this happen because no ideally my expression should be undefined because of that that is what is the question of one of you now see here there is a special term which i had used in our limits chapter when i was doing in the bridge course some of you may have missed it but there is something called indeterminacy remember indeterminate form indeterminate form indeterminate forms can have a finite value also that is what limit actually did limit helped us to find a answer to an indeterminate form actually if you put y as zero you will realize that on the numerator you will get one plus minus under root one divided by zero and if you take a minus sign it will actually become a zero by zero case that's actually an indeterminate form also and therefore in such cases i always advise the students to do a quick check that what you are saying cannot happen just equate it to that given function and see whether are you getting any real x for which it is becoming that value so in this case if you are saying that y cannot be zero by looking at this expression it deserves a check right it deserves a check from each one of us okay so when you do here the check that means if you try putting this as zero you automatically end up getting x as zero which is a real number and you are allowing all real numbers to be put inside the function no so i cannot my dear exclude zero sorry so this this check is good for you know this is good for checking but i will not entertain this restriction got the point so arul that zero that you excluded and sanjay that can actually be included all right let's move on the second restriction that i have to impose is that one minus four y square this should be greater than equal to zero because i don't want this to be negative because being negative here will lead to non real results correct now this is again a wavy curve how it is a wavy curve very simple first of all let me multiply with a minus sign on both sides so multiplying with a minus sign we all know will flip the inequality correct everybody knows this we had discussed it in our offline session now if you factorize this you get two y minus one two y plus one make a number line correct put the zeros of these factors zero of this will be minus half and half put the sign in these intervals so if i take any number more than half let's say i take a one so this entire expression becomes three which is positive number don't have to calculate the value but is sufficient to know that it is positive and since each of these factors have odd powers you can keep switching the sign correct now the question sitter is asking us when is it less than or equal to zero so in which interval do you see a negative sign that interval you write first and i think half and minus half can also be included because they're asking less than equal to zero so there you go range of the function is close interval minus half to half okay let me show you the graph also so that you are convinced okay can you see the graph of this i'm sorry no because when i multiplied with a minus sign i have to flip the inequality so the answer is always the inequality of the discriminant which we get again let's not generalize like that some values here and there can be excluded also like last time you saw that why why was the denominator value was excluded in this yes yes yes yes see avoid making any generalization till you are very sure about it okay i would always believe that seeing a graph by the way this is one of your seniors question moving it so y equal to x divided by one plus x square or x square plus one okay so look at this function yeah this is the graph of the function however you don't have to know the graph how to make the graph you can see that this function is between half to half so i'm just plotting y equal to half see top most is half bottom most that means it'll just grace past the bottom most part yeah see it is restricted between minus half to half and this can very well become a zero see here it is becoming zero here when it is passing through zero so it is between minus half to half this is your range of the function domain is it can go anywhere from minus infinity to plus infinity so can you explain again why it can't be zero i don't understand why can't be zero no i'm saying it is zero it can be zero i'm saying it is zero why can't be zero is removed means why can be zero i'm saying excuse me sir wait one second the person who asks this question i think i'm saying why can be zero my dear i'm not saying why can't be zero you're saying why can't it be zero okay if you're asking that to answer that if i put x value as zero what will be the y value here tell me let's say there's an input x let me make it like a machine yeah if i put a zero here in the input what will be the output zero so i am getting my y value as zero when the input is zero so that is fine zero is acceptable as my output okay so okay this is a false positive i call this as a false positive what is false positive this is a commonly heard term in false positive in my mistake it has come in my answer but it doesn't hurt us false negative is a problem false positive is never a problem like i'm giving you a simple example let's say you know if i go for a covid test and the test somehow shows i'm having covid it's a false positive even though i don't have covid it's showing me i'm i'm having covid so maybe i'll be more you know careful i'll go for another test or maybe i'll consult a doctor that hey i really have a covid but if somebody says hey you don't have a covid and you you were having a covid then that is a more problem because then you'll be careless that okay i'm not suffering and you'll keep spreading it and i'm not creating a lot of issues right oh my god nikhil has a question sir but when you plug in a number let's say five into the function then we get the output five by 26 which isn't half so shouldn't the range be only no five by five by 26 is a number between minus half to half no this is a range of values it is saying your answer will be in that interval five by 25 26 is very much between half to half yes i'm not saying it is these two values you people are still not careful with the bracket see let me for one final time clarify this see if somebody writes x belongs to this x belongs to this x belongs to this and x belongs to this let's say i take one of the cases what is the difference between these three expression this means x can only take two values a or b nothing else so those who are using curly braces incorrectly for them this is an alarm if you put like this you know how will the teacher read this the teacher will read that x can take a or x can take b nothing else what the point if you write like this then the teacher will think as if a and b are two real numbers and you are writing all real numbers between a and b a and b excluded correct if you write like this then how will the teacher infer it the teacher will infer it as if you are writing all numbers between a and b all real numbers between a and b including a and b correct and if you are writing it like this how will the teacher infer it the teacher will infer it as a to b everything between a and b oh sorry everything between a and b except a got the point so please kindly know how to read those intervals which we have already done in our set theory chapter so i didn't understand why you multiplied the inequality why i could have done without it also varun not a not a important act here i could have done it without that also okay so let's let's do it by varun's way so varun is saying why you have to multiply with the minus one couldn't you solve the inequality without that yes i can solve it see varun and this is i think taran jeet's question also he sir why did you multiply with the minus maybe yes taran okay now see even if you had not multiplied with minus one you can still solve the question how this time again you'll make a number line again minus half and half will be your intervals now take a value which is more than half you realize that this will be negative now correct correct this will be positive this will be negative sign has changed now but it will not change the solution you know why because now this time you're trying to find out when is it greater than equal to zero that means wherever you have written a positive sign so again it will become minus half to half so answer doesn't change answer remains the same varun okay sir clear any question but you can solve without using the you know i mean the number line how so can you move to the other one minus and now you'll say i'll do this sir over square like this then what do you do after this minus half no no no no no please don't do all those things you are not solving an equation my dear all those things are okay if you are solving an equation in equation they need a little bit of more analysis okay please do not try to do all those things this is what i was telling in the class also when people solve inequality they are still in equation solving mode no you are not solving an equation you're solving an in equation so please follow the rules and regulations of in equations equation and in equation there is a bit of difference you can't apply equation methodologies for in equations okay so with this i'll stop for today's session thank you for joining in because this could have not been possible in a maybe a you know classroom environment so targeted you know doubts coming from each one of you thanks a lot we'll meet again on friday 16th of june in the school premise bye bye take care good night and stay safe thank you sir nice sir thank you bye thank you sir see you sir bye sir bye bye bye bye bye bye bye bye thank you sir thank you