Let's move on to agenda revision, submission of documents or motions to direct the city manager to add agenda items to future agendas. Council Member Christensen. I would like to move. Since we didn't discuss it last time. I would like to move that we amend the vehicle parking regulations to create to break out unregistered vehicles from junked and abandoned. Junked and abandoned vehicles because unregistered vehicles are not junked and abandoned vehicles. Do I have a second? I will second that. All in favor of putting this on the agenda. Future agenda say aye. Aye. So raise your hand if it was an aye. All right. No, no. All right. It passes four to three with Council Member Christensen, Council Member Ferring, Council Member Rodriguez and Council Member Peck for and the three of us against. All right. Thank you very much. Okay. Let's move on to city managers COVID-19 report. All right. Great. Thank you for having me. Longmont City Council and Harold. You can go to the next slide. I just wanted to start off the presentation by giving an update on the CDPG dial and mitigation metrics. So just for everyone's reference, I wanted to share the existing CDPG dial levels. So these three criteria. So our two week cumulative incidence rate for COVID-19 is 124.1 cases per 100,000 and that really puts us right in the middle of the new cases, the incidence rate for safer at home level to next. Let's go ahead then and move on to the consent agenda. Don, do you want to go ahead and read us the. Thank you. The riot act. It's a long one, Mayor. 9A is ordinance 2020-44 a bill for an ordinance making additional appropriations for the expenses and liabilities of the city of Longmont. For the fiscal year beginning January 1, 2020, public hearing and second reading scheduled for October 27, 2020. The ordinance 2020-45 a bill for an ordinance fixing and levying taxes upon the real and personal property within the city of Longmont for the year 2020 to pay budgeted city expenses for the 2021 fiscal year. Public hearing and second reading scheduled for October 27, 2020. 9C is ordinance 2020-46 a bill for an ordinance fixing and levying taxes upon the real and personal property within the Longmont downtown development district. For the year 2020 to pay budgeted expenses of the Longmont downtown development authority for the 2021 fiscal year public hearing and second reading scheduled for October 27, 2020. 9C is ordinance 2020-47 a bill for an ordinance conditionally approving the vacation of a pedestrian trail easement associated with the 110 Emory minor subdivision plan and site plan, generally located south of second Avenue and east of Emory street. Public hearing and second reading scheduled for October 27, 2020. 9C is ordinance 2020-48 a bill for an ordinance amending section 15.03.80 of the Longmont municipal code on zoning districts measurements and exceptions. Public hearing and second reading scheduled for October 27, 2020. 9F is ordinance 2020-49 a bill for an ordinance approving the concert plan amendment for the bone farm rezoning and the annexation agreement located at 1313 spruce Avenue, public hearing and second reading scheduled for October 27, 2020. 9G is ordinance 2020-50 a bill for an ordinance amending title 10 chapter 10.24 creating new section 097 of the Longmont municipal code creating a temporary prohibition on rental late fees due to COVID-19 related hardship. Public hearing and second reading scheduled for October 27, 2020. 9G is resolution R 2020-93 a resolution of the Longmont city council approving the intergovernmental agreement between the city and the University of Colorado for socio technical design for a middleware information exchange hub. 9I is resolution 2020-101 a resolution of the Longmont city council approving the intergovernmental agreement between the city of Longmont. 9K is resolution 2020-103 a resolution of the Longmont city council approving the Longmont city council approving the intergovernmental agreement between the city and the Colorado historical records advisory board for grant funding for digitizing 9L is resolution 2020-104 a resolution of the Longmont city council approving the business lease between the city and Dennis L Blomker and Carolyn A Blomker for 500 Kimbark Street sweet 101. 9K is resolution 2020-103 a resolution of the Longmont city council approving the intergovernmental agreement between the city and the Colorado historical records advisory board for grant funding for digitizing videotapes. I just like to pull E for a brief comment. All right, great. Let's go ahead then and have a motion for the consent agenda less FLG and E. So moved. Second. All right, it's been moved by Councilor Martin seconded by Councilor Farrell fairing. All in favor say aye. Aye. Aye. All right, the motion passes unanimously. Let's move on to ordinances on second reading D ordinance 2020-41 less likely that someone will call in late ordinance 2020-41 is a bill for an ordinance authorizing the city of Longmont to lease the real property known as Vance brand municipal airport hanger parcel h 74 to best steel LLC can I have a motion for ordinance 2020-41. I'll second it. Oh, Councilor Christensen will second. Okay, great. All right, I'll see no further discussion or debate all in favor of passage of ordinance 2020-41 say aye. Aye. Opposed say nay. All right, 2020-41 passes unanimously ordinance 2020-42 is an ordinance authorizing the city of Longmont to lease the real property known as Vance brand municipal airport hanger parcel h 31. To George and Evelyn Grell. I'll move ordinance 2020-42. Great. It was moved by Dr. Waters seconded by Councilman Martin all in favor say aye. Aye. Aye. Aye. All opposed say nay. All right, the motion carries unanimously. Can we have a motion. So moved. I'll second that Joan. All right, all in favor of passage of ordinance 2020-37 say aye. Aye. Aye. Opposed say nay. All right, that motion passes unanimously. Go ahead ordinance 2020-239 a bill for an ordinance amending chapter 6.70 of the Longmont municipal code to permit medical marijuana delivery. Can we have a motion for ordinance 2020-39 a bill for an ordinance amending chapter 6.70 of the Longmont municipal code to prevent medical marijuana delivery. So moved. I'll second. All right, we'll go ahead. It was moved by Councilor Martin. It was seconded by Mayor Pro Tem Rodriguez. All in favor of ordinance 2020-39 say aye. Aye. Aye. Opposed say nay. All right, the motion Council's council. The motion carries and passes unanimously. All right ordinance 2020-40 a bill for an ordinance amending chapters 2.68 on local licensing authority 6.70 on marijuana stores chapter 9.60 on medical and recreational marijuana. All right, is there a motion or debate or dialogue from Council. I'll move ordinance 2020-40. Second. It's been moved by Dr. Waters and seconded by Mayor Pro Tem Rodriguez. Seeing no further discussion or debate. Let's go ahead and vote. All in favor of ordinance 2020-40 say aye. Aye. Aye. Opposed say nay. All right, that motion carries unanimously. Let's go ahead to last second ordinance is ordinance 2020-43 a bill for an ordinance conditionally approving the vacation of access utility and drainage easements associated with the Highland subdivision generally located north of Highway 119 and west of County Lane Road. Would someone like to offer a motion or offer dialogue or debate on this particular matter? All right, I'll preferably the motion first. I'll move the ordinance 2020-43. All right, the motion has been moved by Dr. Waters and seconded by Councilmember Martin. All right, so all in favor of ordinance 2020-43 say aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye. Aye appreciate them catching both these items. It'll make a big difference in the long run. Thank you. All right, great. Would you like to make a motion, Council Member? I move ordinance 2020-48. I'll second that. All in favor say aye. Aye. Opposed say nay. All right, the motion carries unanimously. Let's go ahead and Marshall, let's do yours now. F, let's go ahead and vote. We have a motion to pass ordinance 2020-49. All in favor say aye. Aye. Opposed say nay. All right, it passes unanimously. And what was your other one, Council Member Martin? It was 9L. Yes, thank you. So let's go ahead and vote. We have a motion on the floor to resolution 2020-104, resolution in support of repealing the Gallagher amendment, also known as amendment B, on the state's ballot. I voted today. So let's go ahead and vote. All in favor say aye. Aye. Opposed say nay. Nay. All right, and they said it in unison, but they both said nay. So it passes 5-2 with Council Member Christensen and PEC against. All right. And then last but not least, I wanted to pull the G, yeah, temporary prohibition on rental late fees due to COVID-19. I'm not going to argue against it at this point, but I'm going to actually move that we table this for two weeks until our next regular council meeting. Second. Second. All right. It's been moved and seconded. We now have a motion on the floor. Let's go ahead and talk about the motion too. We're not talking about the ordinance itself now. We're talking about the motion to push it off two weeks. So we have, so staff can bring back unintended consequences. All in favor say aye. Aye. Aye. I'm okay. Aye. All right. Raise your hand if it was aye. All right. Raise your hand if it was opposed. Nay. All right. So the ordinance passes 6-1 with Council Member Christensen opposed. So yes, it will be back, but in two weeks. Looks like we have another phenomenal 2021 budget presentation. Good evening, Mayor and City Council, Theresa Malloy, Budget Manager. And this evening's budget conversation is final direction from you all and then our second public hearing on the budget. What we are going to ask for direction on one of the items we're going to ask for direction on is the remaining one-time funding that is available that has not been allocated in the budget yet. Comment. So is this the time, Theresa and Jim, that you would like motions on what to do with one-time funding? That is correct. I'm going to make motions one by one on all three. I'm going to move that we approve $50,000 in one-time funding to complete the library feasibility study. Second. Thank you. Anybody? All right, let's just go ahead and vote on it. All in favor say aye. Aye. Aye. Aye. So the motion passes unanimously. All right. The motion would be the budget $50 per kindergartner to follow through on our commitment to the 529 Jump Program. I'll second that. All right. Let's just go ahead and vote. If Mr. Hardcore discussion on this. All in favor say aye. Aye. Opposed say nay. All right. The motion carries unanimously. So I'm going to move that we add, we budget $250,000 and additional $150,000 in one-time funding for childcare and early childhood education or early childhood learning programs in 2021. All in favor say aye. Aye. Aye. Opposed say nay. All right. The motion carries unanimously. Let's go ahead and move on to Mayor and Council Comments. Council Member Christensen and then Council Member Martin. Everybody should have their ballots right now. Please vote. Just vote as soon as you can and vote. It's important. Yes, I'm always frustrated when I, we, we can't engage with people in public invited to be heard that need to be engaged with. And I would like to point out, first of all, that people who don't live in homes that are attached to net next light can still call into council meetings and speak using their cell phone. And there are not that many people who don't have one. So I just want to say, if you feel excluded from, um, from public invited to be heard, you shouldn't be because I think we can find a way to hear from you no matter who you are. All right, great. Dr. Waters. Thanks Mayor Bagley. We got, we received from Michelle a couple of days ago, maybe it was yesterday, um, uh, an update on the status of recruitment in the application process for residents interested in serving on boards and commissions. Uh, the people for whom we did none of that is us in terms of accountability, um, clarity on what our responsibilities are as liaisons and what what what provoked me for this was looking at the at the fact that we're going to recruit again. And then I asked on, do we have, I'd never seen anything in my orientation about any written statement about what what my responsibilities should be as a liaison. I've kind of made that up as I've gone along. Uh, then I asked if we'd ever collected any feedback from boards and commissions on what what council members could do or do do to add value or could do to add value or what do we do that is distracting and that that it would be, um, prudent in my mind to put together some guidance, uh, some agreements on terms of how we fulfill those responsibilities. Um, so as we think about interviewing folks and we and we've taken steps for their accountability. Um, I'm one that thinks we ought to take a step or two in a direction of creating more accountability for us by collecting data, translating those data into some expectations or some agreements for ourselves so that there's some level of I have no idea whether or not I'm doing the job of liaison in ways that are similar to other members of the council. If anybody else cares about that. Um, but I'll think about that as we anticipate interviewing folks who we'd like to serve and who would like to serve. Councilor Peck. Um, thank you. Uh, I am actually going to reach out to the people who are having the Saturday vigils at sixth and main on both sides of the street. Um, it is, it is the right of everybody to protest, but I am reaching out to ask you to respect each other's ideologies. Um, and your right to peacefully protest. And I would like to also, um, remind the police department that as council members, we are neutral. We do not come in wearing anything that addresses our ideologies or our philosophies. We are here to represent everybody and do not push our our ideologies. Um, let's all just live in the city peacefully and respect each other's points of view. Thank you. Great. We have a motion to adjourn. Councilor Peck, do want to make that motion? I move to adjourn. I'll second it. All in favor say aye. Aye. Aye. Aye. Opposed say, airplane. Look at that. Look at that. What solidarity. Offences unanimously. Yeah.


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The Center for Adolescent Renewal Care and Upton Garden Girls Center are receiving support from corporate entity CIBC First Caribbean International Bank Limited. Details in this report. At-risk adolescents are the latest beneficiaries under CIBC First Caribbean's Adopter Cause program in St. Lucia. As the bank continues to adapt its corporate social responsibility efforts to meet the changing circumstances brought on by the COVID-19 pandemic, it has shifted focus from the primary schools to rehabilitative and development institutions on Island. This year represents a departure from the past. We have selected to concentrate our programs on the Center for Adolescent Renewal and Education or Cure and the Upton Garden Girls Center. We're therefore very proud to announce that the bank will be providing tablets for four centers across the island for use by the students for educational purposes. The center requested eight devices, however, we undertook to increase that number by two so as to ensure that each center is covered. The ten devices donated to the Center for Adolescent Renewal Care for its skills development program are valued at $6,000. CARES Executive Director Carleen Mason says this donation will complement the center's blended learning activities amid the pandemic. So many of our trainees are not able to connect to our online platforms because they do not have the devices. So rest assured that these tablets will be greatly used. What we have found is that even though some of the trainees may have internet access because they do not have the correct devices, the instruction is placed and they are unable to get it. But this gift will greatly enhance what the trainees would now be able to do. CIBC First Caribbean also presented a check to the Upton Garden Girls Center for $5,000. On hand to collect the check is Jacqueline Messiah Simian, director at the center. Upton Gardens Girls Center is positioning itself to become a viable second chance for young women in the country. We have introduced our CSEC and CVQ programs effective January 2020. Of course, COVID has brought some interruptions to that. However, this generous donation will go directly to our scholarship fund. That fund covers school uniforms, transportation and in the future it will cover examination fees as we prepare young women for CSEC and CVQ courses. CIBC First Caribbean has pledged continued support to both institutions going forward. We intend to augment this effort with other training delivered by officers of the bank such as mentorship for the care program and for Upton Gardens Center. We have made a job interview, coaching and workplace preparation sessions. We truly hope that in doing so we will make a meaningful contribution to your personal development and these tools will be a catalyst to spur learning and career opportunities as we look forward to hearing the testimonials in the near future.


Today, we are going to talk about relations and functions composition of functions and inverse of a functions Now composition of function is basically everywhere Okay, almost 99. In fact, I claim that almost all the functions are basically composition of two or more functions Okay, now, what are the composition of a function? Let me define it first composition of functions Let me define this in a simple layman pictorial view to you. Let's say there is an input X Okay, let's say there's an input X You put it in a function. Let's say G Let's say the function name is G What do you get as an output? We get G of X Correct, so this is the output that you get when you put an input into a function G, right? When this input is put into another function, let's say a function name is F Okay, then you get something which we call as f of G of X Right, so it's basically a function of a function Right, this is a function of a function Now if you could devise Another machine, let's say G and F were machines So if you put an input X into the machine G, it will give you G of X Let's say if you put the input G of X into another machine Right, that is the output from G into another machine. Let's say F. I will get F of G of X Now let's say I devise another machine which directly gives you this output Then that machine will be called as Fog machine This is the symbol that we represent used to represent F of G of X F of F of G of X. This is how we pronounce it fog Okay, not the DO one, but yes, it gets its name as F o G Now many people ask sir could there be further more, you know Nesting of functions. Yes, let's say if you put this into another function, let's say I put this to another function H H Then I would get something like H of F of G of X as you can see there is not three levels of nesting F for H of F of G of X Okay, so if I devise a machine Which directly gives you this output from X Then I would call that machine as Hofog Pronunciation is quite weird H O F O G Hofog Okay, I'll give an example from example Probably you'll be able to understand that what is composition of function Now, let me tell you my dear students when I say F of G of X It doesn't mean F of X into G of X now see they're normally five types of operations. Normally we perform between functions We can add them Right, you can add two functions You can subtract two functions You can multiply two functions You can divide one function by another function and the fifth one is you can take a composition of functions So addition subtraction multiplication Division what we call as quotient also sometimes and the fifth one is composition of functions So this is what we are learning today the fifth operation between function, which is the composition of two functions Let me give you a simple example. Well, you from that example, you'll be able to understand it very clearly let's say Let's say I have a function defined from real numbers to real numbers as as Let's say e to the power x e to the power x. Okay, and I have another function G of X Let me write from where to where it is defined. So let's say. Yeah, this function is Defined as X to the power three minus five Okay, so these are the two functions f and g f is defined from r to r. What does this are mean? What does this are mean? The domain domain. Yes What does this are mean? Codomain Okay, so domain and codomain both are real numbers for both of these functions, right? Now if I say find f of g of X Okay, if I say f of g of X Okay, what will be that function? Normally we write it like this also or you may write it like this also it all depends upon how you present it Here you don't have to put unnecessary brackets here probably you'll have to put you know two brackets one inside the other Yes, can anyone tell me what is the f of g of X function? So see f is eating e to the power X That means in place of X over here In place of X sorry e to the power X is eating X Q minus five. So basically this will go in place of X over here That is the meaning of f of g of X It just means that replace your X with g of X So this entire X you will remove from there and what will you place in? In the view of that you will place X Q minus five there and this is what you'll end up seeing This is going to be your answer Are you getting my point? Any questions here? I rate is not corona inside the house. Yeah, don't wear mask inside the house Okay, yeah Is this clear? Okay. Now should I ask you a question? tell me Tell me what is golf? What is golf g of f of X? Please type it in your chat box. I want to see if you have understood it. What is golf? Golf means you're putting f of X in place of X itself in G So in G the X is to be replaced with what f of X itself Right. So what is it? Yes, I show me so Smithy What's the answer going to be tell me are you finding it difficult to write or something?  Okay, let me help you So it will be so what do you do is the entire e to the power X will go and replace the X over here See this entire e to the power X will go and absolutely Pranavasi will be e to the power X Q minus five You can say e to the power 3x minus five Getting the point nothing difficult about it. Okay, try one more try one more. What is fof? Yes, the function can also feed itself Okay, this is called recursive. You're calling the function itself. What is fof? fof This X will be replaced by e to the power X itself. So it'll become e to the power e to the power X Is that fine? Is that fine? Okay What is Gogg what is Gogg So this entire function Will be fed in place of X in the same function This entire function watch my listen to my words carefully. This entire function Will be fed in place of X itself. So it'll become x cube minus five cube minus five Are you getting my point? Trust me most of the functions that you will get my dear students would be composition of functions. Okay now right now Whatever examples you have seen whether this is this example this example this example or this example You must be wondering That if I make a composition of function, what is going to be the domain and what is going to be the range of it? That's actually a very, you know important thing to address because we cannot write a function as Like as we have written it here. We should mention its domain and range or at least domain and co-domain Correct now here is something which we need to understand All of you please listen to this This is the most vital part of this discussion. I know you are finding this easy. It's very simple Okay, but try to understand a very vital part of this discussion. I'll write it over here very important From the previous example, it must be appearing that it's a cakewalk, right? Pick up a function put it in the place of X in the other function and you are done. That is the composition of functions They're not completely wrong. You're not completely right also When we are finding fo g of X we need to exercise a precaution What is the precaution? I'll just tell you let's say there's a function f which is from a to b Okay, and there's another function g which is from let's say c to d Okay, a and b are their respective domain and co-domain for f and c and d is basically the respective domain and co-domain for g Okay Now does fo g always exist? Can we always find fo g for any function given to us? The answer is no Right, so what restriction or what criteria I must fulfill if my fo g has to exist So listen to this criteria if fo g has to exist if fo g exists Then remember range of g Range of g g which is g function the function which has been fed Right, so this is what this is f of g right f of g So the range of this function, which is inside the bracket that must be a subset of Domain of f this condition is very important If this condition is not fulfilled You cannot find fo g are you getting my point any questions here? Now people ask me why? The reason to this is very simple if the output given by the inside function is Beyond the taking capabilities of the outside function Then it will not know what to do with those values I'll give you a simple example. I'll give you a very simple example very simple example let's say I Have a function g Which is defined as x plus 1 Okay, and it's domain is Domain is from 1 to 2 Okay, so only in the domain 1 to 2 this function is defined and how does this function behave? It behaves as x plus 1 in that particular domain of x. Okay Now you tell me what will be the range of g anybody? Please tell me the range of g write it down on the chat box if your input is from 1 to 2 output will be from Range means output range means output, right? Yeah, so what will be the output 2 to 3? Thank you, Aishwarya So it will be from 2 to 3 Correct. Okay. Now, let's say I define another function. Let me write it in yellow I'm giving an example over it Let's say I write another function f of x which is x square Okay, and this one can only works for 0 to 1 Okay, beyond this the function does not exist Right, is that clear? Okay, I understood the definition of g and understood the definition of it. Okay Now, what is the domain of f? You'll say simple sir. You have already written it. Why you're asking us? Just to confirm the domain is 0 to 1 Now, can I find f of g of x? Can I find f of g of x my dear students think and answer yes or no yes or no press on the poll button yes or no Can I find f of g of x? Yes versus no, they are very close to each other. Come on. Come on. Everybody should vote. Come on fast fast fast Vega Vega, okay, I'll close the poll in another 10 seconds Okay at the count of five you must all have voted five four three two One go for it Are you ten people still left to vote past? Okay, fine Why why morning morning you people are giving me a shock The answer is no my dear the answer is no we cannot find f of g the reason being The reason being the output of g is from two to three and The input capabilities is only from zero to one Correct. So if you let's say the output is comes out to be 2.5 And you put in the place of x in f Okay, so can I find f of 2.5? Can I find this out? We cannot we cannot find it out because this guy will say hey I'm only designed to work between zero to one. Why are you putting 2.5 inside me? See simple.   Papa and mama there dad and mom are there. Okay? Let's say you open the door and you see your brother or sister coming in Will you know what to do with them? No, because I have not designed you or I have not given any instructions to you that when you see your siblings What to do with them? Should you give them tea? Should you give them coffee? Should you give them milk? Okay? So you are only designed to work under two conditions when you see your mom and over when you see your dad In the same way this guy f Think as if it's a function, which is only designed to work for zero to one domain means what domain means These are the only inputs for which it will work Beyond that if you put anything it will refuse to work That is the meaning of domain. Yes or no so When you geofx output is from two to three it is not Under the domain or not under the processing capabilities of f so it will say boss Please do not try this out. I cannot process this so it does not exist It does not exist Right, please do not start writing in a simple Please do not start doing substitution because half the people here if I ask this question to them They will just say X plus 1 the whole square. No, it is not that simple. It is not that simple If you don't meet this criteria, if you don't meet this criteria, you cannot find f for G Have I made myself clear? Have I made myself clear? This is a very vital part of composition of functions Any questions I'll pause here and I'll take a question from you Now one question that probably may arise in your mind is what if there is a partial overlap That is something very interesting. Let me take an example of that Okay, is this example clear anybody any doubt? Please ask or type clear on your chat box if it is clear CLR just type clear so that I can Understand that you people have understood it. Thank you so much. Thank you so much. Now. I'll give you another example Just listen to that also very carefully Let me go down. Let me just put a dash Now I'll define a function like this Geo fix function is X plus one. Okay For X lying between again, let's say the same thing one to two one to two One to two and f of X is another function which is defined as X square X square when X lies between Let's say One to One to you can say one to four or let's say one to you can put a 2.5 Okay, fine one to five by two you can think like that. No Let's again repeat the whole process for this function What is the range You have already told me so no need to tell me again two to three correct two to three Now the other function is designed to work for One to two point five Right, so there's a partial overlap happening. Where's the partial overlap from two to two point five So in such cases, how do you find? How do you find f of g of X? That's a question that would have come in your mind right now when there is no overlap That is fun fog doesn't exist. No problem with that right the previous example But when there is a partial overlap, how do we deal with that situation? So understand this When there is a partial overlap What we need to do is we need to check All of you see here. We need to check When is this function G? Within this capability. So of course one it cannot take when it cannot take because For for it to take a value of one X should have been zero, but zero is not within its domain, right? So the minimum it can take is to Right and the maximum it can you know go till is two point five So when does this particular function become a two? Of course at one Correct, so try to understand this when does this become? When does this become a to the minimum of this? It becomes a two when X is one, right? Yes or no Right and the maximum it can go is two point five. So when does it become two point five? It becomes two point five when X is one point five Okay So now I'm going to find my fog Only from one to one point five why Because under this value of X these output will be from Two to two point five, which is within the capabilities of this function so under this definition of fog or under this interval of X fog will exist and that will be what and that will be nothing but X plus one the whole square So this is going to be your final answer It is slightly complicated because it requires you to think in terms of The range of G must be a subset of domain of F So now you must be wondering What about the other values as you can see? I have I can give inputs between one and two But I can only allow to find fog between one to one point five So what happens from one point five to two from one point five to two the function fog will not exist The function fog will not exist Are you getting my point? Now Many people ask me sir will we get such kind of a questions in our pu the answer is Very very rare. It's a very rare thing that a question like this will appear Okay, mostly the question will fall under this category either it doesn't exist or the examples that we had just now taken Okay, so don't worry too much about it But still I would like you to ask me any questions if you have You can unmute yourself. You can talk you can write any question if you want Okay Okay, one of you are saying please repeat the whole process. Okay. I'm going to repeat the whole process now listen to this very very carefully Your G of X is defined as X plus one When X is between one and two only Right that is called the domain. This is the domain of G Okay, so this is the domain the range is what? According to f of X two point five to two. Yeah, yes Now listen to this carefully a smoothie. You'll understand it from here. Also, so your range of G will be from two to three correct not two to three is Overlapping with one to two point five Right two to three interval is overlapping overlapping means there's a partial overlap from one to two point five Okay So only in those intervals where your G of X lies between one to two point five your answer fog could be found out or can be found out Right now you just have to understand What is the maximum I can allow and what is the minimum I can allow my f of G to go So f of G can go to a minimum of two Okay, which is within one to two point five. So no problem with that. I have no problem with two Maximum it is going to three. I have a problem with that. I Have a problem with that three fellow because that three is beyond this interval correct So when when you put a three in f f is a my god, what is this? I was only designed to work till two point five. Why are you putting three inside me? Right. So this three I cannot go till What I can go till maximum is two point five only that is this guy So now I will see What input here will make it two point five? It's a very simple calculation. They'll say one point five So from one to one point five only you can put into the G of X The reason being then its output will be from two to two point five and then you can find your fog That is why this function exists Okay, any question here. This is a very very important concept any questions You must speak out my dear students also no no need to type because I would understand you will have you know Doubts, which you may not be able to type properly clear Okay, I'll show yes. Yes, please sell me Say according to f of x is equal to x square means two point five whole square, right? No, no see two point five whole square when you're putting one point five Whatever you put inside it you will be adding plus one to whole square for example Let's say if I put one point three you will put one point three plus one the whole square I'm writing a function in general. I'm not writing it for every point For this entire interval Your function definition will go like this x plus one the whole square that is what is your f of g of x Yes, for one point five it will be two point five square for one point four It will be two point four square for one point three will be two point three square So whatever values you put from here, you will be adding a one in squaring this that will be your f of g What I'm interested in you understanding is if I go beyond one point five I will not be able to find the function the reason being if I put one point five in G Sorry one point six in G. Let's say I take a simple example one point six in G My output will become two point six So now you're putting two point six inside f f will become angry and will say why are you putting two point six inside me? I'm only designed to work till two point five Are you getting my point? That's why I have to restrict the input of G So that the output of G is restricted to the domain of f It's a little long thought You have to restrict your input of G Says that the output of G is within the domain of f Are you getting my point? Any more questions anybody This is a concept which many good students also don't are not able to understand. Okay, so if you have understood it hats off very good Okay, when I was studying this chapter almost 15 16 years ago Okay, I equally had the same trouble in understanding but once you know you start solving questions on this you'll be able to understand now Coming back to this coming back to this This is a very vital concept that you have understood right now. So let me erase this Okay Another thing that we need to keep in mind is that The domain of fog The domain of fog Will actually be a subset of domain of G very important As you can see in this example This is the domain of fog This is the domain of fog As you can see one to one point five is a subset of domain of G. It is within that It can be equal to also it can be equal to also but it is within that Okay, so domain of fog will be a subset of The domain of G are you getting my point? So these two vital things you need to understand while finding composition of functions If this is understood now, I will like to move on to the concept to the properties of composition of functions Please ask if you have any questions here Okay, let's take few examples, you know just one more example for our sake of clarity Let's say I Have a function G Defined as X square, okay from one two Three one two three, okay And I have a function G which is sorry f which is defined as 2x minus 1 Okay, if your output lies between 2 to 2 to 5 2 to 5, okay, is that fine? Think very carefully before answering Think very carefully before answering Find fog find fog I'll give you a minute. So when you're defining it Please tell me how it is defined And for what values of X? For what values of X? So these two you have to tell me while you're defining it Okay, think very carefully and then answer Let's have a minute for this. Let's have a minute for this. Don't be in a hurry Take your time That's very important Okay, let me give you options. I think then you will be able to respond to me. So Let's say I give you options These are your options option number a The function will be defined as Like this the function will be defined as 2x square minus 1 for X lying between This Option number B It will be defined as 2x square minus 1 If X lies between 2 to 3 Okay, option number C It will be defined as 2x square minus 1 if X lies between root 2 to root 5 an option number D F4g is defined as This if X lies between 2 to 5 Okay, now see I'm not playing with the definition definition. Everybody knows what is going to be, okay? I'm just playing with this guy these these tables. Okay, so I'll on the poll option for you on that poll option I want everybody to respond. Okay, your time starts now I'll give you one minute for this somebody has given the answer Kirtana very good Kirtana Not saying right or wrong to anybody I'm just saying Very good if you have responded this single question will Make me understand that whether you have understood the concept or not Everybody please vote last 30 seconds last 30 seconds time is running out last 10 seconds My dear students come on come on 20 people still left to vote come on At the count of five I'll stop five four three two one Go, please vote. Please vote. Please vote everybody. I don't get to know what you voted. Don't worry. Tell people still left to vote Okay, never mind Let's see the result. Okay, this is the Verdict of I hope you can see the result. Oh, yeah, now I can see the result So Janta has said B 55% of you have said B Okay Then the next highest word has gone to a okay Let's see what is correct whether a or B or none of them. Okay. Let's see. Let's see everybody Now here comes your understanding when you look at this interval For the domain of G Okay, everybody if you know the graph of x square, this is the graph of x square, right? Okay, this is the graph of x square up parabula correct from 1 to 3 the graph will go from 1 to 9 Right, so this output. Let me write it over here that The range of G will be from 1 to 9 Anybody has any doubt with this if your function is x square and your input is from 1 to 3 The output will be from 1 to 9 that is 1 square to 3 square, which is 1 to 9 any doubt with this anybody No doubt Now the input capabilities for F is only between 2 to 5 Right, so it is not going to work beyond 2 to 5. Right. It's like the working hour for F Okay, no matter how much money you give him. He will not work beyond 2 to 5 Okay But the requirement here is from 1 to 9. So there is a partial overlap. All of you, please see let's say 1 to 5 is this Sorry 2 to 5 is this and 1 to 9 is this so you can see that there is a partial overlap occurring over here Okay Now partial overlap means we cannot put 1 and 3 Inside your G because if you do that the output will Exceed the input capabilities of F So what do we suggest the students very simple if you want to know from where to where the function will work Just substitute x square in place of x over here So if x square is from 2 to 5 X will be from Root 2 to root 5. I mean roughly speaking. Okay, of course, this is not the you know exact interval. Okay So in this interval from root 2 to root 5 The outcome of this function will be within 2 to 5. That is what we have known right, so only in this input Only in this input your function is going to work Else it is not going to work Else it is not going to work So only from root 2 to root 5 You can substitute this x square in place of x over here and option number C will be the only right option Okay, so Kirtan was the first one to get this side. Very good rest most of the people went wrong in this Getting my point So your output must be out. This is the output. No your output must be within the input of this So when you do that you end up getting x lying between root 2 to root 5 Now you should also check whether it is between 1 2 3 or not Yes, root 2 to root 5 is very much within 1 2 3 so you can take the entire interval Is it fine? Okay, let's understand the properties. Let's understand the properties of composition of functions composition of functions the first property the first property the first property that we are going to talk about is Composition of functions is not commutative That means f of g of x Need not be g of f of x Okay Spurti spurti if you can hear me, please do not scribble on this on the Screen, okay, I can know who is scribbling on the screen. Fine Spurti, please do not scribble on the screen. Fine, okay That means fo g is not equal to g o f F o g is not equal to g o f Okay, so in general So do so do also, please do not write anything on the screen This function is not commutative in general Not commutative in general. It's very easy. It's very easy to verify this also Let's say I make a function like e to the power x and The other function as let's say sine of x Okay, of course, I'm considering that fo g and g o f can be found out So fo g in that case will be what? F o g is where you are feeding sine x in place of x That will be e to the power sine x Right and g o f will be what? G o f is where you are feeding e to the power x in place of x over here. That is sine of e to the power x Of course, they're not equal of course, these two are not equal Right, so we cannot say fo g and g o f are equal. That's the first property which we need to understand Okay, they are equal under one situation What is that situation? We'll discuss that in today's class. Don't worry Any question with respect to this property? So fo g is not equal to g o f in general in general means most of the time Of course under few situations, they may be equal next property Next property is F o g o h let's say I have a composition of three functions Okay, it is same as fo g o h It means they are associative composition of function is associative So if you find f o g first Okay, and in that you feed h that means fo g o h this one It is same as if you are feeding g o h to f See basically, this is writing like this f Oh gee Yeah, this is what I mean to say when I write it like this Okay, this means in f of g of h I am feeding h So ultimately you'll see that you have written the same thing both the sides Right, so it doesn't matter first whether you do fo g or whether you do g o h Ultimately fo g o h will be the same Let me give you an example Let's now have three functions Can you all give me any function anybody who wants to give me a function suggest me any function? Any function? Tell me any function which you which you know can of x Okay, tell me another one Let's say log of x or ln of x ln means log to the base e. Okay. Tell me another one. Let's say x to the power 5 Okay, so these are your let's say three functions fine now if somebody says find F o G o h. Let's say somebody says find this function What does he mean? He means that first find f g o h So we have to first find g o h G o h means you have to feed h to g. So this fellow will be fed here It will become ln of x to the power 5 Any question? Okay, now second step is where you are finding fo g o h fo g o h means This function will be fed in place of x in tan. So it will become tan of ln of x to the power 5 Any question here Anybody has any doubt in understanding tan of ln of x to the power 5. How does it come clear? Everybody is clear Okay, now we'll find the other one. Let's say I call this as a Now I'll find B. What is B?  Who will tell me fo g? Please write it on your screen on your chat box fo g. This is f Look at this is your f. This is your g. Okay. What is fo g? Write it down on your chat box F o g means you're feeding g to f absolutely print of our tan of ln x very good So fo g will be tan of ln x tan of ln x Okay, we write it properly fo g Okay Now second step is finding second step is finding fo g Bracket o h means h is fed it to this function What is h here h is x to the power 5 so if I feed X to the power 5 to this function it means its x will get replaced with x to the power 5 Is that fine? Do you want me to take few more questions on these kind of stuff because I'm feeling that most of you are Restricting yourself from answering because the concept is not very clear So what I'll do is I'll give you more questions so that the idea is clear Now let me complete this property as you can see this term and this term are the same These two terms are the same And that is what I meant by the property that F o g o h in bracket is same as fo g o h Okay, now before that I'll do some exercise with you a simple which some questions I will take. Okay, let's say I Have sine x as my f of x. Okay, let's say my g of x is a root of x Okay, and let's say my h of x is Let me take an example of Of another function. Let's take Five to the power x. Okay Now all of you Please answer the following questions What is f o h? O g.  Okay, is it option one Root of five to the power sine x Okay, is it Sign to the sign of five to the power root x okay, or Is it sign to the power Under root of five to the power x or is it Five to the power sign of root x Okay, so four options are given a b cd. Please choose one of them. Okay, your time starts Now I hope you can see the poll Okay, think carefully and answer I'm asking what is f o h o g f o h o g Nine of you have voted so far very good Everybody please vote. Don't worry. I don't get to know who is voting. What okay. It's completely anonymous to me But don't start scribbling on the screen because then I can know who is doing that Last 15 seconds my dear. I don't think so. It's more than a minute question last five seconds five four three two one go Okay, let me stop This is what Janta has said Janta has said option B. Okay, so Janta. Let me tell you option B is correct Okay, very simple. You first have to feed G to H So when you feed G to H, you can find it in any order you want When you find G to H, basically it becomes under root of five to the power X Oh, sorry G to H you have to find. Sorry, sir But you have to feed this to this guy So it'll become five to the power root X and now you have to feed this to F So now you have to feed this to F So f o g o h will be sign of five to the power root X. So that is option B only very good Okay, now you have understood one more will take up so that I'm confident that you have understood it Okay, let's find H o f o g hafog Okay These are your options five to the power sign of root X sign to the power So sorry sign off under root of five X five to the power X Okay under root of five to the power sine X or five to the power of under root of sine X Okay, let me check whether the right option is there inside it or not. Yeah So let me launch the poll now Think carefully and answer. I want hundred percent, you know, correct result this time. Okay voting is on think carefully Then answer your dears. Very good. Nice. Please vote. Please vote 22 of you are still to vote last 20 seconds left Fast fast last 10 seconds last 10 seconds last 10 seconds last five seconds five four three Two one go Please vote. Please vote 14 of you are still waiting. I don't know why what Okay There you go 69% Janta has gone for a Janta you are janaadan Janta is correct option is right. Okay. Very simple first you feed g2f First you feed g2f when you feed g2f It will become sine of root X Okay, and then you are feeding this to H. So the sign of root X you are feeding to To H so it'll become five to the power that them so five to the power sine of root X option is Okay Very good. Now, I don't think so. You'll have any doubt in understanding this property. Now. Let's go to property number three property number three is this is a slightly theoretical property if f of X is 11 okay, and G of X is also 11 One one means one one function. Okay, then please remember F4g or G of whatever you want to call it that will also be a one one function That will also be a one one function If f4g is one one If f of f of X is one one G of X is one one then f4g will also be a one one function Undoubtedly Undoubtedly is that fine? Okay now Let's take Few questions that are there in your No chapter Will start with Let me start with a very simple one. Let's say your f of X is defined from this set To let's say this set one two five What does it mean? It means this is basically your domain and this is basically your co-domain G of X is defined from One two five to one three Okay Now f of X is known to be F of X is known to be in fact. I should not write f of X. I should just write The relation f okay the relation f is known to be one comma two Three comma five Okay, four comma one okay and and G is defined as one comma three Two comma three and five comma one Okay my question is find Gof in this find gof in this Now before you start solving this question, I would like you to pause for a minute and listen to me Is composition of function only allowed in function? The answer is no it is also allowed in relations. In fact, whatever is applied to functions The same concept is applied to relations also Okay, because functions are within relations relations are a bigger set Function is a small subset out of it. So even you have composition of relations also Okay, now think as if these are two relations f and g are two relations You have to find gof. How will you find it out? In case you are done with it, please type the Ordered pairs that you think please write this in the rooster form. Okay, I think You are aware of what is a rooster form. So write down the rooster form for this set That means which all elements will be there in Gof Think carefully and answer my dear think carefully and answer this time. I'm not going to give you options Again hint to solve this You must be guided by what is the see this is g of f of x, right? right, so what is the output of this and What what output g gives for that input? I'll give you a simple example in this. I'll not complete it I'll just give you one ordered pair which is there in your answer. See When you put one inside this function the output is to so this is your input and this is your output, right? Correct. Yes or no Now this output This output is now going inside g correct, so check Which of the ordered pairs starts with a two so out of these three which starts with the two yeah there this go this Go this starts with the two correct So when you put two inside it, it'll give you three isn't it so the answer is When you put a one you should get a three as the answer. So here one comma three will come out as an ordered pair Absolutely correct. Madiha. Well done. Madiha has absolutely given the right answer. Very good Now I've given you a hint. Please complete this now. What will be the next ordered pair? It's very easy. I just understand this you'll be able to do it like this Ashwarya is this I mean is your answer exhaustive? Have you missed out anything just check? Smoothie also, please check smoothie Pranava also Absolutely correct Pranava very good So two of you have given perfect answers Others I'm sure you can do it as well. Come on Let's have, you know 30 seconds for this not more than that 30 seconds very good Yes, Ashwini Wonderful you guys are awesome Keith and also correct. Okay. Now see Next is three comma five. So three is the input to this function Five is the output. Now see which of them starts with the five. Oh, yeah, there you go. This starts with the five So three and one will be paired up simple Now four and one see which starts with the one. Okay, this starts with the one so four and three will be paired up Simple over this is the answer Okay, I Need out here any questions here, please ask Please ask like to do one more of this type Should you do one more just one more one more one more? Hmm. Okay. Let's do one more. So next question I'll write it over here. Huh? This time I want everybody to answer. Okay. Think carefully and answer, huh? So let's say I have a function directly. I'm giving you the rooster form for this See functions may be given in two ways to you. They may Functions may work on Continuous domain. They may also work on discrete domain. It is not necessary that everything works on continuous domain Okay, some functions they work for discrete domain also discrete means there is no continuity in the values of the input Okay Let's say this is your FNG fine These are functions. Okay, these two are functions number one Write down gof Okay, number two Write down fog Okay, so both fog and gof we need to write down So let's have the answer for the first one. So put one and then write your answer. Okay, I'll give One minute for this good enough one minute time starts now I don't be under the pressure of time. Okay, if you feel that you need more more time You can ask sir 30 seconds more or 15 seconds more. I can help you with this. Don't worry about time correctness is more important Correctness is more important. Please write ordered pairs. These are all ordered pairs. My dear students Write it as ordered pairs. So what all ordered pairs will be there? Write down here. Also, what all ordered pairs will be there? Write down Okay, ashwarya has given one response very good ashwarya Shusti Smutty very good Very good. Yes, Shrini very good. Nice Okay First one, I just want the answer for the first one. Don't give me the answer for the second one I'll lose out on that information. Okay only the first one answer. I need No second one answer second one. I'll give you time. Okay last 20 seconds Last 20 seconds, please wind it up. Nice. Very good. So Anna Yes, yes, Shrini. I saw your answer. Yeah, please respond others. There's so many people in this group Everybody should write down Should I start taking names? I don't want to start taking names here. Let it be wrong. Let it be wrong I'm not I'm not sitting here to judge anybody. I'm sitting here for Helping you out Okay, let's see You're finding G o f first so focus on f When you're finding g o f focus on f first f is this one Now 0 4 see which of them starts from 4 or this starts from 4 so 0 and 2 will be club So first element will be 0 to 2 5 which starts with 5 this starts with 5. Yeah, this starts with 5 So 2 and 0 will be clubbed up Okay, 3 and 7 which starts with 7 are this starts with 7 So 3 and 2 will be clubbed up. So the answer for the first part is 3 comma 2 Okay Give yourself a pat on the back if you have got it. Correct. Very good. I can see many of you getting the right answer Okay. Now f o g another one minute for it Okay, I should show you I saw your answer Pooja Pooja was it for So Pooja is given for one. Okay, Pooja. Give me for second one Kavya, I can see the answer second one. Please write second and then give your answer else I will be confused whether you are writing it for one or two Good second one second one peace. Mm-hmm. They just when you don't write it on the screen Write it on the chat box. Yeah, lick it very good. Now people are participating Panawa. Where were you in the first one? Sanat very good. Okay Let's discuss. Where's this case?  So for two is there Then see which is starting from two our second one starting from two. So four and five will be clubbed So four five simple as that simple as that. Okay, it walk Five zero now, which is starting from zero first one starting from zero. So five and four will be clubbed up five comma four Okay, seven and two which is starting from two which is starting from two the second one So seven and five will be clubbed up seven and five will be clubbed. That's it done That's it done. Is that fine? So this is the basic question that you will be getting in this concept. Let's take another one. Let's take another one Let's take a slightly serious one. Let's take this question. Okay. Hope you can read this question question is There is a function from R to R Defined as x cube. There is a function from real numbers to real numbers defined as x cube Okay, there's another function from again R to R Defined as x square Defined as x square What do we need to find number one? We need to find g o f So let's find g o f first then we'll come to g o f of three. Please write g o f fast It should not take you more than five seconds five seconds questions G o f You can type it also on the chat box. It is so easy. No x to the past six All you need to do is put this in place of x over here So it's x cube square that is x to the past six over Okay, okay Now what is the domain here domain is for all real numbers now here? You don't have to check many people ask me sir You did not do that check that Domain of f should be within the range of sorry range of f should be within the domain of f Range of this should be within the domain of g correct range of f should be within the domain of g See, I don't have to worry about because all of them are from real numbers So whatever output I get from f whatever output I get from here is well within the input capabilities of g So that is fine. So you can directly write down the answer x to the past six. So what is g o f three? What is g o f three? Three to the power six absolutely. Yes, which is nine cube seven twenty nine Simple question see let's take the another one Let's take another one. Yeah, let's take this one. Hope you can see the question clearly question says f is a mapping from R to R read the question carefully f is a mapping from R to R Defined like this. This is the definition of the function G is a function from R to R Defined like this Correct. So first they're asking you what f o sorry g o f for they're asking you g o f Tell me quickly here. You don't have to worry about seeing the range of f being a subset of domain of g Because all of them are going from R to R problem comes when the the domain and range etc They are you know some limited sets Okay So tell me the answer quickly without much waste of time.  Okay, and this is also valid for all real numbers See make it a habit whenever you are writing a function mention its domain Right. It's a good practice.  So it's four times x cube plus two minus one Absolutely a shashwani Okay, I think a Madiha slight mistake has happened in your answer. Please check. Okay. This is also true for all real numbers Now are they equal are they equal? I don't think so. They are equal. I don't think so. They are equal Okay, you can test some for some value. So these two functions are not equal Okay, well, you can test for a zero if you want so G o f of zero that will give you minus one cube Plus two which is actually a one But f o g of zero will give you what four into zero plus two minus one which is seven Right. So even if for one value if it fails, we can conclude that g o f is Not equal to f o g And we also saw this in our properties also, right? So in general, they are not commutative in general F o g and g f are not same. Okay Now boys and girls, dear students, we are not going to talk about the last part of our discussion Inverse of a function Inverse of a function Yes, I know you would like to do more questions, but I think one of the sessions will devote to solving questions on inverse of a function And composition of function. So let me take this up Now what is inverse of a function? Okay, so let's say I have a function f defined from a to b Okay, so this is my function Then what is its inverse? What is its inverse? Let me explain this in a very layman language very simple words so that you will remember it for the rest of your life Let's say another function g Okay, is the inverse of f Then what actually happened see let me make a schematic representation of it So let's say this is your machine f Okay In this machine If I feed something Let's say, you know, this is a machine which takes water Okay And it gives me juice Let's say mango juice Okay If I device another machine Let me draw it in yellow If I device another machine Let's say the g machine Right Where if you feed this juice It will give you a water back It will give you the water back Okay I don't think so there is a machine in real life like this I have made up a machine on my own So let's say I invented a machine g In that machine if you put your juice mango juice, let's say whatever juice you want It will return the water back to you Okay So then this machine is playing the reverse role of this In fact, both the machines are inverses of each other This isn't plain and simple words If you want to understand this concept This is a simple example to make you understand this So they are called inverses of each other Right So something which converts the output of f To input of f That will be the inverse of that machine Right So you can take another example Let's say If I put Ata Ata Into this machine It gives me chapati Let's say Okay Then inverse what happens If you feed chapati What will it give it to you It will give you the Ata back Right Hope there was a machine like this Where if you put chapati Your Ata will come back Which you had used to make that chapati Okay But unfortunately most of the processes are irreversible in life Isn't it This is just for your understanding purpose Okay So if you see from this example It's very clear that If you feed x to this And it gives you f of x Okay Then this function feeds f of x And returns you the x back Plain and simple language Correct That means your g will work with Your output of f To return you the input back So see here I have reversed the positions here Now Let me give you some real life mathematical example e to the power x Log of e to the power x Or log of x to the base e Both are inverses of each other Both are inverses of each other Okay Another example I can give you Why does inverse of each other Very simple If let's say if I feed 5 to this function I get e to the power 5 right Correct If this e to the power 5 I feed it to log What do I get I get a 5 back So what you had input over here You have got that back as the output Okay Both functions are inverses of each other You can do the reverse activity also Let's say if you feed You know let's say 5 to this You get log of 5 to the base e Right And if you feed this to f So let's say you are finding f of Log of e to the power 5 Means e to the power This I'm sure you would have learned log properties It gives you 5 only What does log property say A to the power Log of anything to the base a Is the same thing right Hope you are aware of this property This is called the fundamental logarithmic identity This is called the fundamental logarithmic identity Okay So these two functions are behaving as inverses of each other Can you give me another example Another example Of two functions which are inverses of each other Anybody Any example Give me correct only Okay Let's talk about this X square root of X R plus to R plus So let's say g of X is root of X They are inverses of each other So if one squares other does square root of that Right Absolutely better Now A very important part point over here Please listen to this Can I invert any function in this world Can I find of inverse of any function in this world The answer is no Not every functions are invertible So which functions are invertible Let me talk about it Only those functions Which are bijective Okay What is bijective Both one one and onto Remember we did these Classification of functions or types of functions Only functions which are simultaneously one one And onto also called bijective functions Okay Remember the name One one was called Injective Onto was called Surjective So when both injective and surjective functions are No Or both of these characteristics are present in a function We call it as a bijective function So remember Only one one And onto functions are invertible Now the question comes Why Why only one one and onto functions are invertible Okay Now let me give you A case where if the function is not one one What will happen? And if the function is not onto what will happen Okay let's take a case Let's say I have I have a function which is not one one Let's say Let's say A B and C and I have One and two Okay now this function Okay let me write it as set A set B So this is my function F So the mapping of the function is like this Okay can we have a function like this? Yes we can have many one is allowed Now what does it mean? It means that when you are feeding A to the function F It is giving you one Correct See the arrow diagram Arrow diagram will tell you everything When you are feeding B to the function you will get a one And when you are feeding C to the function you are getting a two Correct Now let me make its inverse Let me call the inverse by the name of G So G will work with the output And give you the input back isn't it? Everybody please see this definition This one Okay I am basically following this Okay So now If I start using the same you know mapping So when I had put A I got the output as one So when I put this one in the inverse function I should get A back Correct And I should also get B back And I put two into the function I should get a C back But let me tell you my dear students This is not a function at all This is not even a function Remember the functions criteria The function criteria was A pre-image must give you a unique image Right Here one is having two images to it A and B So forget about it being inverse of something It is not even eligible to be called a function Right That's why in such cases In such cases we will not be able to find the inverse of a function Therefore being one one is very important Therefore being one one is very important Excuse me Yeah Now why should be onto Why should be onto Let me give you an example Another case Let's say it is not onto let's say I take a dummy example Again A B C and let's say one two three four Okay, let us say it's one one But it is not onto Okay, so here I took an example of many one And I saw what went wrong now I'll take an example of into And I will show you what will go wrong in this case also So you can see here that When you put a into the function you get one When you put B into the function you get a two When you put C into the function you get a three Right Let's make the inverse of this function Let's make an inverse of this function Can you all hear me Sorry there was a brief power cut can you all hear me now Am I audible Yes sir Okay, thank you thank you Girtana Alright, so now I will see that if the function is into what will go wrong Now let's make a function Let's make the inverse of the function G from B to A So one two three four and there will be ABC So see domain range positions will slip in case of inverse Very simple because inverse will work with the output right So output will be its input That's why B set will come as a domain now Okay, now see here When you had put a into the function it gave you one So when you put one into the function G it should give you a Correct Similarly when you had put B into the function you had got two So when you put two into the function you'll get a B Similarly this will match with C Now you can see that four is left unmapped Four is left unmapped What does this indicate this indicate that boss it is not a function It is not a function so forget about it being inverse I know function it is not even eligible to be called a function Right so even in the case where you have an into function We cannot have an inverse that's why it becomes very important That I should have a bijective function for it to be invertible Right yes or no now we'll talk more about it But first few things the notation that we normally use For the inverse is f with a superscript of minus Please do not read this as one by F. No, it is just a symbol It is just a symbolic thing. It doesn't mean one by F and all Okay, it is just a symbol used for expressing inverse of a function Okay few examples which we have already seen And I like to show you one more example Let's say I talk about sine x function Okay very simple thing that you need to understand from whatever discussion we had Now if I define sine x from R to R Okay all of you have seen the graph of sine x function Correct it's a sinusoidal curve like this Yes or no everybody has seen sine graph right Okay now is this function invertible Just press yes or no right now Can you find inverse of this function This is the definition of sine x which I have given you right now In the present definition in the present form Can we find the inverse of this function Think and answer think and answer you have 15 seconds If I've only answered nothing can be done But those who are supposed to vote now I think 19 of you are supposed to vote I think 19 of you are supposed to vote okay Please take a critical analysis and then you know vote Okay another 15 seconds I'll stop the poll Okay last five seconds five four three two one Okay, this is the verdict of Janta Janta 56 percent That says no you cannot find the inverse of this You are absolutely correct You cannot find it out because in the present form Your function is neither one one nor onto why it is not one one Because if you draw a horizontal line It is cutting the function at so many points Okay, so it is not one one Why it is not onto because for onto its core domain Should be equal to the range range is only from minus one to one So it's not onto also right? So if it is not one one and if it is not onto We cannot find the inverse of this function But many people say sir but sin inverse exists no How are you finding sin inverse if sin function was not invertible We have a whole chapter on that inverse trigonometry function That means that chapter itself should not have existed Okay now hold on in the present definition It is not one one nor onto But by changing your domain and co domain You can make it one one and onto Basically you are empowered to change its domain and co domain So in the present stage f inverse X does not exist dne Okay, but if I redefine this function like this everybody please see Why not onto see should see onto means what What is the definition of onto when do you call a function onto Oh you forgot it means your range should be equal to co domain Correct, but range of sine function is only from minus one to one No, I'll show you a sine function. Let me show you a sine function. I hope you can see the graph Y is equal to sine of X. Okay, you can see this function is only from minus one to one range Means for finding the range of a function you should see its Y axis span For finding the domain of the function you should see its X axis span Span in the X axis span, you understand right the coverage Right, so this is for domain. This is for range Okay exercise domain range domain range Okay, so range of this function is only from minus one to one Getting the point so if it is a minus one to one Your co domain is all real numbers Right, so range is not equal to co domain. So it is not onto So if it is not onto and it is not one one means it is not bijective If it is not bijective, it is not invertible simple as that Any questions here? Okay. Now, what have I done to the very same function sine X I have curtailed the domain so from our I have made it like this This is now my domain of the function. Okay, and I purposely made my co domain As the range so this is the co domain of the function This is the co domain of the function and I made it as the range of the function Correct now you must be wondering why did I choose this interval Because if you just draw the graph Sorry, sorry if you just draw the graph of sine X from minus pi by 2 to pi by 2 So minus pi by 2 is let's say here Pi by 2 is let's say here your graph is only this much Okay, that means only this part of the graph I have made only this part of the graph I have made I'm just bubbling it up with white so that you can understand Okay, if I make that then what happens is any horizontal line that you will make will only cut it at one point It only cuts it at one point it becomes a one-one function Okay, and I have restricted my range Sorry, I've restricted my core domain to minus one to one Okay, so I have made it both one one and all two Right, yes or no so now this function is invertible So when you're defining the inverse, please remember They both will interchange their position so now it will become like this Okay, and definition wise it will become sine inverse of X So this entire chapter of inverse trigonometry that you are going to study in your class 12 Is when you have restricted the domain and core domain of all these trigonometric function To make them one one and onto Don't worry about it right now We'll talk about this when the chapter has been taken up Okay, just a second Okay, is this fine Now Question comes now How do we find inverse of a particular function? Okay, let's say I know that the function is now one one And I know my function is onto How do I find its inverse? Okay, so let me take an example Let me take an example Let's say my function is defined from R to R Okay as 3x plus 5 Okay The two type of questions will come in your pu exam Number one prove that this function is invertible And hence find the inverse Okay Number two is Find the inverse So if you are asked to prove that it is invertible and hence find the inverse It will be a big question It will be almost a four or six marker question Because not only you have to prove that the function is one one Onto But you also also find its inverse Right so this is one type of question that comes Second type of question is just find the inverse Okay, so they say assuming it's one one or onto Or it's given that it's invertible find the inverse There you don't have to prove one one and onto Now many students ask me sir if it is not written there in the question Then what do you do in that case? Then see the marks If it is a very high mark question For example six mark and they have just said find the inverse Then also prove it's one one and onto Okay, so right now I'm assuming that I don't know anything about this function The question setter has asked me number one Show that f of x is invertible That means it is a bijection or a bijective function And second is find the inverse of f So let's say these are the two things which have been asked to me in the exam Okay, so what I'll do is first I will show it's an invertible function So I'll write it down For bijection or for bijective Okay under bijective first I will show it's one one The other day I think you have done a lot of questions with money set also Regarding finding a function one one or not So what do we do? We'll say let x1 and x2 be two inputs coming from the domain of this function This is the domain of the function and this is the co-domain of the function Such that f of x1 is equal to f of x2 That means you are trying to say 3x1 plus 5 is equal to 3x2 plus 5 Now 5 and 5 will get cancelled 3 and 3 will also get cancelled because they are non-zero quantities We can always cancel them off Leaving you with x1 equal to x2 as the only possibility Only possibility Remember when is a function one one When assuming f of x1 is equal to f of x2 Gives you x1 equal to x2 as the only possibility Then it means f of x is one one Okay, I don't want to repeat the same stuff We had done enough questions on this Next is I want to prove it's onto Onto what do you show? You show that your range and co-domain are same So we have to find range So remember how do you find range? First we write y as the function Domain is already known to us Domain is real numbers So what I am going to do is I am going to make x the subject of the formula So I am going to make x the subject of the formula Now if I want to ask you For what value of y does this function give you a real value of x You will say it gives you for every real values of y That means your range of this function is all real values Right, so now check here Co-domain is also all real values Range here is also all real values Both are equal Both are equal And hence the function or therefore it implies that f of x is onto function Right, so this is something which you need to do if you are asked to show that the function is invertible Right, but don't forget to write the final comment Since f of x is both 1 1 and onto Both 1 1 and onto It implies f of x is a bijective function Is a bijection Bijection or bijective function both are same things Okay, and hence And hence f of x is invertible f of x is invertible Okay, so first part of the function you are done First part of the question you are done Any questions here please ask Any questions please ask Regarding proving 1 1 regarding proving onto any questions No questions clear everybody Okay, so first part is done First part of the question is done Let's do the second part of the question Second part of the question is finding the inverse of this function Now all of you try to understand here Try to understand here You have been given this function Correct, yes or no Now what are you trying to find? You are trying to find I am trying to find this function Correct, yes or no Now everybody here knows that Everybody here knows that The output of the inverse When it is fed to the input of the function I will get the input back This is by the very definition of inverse This is from the definition of the inverse only See f and f inverse Both are inverses of each other Right, inverse is not only one direction Right, so if machine f If machine g is the inverse of f machine f is also the inverse of g machine So e to the power x is inverse of ln x ln x will also be the inverse of e to the power x vice versa happens Not one sided So if a machine is taking water and giving you juice and another machine is taking juice and giving you water Then both the machines are inverses of each other Getting my point So from this definition it is very clear that This output should have been y Correct, yes or no So this is actually your y right So f of y is going to give you an x Correct and vice versa That means if you put f into the inverse function I should get x back So both these relations are true Correct, that means f inverse of y will also give you an x Right, so what I am trying to explain you over here Just get it in a simple word, let me take it to the right When you are finding the inverse function When you are finding the inverse function You just have to find out Basically this Because this is as good as saying Y is equal to f inverse x Are you getting my point? Yes or no? So what I am going to do here is that In order to find this I am just going to See the process I am going to start with the given function Correct I am going to make x the subject of the formula This is method number one There is one more method I will tell you Method number one So first of all I am going to write x in terms of y The very same step that you had written While finding or while proving it was on to this step This step I am repeating once again Correct Right Now basically what have you done You have come to this step Right You have come to this step Correct So this function here Okay Is basically a function which is giving you f inverse y Okay Let me again rewrite this so that you are able to understand We need to find this isn't it? We need to find this Correct Yes or no? Correct And from your function If you are doing this term If you are doing this activity Let me take my camera little up Or camera to the right So this is what you have done right now This is what you have done right now So you have found something called f inverse y Right But I need to find f inverse x So what do I do? In the same thing I replace my I replace my Y with x And we are done So x minus 5 by 3 Will become your inverse of this function So if this function is your f of x This function will be its inverse Simple as that So I will repeat the steps Just follow this step Step number 1 Right Y is equal to that function Step number 2 Make x the subject of the formula Step number 3 This part of the function You replace your y with an x Let me write it over here And that is going to be your answer Okay So here make x the subject of the formula Make x the subject of the formula And here just replace your y with x Replace y with x in this part Okay And this is done This is your inverse of the function Right And if you want to test it out Whether you actually got the right answer Try doing this Remember we have done composition of functions Okay So try feeding one function to the other Okay let's say I do this That means I am feeding this to 3x plus 5 So see what will happen 3 into Okay see what will happen It will give you the input back So this input has come outside as the answer Okay Here also this input will come out as the answer Check Now I am putting this entire function to this guy 5 5 get cancelled 3 3 get cancelled So you end up getting an x back That means your result is absolutely correct Your result is absolutely correct Let me give you one more question Any questions on this Any questions on this on how to find the inverse I think one more practice we will do Then we will be able to understand it in a better way Okay let's say My question is It's known that this function is invertible Let me give the function as 2x minus 1 Okay It is known that it is invertible Find its inverse Find its inverse Okay Now what are the steps We will follow that only And we will be done with the answer See what are the steps Number 1 First write y is equal to that function Okay Step number 2 Make x the subject of the formula x the subject of the formula means Write x in terms of y Any problem so far I just brought the one on the other side Divide by 2 Correct Now this fellow Focus on this fellow Focus on this In this function If you replace your y with an x Then the answer that you got Is your inverse of the function And we are done Right Happy with this Simple steps to be followed Nothing you know Challenging in this Okay You can check this out also quickly Verification also you can do So let me do a verification also If you feed 2x minus 1 into this See what will happen Let me do this step f of f inverse x And I'll also do f inverse f of x Both of them I'll check whether it is giving me x or not If it doesn't give me x means Whatever I've done will not be correct So I'm feeding this function Now to this function So it will become 2 times x plus 1 by 2 minus 1 That will give you 2 2 cancels So x plus 1 minus 1 That will give me x Wonderful So what I did was correct Now I'm feeding this function to the x over here Okay So this one this one gets cancelled so 2x by x Sorry 2x by 2 I'm back to x again so this is also correct So whatever I've done is absolutely absolutely correct Any doubt in this three-step mechanism Of finding the inverse please highlight Any doubt Okay So in the interest of time I will take one more question Then in the problem solving session You can solve many questions on this So let me take a question on this Okay This is an option based question Let f be defined from r minus 4 by 3 to r As 4x by 3x plus 4 As 4x by 3x plus 4 So this is how my function has been defined Okay The inverse of f is g Okay Now you have to find which of them is g of y Which of them is g of y That means you have to find f inverse y That means you have to stop at the second step Let me go back to the previous slide This thing is your f inverse y So you have to stop at this step Okay you don't have to replace the x back Understood what is supposed to be done Let's do it Let's everybody work this out and tell me the result I will switch on the poll Let's have the answer in 2 minutes time Let's end the poll now Most of you have voted No not most of you Okay please vote Please vote everybody Last 10 seconds I am giving to vote Please vote In the count of 5 I will stop At the count of 5 I will stop 5 4 3 2 1 I am going to stop now Okay So maximum people have gone for option B Let's see option B is correct or not Let's see option B is correct or not See Follow the steps So the question is pretty simple Pretty understandable You have been given a function You have been given an inverse You have to find g y Earlier we used to find gx Or f inverse You can write it as f inverse y also Okay So the steps are pretty simple Next step is write y is equal to 4x by 3x plus 4 Next step is making x the subject of the formula X the subject of the formula So how will I do that? Simple cross multiply So 3x plus 4 times y is equal to 4x Okay So 3xy plus 4y is equal to 4x Correct So you can say 4y is equal to 4x minus 3xy Take x common It will become something like this Right It is not as easy as we had taken in our examples It is slightly more complicated But nevertheless it is not difficult Okay The moment you have got this my dear Your job is over This itself is your f inverse y Right Here you don't have to substitute y with x Because they have not asked They have not asked for it Okay So this itself is your gy So 4y by 4 minus 3y is your answer 4y by 4 minus 3y Absolutely correct P was absolutely correct Very good everybody Very good So is this idea set in? Okay Now next class before we start a new chapter Which chapter is going on in your this thing Before you closed Relation function was over Any other chapter was going on Or should I pick up a chapter on my own Any suggestions from your side


I tell you guys all the time adversity is coming for you and you know we've we've certainly hit some here these last few days but the season stops for no one and and we just got to keep on competing and fighting and that was that game embodied that you know get out to a lead lose the lead scratch and claw and get back into it and then you know fight to the end there to pull one out with some real winning at bats by some guys you know Maggie you know stay into that side of the field getting them over to third base Kiner with two strikes getting a huge hit you know Marwin Marwin getting a big hit and then Hosey finishing off I mean those are that you know that guy's pretty good out there and they strung some really good at bats against them what is the level of concern and what can you further tell us about DJ and John Karla um DJ DJ got MRI today and and it was clean so that was really good news he got I think he got a quarter zone shot tonight and so we're pretty optimistic that you know hopefully he's back if not tomorrow the next day probably what was the other one yeah G just somewhere lower leg so he's going for MRI tomorrow not sure like if it's calf it's a part of the calf or what but won't we'll know more in the morning who else anybody else Dave is that something he did at that at bat Aaron on that no he actually did it warming up before the bat he came to me right before the inning said you got a hit for me so I think it was as he was warming up in the cage for that at bat yeah and what did you see from Michael King tonight it seemed like he didn't didn't quite have the same yeah I mean he loses Arias there and then you know probably was a little too aggressive there with Odor and left one look like middle down which is you know Rudy's power slot there and you know he smoked it so you know just not quite as sharp and they got him tonight but that's part of it too Pete there is something it may he's been dealing with for no couple days no no it actually he noticed it when he went to pinch it last night when he was getting ready to hit because he did he said he didn't do anything to it he doesn't you know he just said man that was weird and then today came in and before batting practice just you know said couldn't go so so as that was you know anytime DJ can't go you're like whoa so I'm hoping it sounds like the news is good though based on on the images and everything so hopefully we dodged something there