 Okay, so when you factorize it like this you end up getting two possibilities out of it my dear students What is one possibility? One possibility is x1 x2 can be zero. Okay, that's number one possibility Other possibility is x1 plus x2 is equal to zero. Okay Now remember for you to show it so one one function x1 equal to x2 should be the only possibility Right, but in this case you see that x1 equal to x2 is One of the possibilities that is not the only possibility There is one more possibility of x1 plus x2 being equal to zero now Many a times You will end up getting more than one possibility This will definitely be there always right, but apart from that whatever you get Try to first negate it negate it means try to prove it is not possible Okay, if you're unable to prove that the other condition is not possible Then you have to accept. It's a one-one function. For example in this case x1 could be negative of x2 and I cannot negate it because there can be two real numbers Where one real number is the negative of the other that is possible, right? So in this case even this is possible Both are possible So x1 equal to x2 is not the only possibility and Therefore we have to accept that this function is not a one-one function It is not a one-one function Okay, if you come from graphical angle also, you'll be able to see that very clearly see we all know that X square graph is a parabola. It's a parabola like this, isn't it? Okay, so if you choose a particular output, let's say I choose the output four Okay, now this four can come from two inputs One is two and other is negative of that two which is negative two that is what this expression is actually trying to say It is trying to say that boss any particular output can come from two different inputs Right. It is not always that they have to be the same. It can let's say four can come from two and minus two Nine can come from three and minus three Twenty five will come from five and minus five. That is what this expression is actually trying to say and We cannot negate it. We cannot say that. Okay. No, this is not possible It is very much possible. It is very much possible and therefore we have to accept that this function is Not going to be a one-one function Okay, more over graphical angle is always there to supplement your cause So if you draw a horizontal line like this, okay, it's going to cut the graph at more than one point, right? Okay, now I will tell you something very interesting over here After seeing this example Many people would get an image in their mind that oh x square is always a you know It's always not a one-one function. See whatever is not one one is called many one. I'll talk about it little later on Okay, so many people many students after seeing this example Create a wrong notion in their mind that oh x square is always a many one function Or it is not a one-one function. No, it is not like that. I Can make x square also a one-one function? Can you tell me how? Can somebody tell me how can I make x square a one-one function? Anybody can I make x square a one-one function? It is all dependent upon the domain my dear friends domain has a very vital role to play here Please do not Exactly Gagan, okay Your your function must be read in light of the domain many people we don't read the domain and all We just go to the definition of the function. This is the definition of the function, right? Don't judge a function by its definition Okay, I've copied it from don't judge a book by its cover. Okay, don't judge a function by its definition Always read the function in light of its domain if possible co-domain also We'll tell you where to read co-domain where co-domain important becomes important So if I take the function to be from r plus to r Then let's see what happens Okay. Yes, so if you can take that also, okay now if I start the same process Let x1 and x2 be from your domain, which is r plus Okay, such that Such that f of x1 is equal to f of x2 Okay, that means you are saying x1 square is equal to x2 square Okay, that means x1 minus x2 times x1 plus x2 is equal to zero That means x1 is equal to x2 is one possibility Okay, and x1 equal to minus x2 is another possibility x1 equal to minus x2 is another possibility But you can say that two is not possible Two is not possible. That means the second possibility is not possible Second possibility is not possible. Why it is not possible. How can we have two real numbers? Sorry, two positive numbers being negative of themselves It is not possible See here your x1 x2 come from r plus We write at the top of the screen How can two positive numbers be negative of each other? Is it possible? Is it possible that two positive numbers are negative of each other? Okay, zero is a possibility, but zero again makes x1 and x2 equal And you can say sir Zero zero is a possibility x1 x2 both are zero Okay, but zero them zero zero themselves make it come under this in a first criteria So my dear friend, what is what I'm trying to say is that if x1 equal to x2 is the only possibility Is the only possibility? Okay Then only you can say this function is a one-one function. This function is a yes It is a one-one function Okay, so please read a function definition in light of its domain domain is a very powerful thing Later on you learn a chapter called inverse trigonometric functions Okay, where you will make all the trigonometric functions that you have learned sine cos tan c cos c cot Etc as one one function and deal with them. Okay by curtailing By curtailing their domain curtailing means making it smaller or making it shorter Any questions here? I'll take a pause and you know, I'll take some questions from you any questions here my dear Clear is this clear? So if this is clear, can I ask you some questions? Okay, let's do some problem practice on this so that we get confidence about it. Okay, my first question to all of you is is is Xcube a One-one function. Okay Is Xcube a one-one function? I was waiting for somebody to ask me a question I'm waiting for somebody to ask me a question I'm waiting for somebody to ask me a question Let's see who asked me that question Who asked me that question? Okay, nobody's asking me that question. So let me ask myself that question. See whenever I define a function You should always ask the domain or the code domain or You ask me. Okay, so switch it that's something which you need to inquire from the question Right now the question is given in such a way that I have not mentioned any domain and all If you want you can take the Entire exhaustive domain But always first try to see whether the domain is already provided to you by the examiner or not Okay, so let me take it from R to R This is an important thing So I was waiting for somebody to ask me the question what domain to take what domain to take So you can take all real numbers to be a domain now, please solve this Okay, and substantiate this or supplement your answer with a graph Yeah, anybody any idea? I'll put a poll button on for you. Just press yes If you think it is a one-one Press no if you think it is not a one-one. Let's have a minute and then we will discuss the same Let's see. What is your response? Okay, Ashwarya, we'll discuss it here. Let's see what happens Up till now 27 of you have voted It is 20 of you. You have 10 seconds more dear. Come on everybody at the count of five. We should all vote Five four three two one go everyone Everyone please Don't worry. I don't get to know who has voted what okay So you are anonymous for me right now Okay, so I can see some 36 of you have voted. Okay, let me end the poll and This is a close call. This is a very very close call 58% of you have said yes Okay, 43% of you have said no Okay, so it's a very very, you know close situation for a contestant like me Okay on the KBC seat. So let us see what happens it so Let me start with the uniqueness test Of course, I will also draw the graph for you to know this So let X1 and X2 be two inputs taken from real numbers. Okay, such that F4 of X1 and F4 of X2 are same. Okay, again, I'm repeating. What does this step mean? This step means That the two inputs X1 and X2 are giving you the same output is giving you the same output Now, let us see whether the inputs themselves are same or not or is there any other possibility which we cannot negate Which we cannot negate. Let's talk about it So f of X1 equal to f of X2 means X1 cube is equal to X2 cube Okay Now I have seen even teachers doing this mistake and I'm requesting you don't do that Don't cancel out the powers from both the sides. Please please please please please, okay? So instead what you should do you should bring both the terms to one side Okay, and factorize it. I'm sure all of you know just recall Okay from your childhood days When you learn the factorization of aq minus bq. What is it my dear? a minus b a square plus a b plus b square, right? Did you recall this everybody? Okay Good old days, huh? So now this can be factorized like this Okay So now this gives me two possibilities my dear One possibility is this is zero That is X1 is equal to X2. That is one possibility Other possibility is this guy is equal to zero. Okay Now somehow, I don't know how but somehow I Need to show that this is not possible Unless until X1 X2 are both equal to each other and zero each Are you getting my problem statement now? I have to show that this is not possible Not possible till X1 and X2 are equal to zero each That means it takes you back to the same position Otherwise, it is not possible. How can you show this to the examiner because without showing that you cannot? Justify that this function was a one-one function. So how will you show this to the examiner? Just don't write statement. Oh, this is not possible. Okay. He will not buy your statement You have to justify it with proper reasoning Can anybody do that for me? No, no, no, no Kirtana. That's a good thought that you have. I really appreciate but example cannot prove something Okay Last class when I was talking about relations proof. I told you examples should be used to disprove things not to prove things Okay, what if one example doesn't give you zero, but some other example gives you zero Right, we cannot cite infinitely many examples, right? It's beyond our scope to do that Okay, so in general, how do you prove that this is not possible? That means this how do you prove that? X1 square plus x1 x2 plus x2 square cannot be zero Unless Unless x1 and x2 are equal to each other and zero each, how do you prove this? Now? Let's say this is our question How do you how do you prove that? Okay, don't worry. There's a way to prove that also If you write this term as just you know, correct me if I'm wrong x1 plus x2 the whole square Okay Okay, or you can write x1 plus half x2 the whole square Okay, let me write it like this in a better way. Let me write it like this x1 plus half of x2 the whole square plus Three by four x2 square So what I've done the left-hand side, which is this term. I have written it like this Right, you can square it and check that it will give you the same result Okay, if you square this it will give you x1 square This will give you one fourth x2 square And it will give you two into x1 into half x2 Okay, and there's already a three by four x2 square waiting outside. Okay, so these two will combine to give you x2 square Half and two will get cancelled. It'll give you x1 square plus x1 x2 Okay, so this thing which is on the left-hand side could be written like this Now it is all made up of perfect squares. You can see there's a perfect square here Okay, so this term is greater than equal to zero always This is also a positive number and this is also a perfect square So this is also greater than equal to zero, right? So how can two terms? Who which are greater than equal to zero give you a zero unless until both are zero if both are zero? That means you are saying this is also zero and this is also zero In which case what will happen x2 will become zero and from here x1 will also come out to be zero That means unless until x2 both are equal to zero each We cannot have this expression equal to zero getting my point. I am again repeating it I'm again writing it once more so that you know Everybody is clear regarding that. So I'm just erasing whatever I have written so far Try to understand once more Okay, this is a very good problem actually Now what we do is we start from the left-hand side and we write this term on the left-hand side as x1 plus half x2 whole square Plus 3 by 4 x2 square Okay, now you're adding two perfect squares to each other We always know two perfect squares are always greater than equal to zero Okay, so you both are greater than equal to zero The sum will also be greater than equal to zero correct and they can only be zero you can further write that this can Also, this can only be zero if x1 and x2 both are equal to zero correct So if this term has to be zero it can only happen when x1 and x2 both are zero which bring it Which brings it back to the fact that x1 and x2 must be equal Else else Else this term will always be greater than zero Else this term will always be greater than zero it cannot be equal to zero Are you getting my point? So hence this possibility that you have that is x1 square plus x1 x2 plus x2 square equal to zero is not possible is not possible Till x1 and x2 both are individually zero and if they're individually zero means they are equal anyways So x1 and x2 equal to each other is the only possibility This is the only possibility Okay, and yes Since it is the only possibility happening we say that the function The function is a one-one function. So the answer to this is yeah Yes Okay, but let me substantiate this with a graph also Let me substantiate this with a graph also graph of x cube I'm sure most of you would have seen this it is like this. Okay, let me show this on the tool to you Okay, I normally use a tool called GeoGibra for that. So let me show that on GeoGibra to you. Yeah This is a software which I had asked in the first class to download GeoGibra classic 6 so y is equal to x cube There you go. I hope you can see this So any horizontal line that you're going to draw hair any horizontal line that you're going to draw hair is Always going to cut the graph only at one point That means every image Every image will come from a unique pre-image Okay, so two different images can never give me the same image that is what we have just now proven Okay, any questions here Any questions here, please ask if you don't ask me questions, then I'm going to ask you questions Okay, so please do not cancel the power then say x1 equal to x2 is the only option that would not be correct Now a little while ago I was I was trying to say tell you a very important thing when you are proving something is not a one-one function When you're trying to show that a given function is not a one-one function Okay, you can always cite example Okay, so in order to you know disprove In order to disprove one-one nature of a function Okay, you may always cite an example an example is sufficient. Now. How do you cite an example? You may say that there are two different numbers from the domain which is giving you the same output If you're able to show such an example You are done with proving it that it is not a one-one function Okay, so you can you can just quote some two different, you know inputs belonging to the domain and say boss see For these two inputs. I'm getting my same answer So it is not a one-one function But to prove it is a one-one function But to prove it is a one-one function Let me tell you my dear students you have to go by the generic proof you have to give a general proof for it You cannot cite example Okay, okay. Let's take few questions very simple question is Mod of X function is Mod of X function, okay Defined for all real numbers a One-one function, let's have a poll. Yes, if it is No, if it isn't let's have one and a half minutes for this take your time. You still have one minute. Think about it Dear all you have last 15 seconds to give a response last five seconds everybody please five four three two One please please please vote everybody. Okay guys you have done this See the result that is there 50 50% vote this is a rare event actually, you know, you guys have done this Right, okay now the answer to that to this question is No, it is not a one-one function The reason being I can easily cite two example, let's say minus two and Two mod will give me the same answer which is two Correct, so there are two separate inputs. There are two inputs where X1 is not equal to X2 But still is giving you the same answer Okay, just a simple example like this and you could have said that boss. No It's a not a one-one function It's not a one-one function. Right. This is what I was talking about citing an example Siting an example. So if you just show that for two separate inputs Minus two and two you are getting the same output That means it is following one many to one not a one to one Correct and secondly, I'm sure you would have also seen the graph of mod function graph of mod function is like the V symbol Correct. Have you all seen the graph of mod function? I'll show you on GeoGibra also. Okay on GeoGibra the command is apps apps X Okay, this is a graph of mod function V shaped So if you draw a horizontal line like this If you draw a horizontal line like this You can see that it is cutting the graph at more than one point. So see two points are getting cut Okay, so to get a same output of two you can use minus two also and you can use two also That is what is what that is what we say as not a one-one function. It's a many one function So why is such confusion? Everybody made a mistake almost 50% of you made a mistake Is the idea clear first of all any question, please ask, okay, let me give you another question F of X equal to let me give you a Defined as X by X square plus one This function is defined from R to R This function is defined from R to R. That means domain is all real numbers Even the core domain is all real numbers Is this function one one function? Okay. I'm giving you two minutes time Okay, only after you're done with the you know uniqueness test Please Put forth your response on the Pole over here. Okay two minutes for this time has already started. Okay Don't be in a hurry to vote. Don't be in a hurry to vote. Take your time idea Okay Last 20 seconds everybody. Okay, the time has already crossed Uh, only 27 of you have voted. I would request you to vote in the next 15 seconds Then we'll decide whether it's one one or not a one one Let's end the poll and see what is the verdict by given by the janta There you go There has been some calculation mistake by the software. How can it be 53 percent and 50 percent? Okay, it must be 53 and 47. Anyways So again, there's a very very close call. There's a very very close call almost 50 percent of you have said no So there's a split of opinion. Okay I don't understand why is this so Okay, let me just quickly give you the final verdict over here. See Many of us May not give an example in this case Okay Some of you Right whose brain work very fast They can cite an example and say sir if you try putting two And if you try putting half Okay Let's check. What is the answer when you try putting two into this function output will be two two square plus one Which is two by five Okay, when you put a half it becomes one fourth plus one Okay, which is nothing but half divided by five fourth Which is half into four by five which is again two by five right Right. Now. What do you see here is that for two separate inputs two and half See i'm just drawing a rough kind of a arrow diagram Okay so Apart from many other inputs. There are two inputs. I'm just showing them two and half Which is both pointing towards two by five Okay, when such a thing is happening, we don't call such function as one one. So the answer is no this function is not a one one function Okay now I understand your question over here. I understand the feeling over here. How am I supposed to think of such examples? Right, my brain is not a calculator, right that I will you know, automatically get two inputs, you know, I'm not you know uh A robot. I'm a basically human being. How can I think of such two examples? Which is give me the same output Even if you're not Don't worry. Don't worry. This is not the end of the world We'll go by the regular approach. What is the regular approach? We'll say that this is one approach Okay, now people ask me is this sufficient enough to show it's not one one. Yes It is good enough an example To show that the function is non one not one one function. Any examiner in this world will accept this as the answer Let's say I go by the normal route I will say let x1 and x2 Okay belong to real number Okay belong to the domain of the function correct such that such that f of x1 is equal to f of x2 That means there are two separate inputs, which is giving me the same output. So I'm going with this assumption Which means this is equal to this If you cross multiply This is what you are going to see Correct, no doubt about it. No doubt about it. Let's bring everything to the one side Let's bring everything to one side good enough From the first two terms take x1 x2 common Take x1 x2 common Now I can clearly see that x1 minus x2 could be taken out as a factor and it will give me 1 minus x1 x2 as the other factor So this leads me to two possibilities This leads me to how many possibility two possibilities one being x1 equal to x2 And the other being x1 x2 equal to 1 Okay Now do you realize why I took two and a half because there could be a case of this nature So even if you put three and one by three it will work Even if you put four and one by four it will work Even you put hundred and one by hundred it will work Any two inputs which are related by this you know relation Will also give you the same output Any two inputs which are related by this relation Will also give you the same output so x1 equal to x2 is not the only possibility This is not the only possibility It is one of the possibilities Are you getting my point my dear students here? So this is something which you cannot negate Right you cannot falsify that you cannot see that cannot say that boss that cannot happen It can happen Why not can't there be any two real numbers which are reciprocal of each other? Right it can be there's so many cases There can be so many cases where one real number is reciprocal of the other And both of them will give you the same output under this function Okay Right so this is also one of the possibilities and hence x1 equal to x2 is not the only possibility if it is not the only possibility We have to say it's not one one Okay Now i'm sure many of you will not be able to plot the graph for this Many of you will not be able to plot the graph for this Okay, because this is not a function which we normally deal with our day to day, you know working It is not the simple function like a linear function or a quadratic or a cubic or a log or a modulus Okay, so here graph approach may not work. Some of you may plot it Some of you may not be able to plot it. So don't waste your time And i'm going for the plot if you know it you can supplement your answer Any questions here, please ask Okay, so this is like a you know matter of luck if it strikes you You are done if it doesn't you have to go by the normal route. Okay. Let's take more questions. Let's take more questions Let me define a function like this So there's a function from n to n n means what? n means what? natural numbers Yeah, natural numbers This function is defined like this my dear students f of x is x plus 1 by 2 If x is odd and it is defined as x by 2 If x is even Okay x here is belonging to natural number. Okay, so needless to write this but i'm still writing it x belongs to natural numbers Is this function 1 1 is f of x 1 1 function take 1 and a half minutes for this and Please press on the poll button. Let me relaunch it Okay, 90 seconds you'll get for this time starts now Don't be in a hurry to you know give your answer. Just wait Solve it This time I want a clear cut distinction. Okay. They should not be like 50 50 or 49 51 like that. Okay Take your time. You still have around 45 seconds more another 20 seconds another 20 seconds Okay time is over, but still i'm giving you another 30 seconds Okay Just 24 of you have voted. I would like everybody to vote by the time it's two minutes Now at the count of five i'll stop. Okay five four three two one Go don't worry. Even if you have not got it. Please press something. Please, uh, put the poll Choose a poll one of the poll options Okay There you go 56 percent of you say It is a one-one function And 44 percent of you say it is not a one-one function, right? Here I can give you a very simple example f of two Everybody, please tell me what is f of two? What is f of two? One one very good. What is f of one? One one End of game This is the end of the game Two different inputs is giving you same output What does it mean? It's not a one-one function Not a one-one function. See most of you made a mistake here. My god. What is happening? Most of you said it is Yes, a one-one function. No, right 53 percent of you said I think 56 percent whatever Most of you were wrong. Why? This is a simple example that you can make out from that it is not a one-one function Two different in we can take many other examples Okay, try taking four and three f of four and f of three f of four will give you two f of three will also give you two Right, so two separate inputs is giving you the same output Another example. There's no dearth of examples in this case I understand in the previous case it is very difficult to guess an example But here it was so guessable Isn't it Okay, till we are confident about this we'll keep doing problems. Okay till we are confident about this we'll keep doing problems Okay Let's take Let's take uh This question. Yeah, there's a function defined from end to end Okay, as x square plus x plus one x square plus x plus one, okay The question is is f of x a one-one function Is f of x a one-one function? Okay, again, I'll put on the poll button This time, please I'm giving you two minutes for this. Paka paka two minutes Please Take your time before you respond. Okay The previous one was a mass blunder Take your time Don't forget your domain is all natural numbers. So your inputs must come from natural numbers Okay, last one minute is remaining Six of you have voted so far good Last 20 seconds my dear last 20 seconds. Please back up Again, I'm timing this out because This is the time in which you're supposed to solve these questions Don't give yourself infinite amount of time to solve questions Okay, while practicing also do not give yourself infinite amount of time. Okay time is up But just to you know, give everybody time to press on the poll button another 10 seconds. Please press on the poll button five four three One go go go go Okay End of poll and of poll. Let's see the result Aiyo Ramakrishna again very very close 42% and 58% why? I would love to see the graph only in one direction only everybody is yes or everybody is no. Okay, if we are split like this Okay, it means you have understood the concept differently. Okay. Anyways, let's let's try to talk about it now Again, let me start with the fact that let x1 and x2 be two such inputs from natural numbers such that f of x1 is equal to f of x2 Okay, that is you're trying to say x1 square plus x1 plus 1 is equal to x2 square plus x2 plus 1 right One one gone for a toss bring everything to one side Bring everything to one side Please do not cancel powers. Please do not cancel factors if you're doing that you are losing on the information Canceling of terms will lead to loss of information. Okay loss of information means wrong decision. Okay So this is nothing but x1 minus x2 times x1 plus x2 And x1 minus x2 Zero okay, so if you take x1 x2 as common x1 minus x2 as common you end up getting x1 plus x2 plus 1 equal to zero x1 plus x2 plus 1 equal to zero So this leads to two possibilities my dear one being x1 is equal to x2 which I told you will always come out Right, no matter whatever function is given to you x1 equal to x2 will always come out Right, so we have to be sure that that is the only possibility for you to prove it's one one okay And the other possibility that comes out is x1 plus x2 is equal to zero But guys tell me frankly Think and tell me is this possible can two natural numbers Added to one give you a zero ever No, sir No, sir To minimum natural number also you pick up it is one and one, right? So I'll give you three at least Isn't it cannot give you zero so this is not possible This is not possible. So this is the only possibility If this is the only possibility Then what is the answer should be? Yes, it is a one one. Yes, the function is a one one function Okay, don't just judge it by oh, it is a quadratic quadratic is not supposed to be one one No, don't judge the book by its cover Look look at the function in light of its domain Look at the function in light of its domain without domain the function is not worth You know approaching Again Slightly disappointed guys and girls. So I'll give you one more question Okay I want you to be confident in the confident in this because This is a very important vital part of your function chapter Let's say I have a function from a real number minus two To real number minus one Everybody understands this. What does it mean? It means domain is all real numbers except for two And co-domain is all real numbers except for one. Okay This function is defined as x minus one by x minus two Okay, is f of x One one Is f of x a one one function? Okay Two and a half minutes for this. Okay Time starts now. Let me put on the poll Yeah, two and a half minutes exactly after two and a half minutes I should have got your response and then we'll discuss it You're one more minute boys and girls from now onwards One more minute for you Last 30 seconds last 30 seconds after that, I'm going to stop the poll Okay Time is up time is up. Let's discuss Let me stop the poll Everybody please press on the poll button. I'm going to share the poll result Fast fast fast fast Okay There you go Ha this time I'm happy that at least, you know, there is a good amount of difference between Yes shares and no shares Okay So 70 percent janta says yes Yes, it's a one one function. Let us see whether janta is right or not. Okay Now first of all before I solve this question, I would like to ask you Why is this condition given to you that two should not be included? And why is this condition given that the range or you can say the core domain cannot be one Very simple Since there is an x minus two here you cannot input anything which is making the denominator zero So two will make the denominator zero. So two is not allowed in your domain Okay Why does not equal to one? So let's say if you are making this equal to one see what blender will happen That means you're saying this That means you're saying minus one is equal to minus two or one is equal to two which is not possible That's why mine one is removed from your domain. Okay So try to make sense from what is given to you in the question. Don't you know accept it and blindly Now let me start with The uniqueness test. Let me say x1 and x2 be two such inputs taken from The domain of the function. That means x1 and x2 are two Real numbers minus two, you know type of numbers Such that f of x1 is equal to f of x2 Correct That means you are saying x1 minus one by x1 minus two is x2 minus one by x2 minus two Correct cross multiply Cross multiply. Okay, open the brackets. You'll get x1 x2 You'll get minus x2 minus two x1 Plus two here also you'll get x1 x2 minus two Minus two x2 Minus x1 plus two So a few terms you can directly cancel out See don't cancel factors. You can cancel something which has been added or subtracted Okay, so do not cancel out factors which you are not sure of the value See, for example, if you have x1 x2 and x1 square People cancel out x1 and x1 Right if you do that you miss out on the fact that x1 could be zero Correct, you lost this information You only got the information x1 is equal to x2 So you lost this information if you cancelled factors But if that's it is 4x1 is equal to 4 then 4 and 4 can be cancelled because 4 is not zero So anything which has a possibility of being zero do not cancel it. Okay This is since a request from all of you Now from here, I can say let me bring terms to one side. So I can say x2 Minus x1 equal to zero. Oh, this is the only possibility. There's no other term, you know expressions coming out So yes, there is this is the only possibility And if this is the only possibility you will have to say yes. Yes. Yes. This function is a one one. Yes Is that fine Any questions here? So well done Janta. You were correct 70% of you who said yes, you were absolutely correct Okay, the third method which I have not told you right now Of identifying a function as one one or not is the calculus method Okay, have you done calculus a bit of you know calculus differentiation in class 11th everybody Somebody please respond. Have you done a bit of differentiation in class 11th? Yes. No Hmm somebody say in the chat Have you done a bit of differentiation? No It's absolutely no response. Okay, let me put a poll button Have you done a bit of differentiation in class 11th? Yes or no Hey, why am I getting split response? Those who are saying no probably you didn't attend that class, huh? 79% of you say you have done a bit of calculus. Okay. Anyways See, never mind. Even if you have not it is just for your information that you can keep this approach. Okay calculus method basically is a method which says If a function has to be one one then The derivative of the function must either be greater than zero Or Less than zero For all values belonging to the domain of the function Or means exclusive or Exclusive or means either it is greater than equal to zero Okay, sorry greater than zero equal to is also sometimes accepted. Okay I'll tell you where it is sometimes accepted. That's why I'm writing it with a dotted section Either it is always greater than Or equal to zero Or or is exclusive or over here or it is always less than equal to zero Basically, this has to do with the increasing decreasing nature of the function And right now just note it down. We'll talk about it when we do application of derivatives Okay, so just note it down as of now Okay, we'll not discuss anything about it We'll take it for the later part of the discussion when we have done Uh application of derivatives. Okay. So as of now Just follow The uniqueness test and if you can supplement it with a graph Nothing like that Okay, so now with this We'll move on to another type of function called many one function Okay many one functions So what's the typical many one function something which is not a one one function plain and simple example is Not a one one is a many one function. So example is your x square function from r to r Okay, what is not one one becomes many one automatically it is either one one or many one It cannot fall in both the categories. Okay So what is not one one is many one Therefore, I will not spend any time discussing about the identification of many one function Because whatever fails the one one function test will automatically pass the many one function test simple as that Okay, so can we just quickly recall in our minds when can a function be many one Many one function will be when it is When a horizontal line will cut the graph of that function at more than one point. So we can say Yes, number one a horizontal line A horizontal line Line cuts the graph of the function That's the graph of the function At more than one point At more than one point Just the opposite of what we learned in the one one function test Secondly, what is the uniqueness test say in one one function x1 and x2 equal to each other is the only possibility here We have to show that and the uniqueness test x1 equal to x2 is not the only possibility Not the only possibility Okay Under calculus test, which I'm going to talk about later. So I'm writing it in yellow under calculus test f dash x could be greater than Or less than zero Right in the domain of the function Or you can say for all x for some x belonging to the domain of the function Okay, as of now leave this third test because we have not done application of derivatives So bottom line is Whatever is not one one becomes many one Whatever is not many one becomes one one. That means they are complementary to each other right Now a question for all of you think and answer think and answer If there is a set a which contains n elements and a set b which has got m elements, okay How many many one functions are possible How many many one functions are possible from set a to set b Think and answer Let me give you two scenarios where your m is less than n And where your m is greater than equal to n. Okay, who will tell me for the first one Just put your response on the chat box When m is less than n how many many one functions can you form from that set? How many many one functions can you form from that set a to set b? Anyone how many one one functions were there? How many one one functions were there Zero zero and how many total functions were there? What are the total number of functions when you have set a as having m elements and set b having an element Sorry set a having n elements and said b having m elements. What are the total number of functions? Remember I told you in the first class m to the power n right remember this formula It's always the number of elements in the core domain Race to the power number of elements in the domain right forgot everybody. Hmm not revising Okay, so out of total functions, which is m to the power n Okay, zero functions are one one that means all the functions will be many one Correct. So the answer here will be m to the power n Because a function can either be one one or many one Since there is no one one that means every function that is getting formed is a many one Yes or no Is that fine Everybody any question regarding how m to the power In the same way you can answer me how many many one functions are there if m is greater than equal to n Think carefully take a clue from this Tell me what is it m to the power n minus mp n Okay, see It's simple mathematics Your number of one one functions where m p n And total functions is m to the power n. So whatever is left that is going to be one one function What is there to be no? Uh, you know unsure about clear any questions This may come as a direct question to you. Okay, let me frame a question on this If you set a contains three elements set b contains Five elements Okay, how many many one functions are possible How many many one functions are possible? Let me see who gives me the response first Just give me the expression no no need to calculate no need to calculate the value exactly Give me the expression for the answer. That's it Yes, I should have got the answer by now anybody can speak out unmute yourself and talk So which which formula will you follow? You will follow the second formula. Sorry, let me move my this thing. You will follow this formula, isn't it? So total functions will be five to the power three And out of that we have to remove we have to subtract the one one functions one one functions will be five p three Okay to be accurate. This will be 125 five p three will be five factorial by two factorial which is 60 Answer is 65 one one functions can be sorry 65 many one functions can be formed Is that fine? Nothing very great. Okay Now I'm not going to take up questions on many one because What is not one one becomes a many one? Okay, I'm going to directly take you to On to an into functions On to and into functions. So let me start with on to first. What is on to functions also called as surjective functions? Also called as surjective functions Okay, many books will also call them as surjections It's a combination of the word surjective and functions. Now. What is an on to function? A plain and simple definition that will help you remember this for lifelong Is where range is equal to codomain? Any function where range is equal to the codomain will become an on to function Normally what happens? Excuse me. Normally what happens? Codomain is a bigger set Is a super set of the range isn't it? Okay But when range and codomain become the same That means Every element in the set B becomes an image Every element Let me write down every element. Let's say the function is from a to b Every element in set B Is an image of some element That means it participates in the mapping process Then that particular function would be called as an on to function a typical example Through aro diagram. I can give you Let's say this is a function. This is a set a this is b and there's a function And the mapping is like this One two three. Okay. You can say okay. So something like this Now this is what we call as a many one on to Okay, see One one and many one they are complementary On to into they are complementary. So you can choose the combination also. It can be one one on to It can be one one into now. What is into I'll tell you here itself Into functions aware Your range is a proper subset I hope you have learned the language of sets Range is a proper subset of the co-domain that means At least one element of set B is left out Again, I'll give you an example for into functions. So let's say Something is like this a b c and one two three four. So let's say the mapping is like this Okay, as you can see here four is left out Okay, it is left unmapped so this is an example of One one into function as if not don't care about one one I'm just giving an example of how an into function can look like Into function is where At least one element of set B At least one element remember at least one element. They can be more than one At least one element of set B is left unmapped Getting my point. Is this definition of onto into clear? Correct, okay, I'll give you an arrow diagram. You need to tell me What type of function is this okay a b c One two three four five Okay What type of function is this type it out on your chat box Type completely with respect to one one many one and into onto Everybody let me see who gives the fast answer fastest Kirtana, can I also have Whether it is one one or many one Along with into onto Very good ashwarya Very good smutty very good Nice nice gagan also nice very good There you go guys hats off very good. So it's a many one Why many one my dear because Two of the inputs are going to the same output b and c are pointing towards two That's why many one Why into because three four five are left unmapped So it is an example of many one into function Is the idea clear okay try this one out Think carefully and answer everybody should participate a b c one two Okay Now tell me tell me many one or one one into or onto so I want a combination Very good puja very good very good. Kirtana ashwarya nice Yashashwini very good. Awesome. Awesome hats off hats off. So it's a many one On to yes because no element of set b is left unmapped So every element of set b is becoming a part of your range That's what is the definition also saying When your range becomes core domain, it becomes onto when your range is lesser than your core domain That means two elements in the set b are left unmapped Then it will become a into function Now how do we identity now the tests for both of them will be the same if something is not onto it is into Automatically right just like if something is not one one it is many one. So I don't have to do separate tests for it See either I have corona virus or I don't have corona virus right Single test is there. No, there is no two tests for testing whether I have and one other test for not for Me not having corona. Okay. So if I if it is positive I have corona if it is not I don't have corona simple as that So one test will be there for onto and into also just like we had it for one one and many one. Okay, so what is this test The test is very simple my dear the test is The test is again look out for the Definition of the function complete definition of the function Look out for this set b This b is your core domain Needless to tell you all of you by this time know that it is called core domain So I'm not giving some new information to you If your range just find out the range Okay, find out the range step number one now for this You have already done this in class 11th haven't you Right, you have done finding the range in class 11th, isn't it domain range, right? Never mind while solving questions. We will revisit that again. Don't worry if your range happens to be Equal to the set b that is given to you. That means set b and range are the same sets Then it is a onto function. Then it implies it's an onto function Okay, else else if range is Not equal to b or you can say a subset of b Let me write it down if range is a subset of b Okay, then you will say it's an into function It's an into function or a non-surjective function Simple as that So what is the bottom line bottom line is This fellow finding the range if you cannot find the range Then this question is beyond your range Sorry for that bad joke Let's take questions. Let's take questions and do Testing whether a function is one one or not. Oh, sorry onto or not. Okay, let me give you a question Okay, just give me a second. I'll pull out one So I have a question for you There is a function from r minus seven by five To r minus three by five Okay The function is defined as The function is defined as f of x being three x plus four upon Five x minus seven. Okay question is Question is Is f of x onto Is f of x an onto function? I'm putting the poll button on. Okay, let's have two minutes for this now If you are not able to find the range, please let me know ASAP. Okay, that's sir. I have forgotten how to find the range We will not waste our two minutes instead. We can discuss that But you have to tell me on the chat box. We have forgotten how to find the range Yes, anybody There's still 30 seconds left In other 10 seconds, we'll discuss so boys and girls, please put forth your response on the poll Okay at the count of five, I'll stop five four three two one You go stop Okay Okay, so janta goes with yes 55 percent of you say yes, but it is a very close call again 55 45 the difference is hardly Anything so let us discuss it. Let's discuss it whether it is onto or not Okay Now all of you please pay attention All of you please pay attention. The first thing is Finding the range of the function Finding the range of the function many of you Who have forgotten to find the range of the function? This is a good exercise for you In order to find the range the first thing that you need to know is the domain Okay Many people they start directly finding the range without knowing the domain See dear students There is no output till there is an input Right, so input is the domain Output is your range Right, so there is no existence of an output Till there is an input So domain decides the output. So this is already given to you Thankfully else you would have to find that out also So if a question comes on finding the range And they have not mentioned you the domain then you have to find the domain of the function as well Okay, but thankfully the domain is given to us in the question why 7 by 5 has been excluded Any reason any special reason why 7 by 5 has been given such steps treatment of being excluded Right denominator should not be equal to zero in any rational function, correct 7 by 5 will make it zero. So remove it Okay, so that's why they have removed it Now how to find the range the process is very simple. All of you please pay attention. We equate the function We equate the function to y Okay, then what do we do we make x the subject of the formula So after this we make x the subject of the formula Make x the subject of the formula Everybody understand the meaning of making something the subject of the formula That means write y or write x equal to some function of y. Okay, how do I do that very simple? All of you, please stay, uh, you know follow the process just do a cross multiplication Just do a cross multiplication. Okay, collect all your terms containing x Any questions so far? So what I did I brought this 3x on the other side, uh, before that I would also like to open the brackets over here Let me open the brackets. It'll be more clear to you. So I made it 5 x y minus 7 y is equal to 3x plus 4 Okay, and I brought the 3x on the left and 7 y to the right Is that fine? And I got this step Okay, from here I can write x as 7 y plus 4 by 5 y minus 3 Okay And treat as if this is the function given to you Okay Now let me ask you a simple question over here If you're allowing your x to take real values other than 7 by 5 That means this should also give me real values other than 7 by 5 Okay, so first it should be real in nature Correct If it has to be real in nature the denominator here must not be zero That means 5 y minus 3 must not be zero That means your y should not be 3 by 5 y should not be 3 by 5 see I've crossed it out right means Your y can be anything any real number But it should not be 3 by 5 because if it is x will become not x will become undefined because your denominator will become zero getting the point This is the trick normally that we follow to find the range of functions So your y can take Any real numbers except 3 by 5 This is your range of the function This is your range of the function anybody having any doubt in this finding the range Okay, now people ask me sir Real number okay. I understood why the 7 by 5 and how did you account for that 7 by 5 in this see Even if you want this expression cannot take 7 by 5 try type putting it to 7 by 5 see what blunder will happen Okay, very interesting thing will happen. Okay. Let's cross multiply If you cross multiply If you cross multiply See something very very interesting will happen. Everybody please pay attention, huh? This will become 35 y minus 21. This will also become 35 y plus 20 35 y 35 y will go off What does it mean? Minus 21 is equal to 20. Is it ever possible? Is this ever possible? Not possible Isn't it? So anyways Your x cannot take 7 by 5 Right because if it is then something like this will come up and which is not possible Okay, so you don't have to worry about the 7 by 5 There cannot be any y possible for such inputs They cannot be any y possible for such inputs So the only thing that we need to bother about is that the denominator should not be zero Okay, so now just check it out. Just check it out So this is your b set And this is your range set. Are they equal? Are they equal? You'll say yes, sir. They're equal domain and range both are equal And if they are it means what your function is On to function your function is an on to function onto function means Whatever is the core domain that whole core domain becomes your range That's what you need to understand nothing else nothing else Is this fine any question here? Okay, let me give you one more Simple question to you know solve very very simple question Let's say. Yeah, any anybody has any question? No, no, no a core domain can include So range can become 7 by 5 Input cannot become 7 by 5 Kirtana Let me go back. Kirtana has a question See Kirtana your Range can include 7 by 5 But domain cannot include 7 by 5 Answer can be 7 by 5 Output can be 7 by 5 input cannot be 7 by 5 And this says output cannot be 3 by 5 input can be 3 by 5 Are you getting my point Any questions here Yes, sir. Okay. All right. So let me ask you a simple question Just to end the session on a very very positive note If a function is defined from Let's say r minus 1 To set a Okay, r minus 1 to set a Okay, a is unknown to me And the function is defined as x by x minus 1 Okay Suggest a set a Suggest the set a this a such that such that f of x is an on to function Suggest the set a such that f of x is an on to function Indirectly, what am I asking you? If I'm asking you a suggestion for set a indirectly, what am I asking you? Core domain Core domain and it is on to so indirectly again, what am I asking you? Range range range absolutely All you need to do is my dear Get the range If you get the range that is your set a Okay, so that could be a suggestion for your set a okay. So range Let's find out the range. Remember, what did I tell you the process? Make x the subject of the formula. So first, let me cross multiply Okay So take x common By bringing this x to this side bringing this y to the other side Okay, so x will become y by y minus 1 Okay, now if you want this function to be a defined function, we can say y should not be one Else everything is fine else everything you can put in place of y correct So your set a Which is nothing but the range of the function will be all real numbers except one That's it. This is done. Okay In the next class that we meet we are going to talk about Uh more questions on onto functions. In fact, I'll send you some, uh, you know questions to solve also Okay, and we'll also talk about Composite functions, that's a very very important part for your function chapter composite and inverse If possible, please read it out so that you know the class flow is not affected