 We are going to study inverse of functions, okay, so we already did composition of functions now we have to Move to the next step now, which is inverse of a function We're talking about inverse of a function. I would like to know from you What do you understand from inverse function? What does inverse function mean? inverse of a function inverse of a function mean inverse of a function mean Okay Anybody unmute yourself and speak out, no worries I think I had given a brief idea about this in the class Right, so function is what function is at the end of the day functions are machines Functions are machines mathematical machines Which take in some input and give some output isn't it? So let's say I make a machine like this Okay, this is a machine and this machine is called the machine F Okay, so let's say I feed in some input to this machine. Let's say input a Okay, and this machine Gives me an output B Okay, you can think any situation as I told you in the class. Let's say I'm there's a machine where you put in water Okay, let's say water And this machine gives out you a juice Whatever juice you feel like, okay Now if I devise another machine, let us say a machine which Looks like this its name being let's say F Inverse now this inverse that I've written is a symbolic name Symbolic name means it doesn't have any functionality Functionality means it is it doesn't mean one by F It is just a name that has been given to it Okay, you may call it as G you may call it as H any function you want it This is a symbol that we reserve for writing inverse of a function. So this is just a symbol Please do not feed this as if I have written one by F Okay I think Anurag needs to mute himself Anurag please mute yourself Or should I mute you? Okay, let me Yeah, thank you Yeah, so this is a symbol which says there's another function by this name Where if you feed in juice, let's say if you put juice over here Or let's say B over here you'll get back a That means it takes in the juice part of the output and gives you the water So whatever you started with comes back Okay, then F will be called as The inverse of F inverse and vice versa Yes, you can say that the function is getting reversed You can say that the function is getting reversed Okay, so we say that F is inverse of this And this is inverse of F So both are inverses of one another So if this is the function then this would be the inverse Okay, and if F is the function then F inverse would be the inverse However both are functions only don't worry about it Okay, some typical examples that you would come across Would be log function let's say if I say there's a function log x to the base e Then its inverse would be any guesses e to the power x Yeah, now many people will ask me how it is the inverse I'm sure you have done log in class 11 Yes or no? So let us say Correct, correct So let's say if I put some input let's say F2 What do I get? Log of 2 to the base e Correct, yes or no? If this I put in the input of F inverse That means if I put this guy in place of x over here What will I get? I'll get e to the power log of 2 to the base e Which everybody knows is 2 Hope everybody is aware of this property very important property of log I'll write it down over here k to the power log of k of let's say m Let's say m To the base k is actually m this is a very important property If you're not aware of this property please note it down You'll come across this many a times in other chapters also like integration Okay, so what has happened? You put some input you got an output Now the same output you put in the inverse machine Or the inverse function you'll get the input back That is what is the role of it And you can do vice versa also For example let's say in F inverse I put a value of 3 Let's say here I put a value of 3 Okay guys do not think it is 1 by F of 3 Please I'm repeating this once again It is just a symbol No functionality should be attached to it Because when we come from junior classes We have a feeling that whenever something is raised to the power minus 1 It is reciprocal of it It is true if it is multiplicative inverse what we are talking about But inverses need not be always multiplicative inverse So this is a symbol only For example sin inverse x It's an inverse function of sin It should not be treated as 1 by sin x Getting my point So don't have this wrong notion that I am writing 1 by F of 3 Is that fine Now when I put 3 into this function what do I get I will get e to the power of 3 Correct If you feed this output to log function That means if you put f of e to the power 3 You will end up getting Something like this Which is actually your 3 back So whatever you had fed over here Has come back out as an output over here We will do that don't worry Aditi I will do lot of questions First understand the basic concepts Is that fine Is this example good enough Can you cite some other example where one Can you give me a function and it's inverse f of 3x plus 2 Okay if let's say f of x is 3x plus 2 Then what would be the inverse I think it's x minus 2 by 3 Very good See it works Let's say I put any input over here Let's say I put f of 1 What would I get 3 into 1 plus 2 That's 5 Yes or no Now this 5 I will put to this function So the output that I have got Which is 5 I am putting to the inverse machine I am sure we will get that Because this is 3 by 3 that's nothing but 1 So very good I think Aditi was the one who answered this Aditi So that's very good example of yours Now guys whatever we have discussed so far Is just looking at the situation from layman's perspective There are lot more into it Which I am going to discuss with you So let us do some analysis of whatever we have learnt In fact we need to know when does When can we find actually inverse Are all functions invertible Let us try to answer such type of questions So please note that f inverse x Exists If and only if f of x is a bijection f of x is a bijection That means your function has to be A 1 1 As well as an on to function In order to have an inverse Okay So not all function in the world Are having inverses Only those ones will have inverses Which are bijections Bijection means they should be a 1 1 and on to Now my question is Why Why only bijections are allowed to have inverses Not the others Why so much of partial treatment has been given to bijections Why not an in to function Or a function which is many one function Can have an inverse Anybody and mute yourself and talk Let's have it interactive You can speak also Skanda Because it won't work for some specific values Okay anybody else If we use If we choose for the function to be 1 1 And if you want to find the inverse of the function Then whatever input you're putting in Also has to be 1 1 Otherwise it will become a 1 to many kind of function If you choose to find what the Very beautifully explained Okay so I'll give you an example over here Let us say I had a function F which is mapping A set A to B And this function is not 1 1 So let's say I choose a many one function Like this So let us say A B and C And let's say I have 1 2 Okay so let's say The mapping goes like this Okay now what happens is When we are inverting this function It goes without saying that The inverse will take elements from set B And map it to elements of set A isn't it Yes or no Because F inverse works on the outputs And output will always belong to the co-domain That is your B So let's say if I had a 1 and a 2 And we had A B C coming over here So inverse So when you put A you got 1 So when you should put 1 you will get A back But you also got 1 when you had put B So putting 1 can also give you B back Okay And in that case you would realize This doesn't even satisfy to be a function Because functions are not known to be one many Yes or no So it is not I am writing it in red Not a function itself because I am writing it in red because this would be Not be called a function And hence many one functions Cannot have inverses Is that fine So that justifies Why my function should be 1 1 Is that clear everybody any doubt So if it is not 1 1 It has to be many one And if it is many one then its inverse would not be a function Because then the mapping would be 1 too many Remember the definition of a function A unique input Should only be giving you a Unique output Okay For example 1 is mapping to A 1 cannot map to anything else Right Now why it has to be onto So 1 1 many 1 1 to 1 is justified Right How is onto justified Let us say I take another example A function which is Not onto That means it is an into function So in into function We know that some of the elements of set B Are left unmapped So let's say I have A mapping to 1 B mapping to 2 Okay So this is an into function Everybody now knows what is an into function An into function range Is a proper subset of the code domain That means some element of the set B Would be left unmapped Like 3 and 5 Okay So now when I am making the inverse of this machine It would work from set B to set A Okay And in set B you have 1 2 3 4 5 And set A will have A B C Okay Now if you do the mapping you would realize Putting 1 will give you back A Putting 2 will give you back B Putting 4 will give you back C But what about these elements which are left unmapped Can it ever happen That in a function Some elements of the domain are unmapped No it cannot happen It cannot happen Right So what about inverse They don't even qualify to be called a function And that's why it becomes very very important That The function that I am trying to invert Must be a bijection Must be a 1 1 and onto And hence The question setter must always give you The domain and the code domain Of the function And we all know that the code domain there In that case would actually become your Range So remember If A function F defined from A to B Is invertible Is invertible Then B is actually your range We all know it's a code domain right But remember since it is onto It will actually become your Range also Is that fine Okay I am going to substantiate this I am going to substantiate this Let's say I have a function Defined from real numbers to real numbers As X square Is this function invertible Is this function invertible No Sankya says no Arushi also says no Everybody should respond No sir Absolutely correct The reason being this function is A many one function Easily I think a graph would help you out So if I do the graph of this function It will be a parabola Absolutely It's a parabola And if you sketch a line Which is horizontal line That is parallel to the X axis You would realize It will cut it at two points And hence it's a many one function That means if a particular output is coming Let's say four It can come from Correct So the mapping is basically following Something like this Okay I am just giving you a Simple mapping So if you invert it If you invert it Then it will Violate the condition This will become a one to many Which is not a function One to many is Not a function Is that fine But can you make it In and co-domain etc Guys moreover this is not This is not a function also As you can see R is basically the entire Real number from minus infinity to infinity Correct Does it take all the real numbers No it will only take positive values So let's say if I want to make it invertible Let's make it invertible If you take all positive You have to make the domain R plus Okay so somebody Let's make the domain R plus Okay and range Range Range it doesn't matter It can remain the same but still No, if range remains the same It is not an on to function then Because X Range should also be on plus Yeah X will only take positive values Yeah but The output will also be positive So it will be one to one No it won't be one to one Yeah but some elements of R will be unmapped So when you are reverting it When you are reverting it Yeah when you are reverting it Then what will happen Some of the elements of R Will not have any image Okay so I cannot claim minus four To come out from square of anything If my X is real Isn't it Both have to be changed Now you have an option Or you could take an R minus also Because you know I just need one of the branches Getting my point So this function you can say Yes it is invertible But this function is not invertible Okay now What is the inverse of this What is the inverse of this Rodex Very good Okay but whenever you are writing the inverse Make a habit that You are also mentioning its domain In that case it will become a range So this will also be from R plus to R plus Guys all you need to do is Just switch the positions That's it The range over here will become its domain And the domain over here will become its range Okay so very important The I'll write it down over here Important point Domain of F Domain of F Is the range of F inverse And Range of F Is the domain of F inverse Okay So if F is from A to B Okay I'm assuming F is invertible Then F inverse will be from B to A Simple as that Is that fine Now just Follow up question on this So for this definition You told me the inverse of the function Would be defined as root X Agreed What about had I defined my function From R minus To R plus As X square Then can you define F inverse for me Minus root X First tell me the domain R plus to R minus R minus And somebody spoke out the answer Can you repeat once again Minus root X Very good shunkin Brilliant This was a very simple example Where we could make out But when the function becomes complicated No no no Aditi You don't have to do trial and error That's what I was about to say When the function becomes complicated We don't do these trial and error things There is a logical way There is a definite way By which we derive the inverse of the function Let us understand that So my next discussion Would be on How do we find Inverse of a function From the given function How do we find F inverse Now In order to find F inverse I will go by the definition Yes shunkin will talk about that We will go by the definition of the function So When you had put the value of X In F It returned Y to you And when you had put Y in The inverse of the function It gave you X back Yes or no Yes or no This is actually Should I repeat this again Yes sir See when we put X value We get an output Y This is let's say input And this is my output isn't it Y is my output How does inverse function behave Inverse function Takes it Takes the output This is the output And gives you X back That's what our inverse function is all about From the very definition of inverse Correct This is X This is the need of my art isn't it I want to find F inverse X Now the answer is actually hidden over here This particular step says If you make X the subject of the formula From your given function If you make X the subject of your formula From the given function You end up getting F inverse Y And in this step You replace Y with X You end up getting Your final result Let me tell you how it works Let's say I have A very simple function I have a very very simple function here And the function is defined as 7 X Minus 1 We all know this function Is a 1 1 function Can you just mute yourself Sorry Should I mute you Some noise is coming from your side Yeah Yeah I have the power to mute people Okay Is that fine Now So let's say I want to Who is that Yeah So now I have to invert this function First of all Is it a 1 1 and an on to function Is it a bijection Yes it is a bijection You can know it from the graph also 7 X minus 1 graph will be a line like this Not passing through Origin by mistake It went very very close like this Okay Let me make it through my instruments Yeah So let's say it's a line like this Okay So it's multiplied by a horizontal line at one position Okay And the entire range is your R So both 1 1 and on to criteria are satisfied Now we'll find the inverse of it So here let us follow the first step The step number 1 is Make X the subject of the formula Make X the subject of the formula Actually tell me You did this step somewhere else also Where did you do this step The subject of the formula Or expressing X in terms of Y Proof of 1 1 and on to Not 1 1 on to Yes for finding on to For finding range you used to do this Remember The same step you have to repeat over here also Okay so I'm putting a smiley just for you to Take care of this So while finding the range you used to do this activity Okay the same activity you have to do While finding the inverse as well Yeah you will come to it So now the second step is Once you have done this So let me do the step over here So let me say it is Y is equal to 7 X minus 1 So I'll write X as Y plus 1 by 7 So this is the first step done I have made X the subject of the formula Now this fellow that you see here Is actually F inverse Y But what do I need I need F inverse X isn't it So what I need to do is I will replace my Y over here Let me write it as Next step That is step number 2 F inverse of Y plus 1 by 7 Right so replace Y in RHS With X And when you do that It becomes X plus 1 by 7 And this is what we will call as F inverse X So this is your answer for this case Is the process clear How do we find the inverse Because we are going to do a lot of questions on this Any doubt whatsoever Please unmute yourself and speak out Yeah yeah we will do a lot of examples Let me take another one Let me take another one Hope you can see this question over here Can you go back to the previous page Steps This one You want me to Drag the screen up and down I will just copy the second step Okay Let me know once you are done So that I can move on to the next board Yes done Alright so here is a question Let a function be defined from real numbers to real numbers And defined as 2X minus 1 by 3 First of all prove that it is a bijective function And hence find the inverse Now please pay attention over here Many a times in the exam You will be given that It is known it is a bijective function Find the inverse Then in that case you don't have to prove it is a bijective function But when it says prove it is a bijective function And hence find the inverse Which probably would be a 4 marker or a 6 marker for you Then you have to start from scratch You have to first prove it is 1 1 Then prove it is on 2 And then find the inverse of it So I would like all of you to give it a try I am giving you 3 minutes We will discuss after that I will be also sending you the assignments on the group themselves So kindly download it from the group Work it out In case you want to ask any questions You can use that group effectively Just take a snapshot of the question Post it on the group And ask for help Either if I am available I will do it Or some of your peers also can help you out So let's make use of this opportunity Okay done So just type done Or you can type out the answer also I don't think this answer is going to be so complicated That you cannot type it out So Sankhya has already typed Aditi can you just type it out Bhuvan has also typed in Okay I can see your answer Bhuvan Don't worry Aditi has typed in Okay guys So only Sankhya's answer is slightly different from others Arushi 100 Arushi 100 Okay Let's figure this out Is it a one one Of course from the graph you can say it's a line Line has to be one one Okay you cannot get a line at more than one point By a horizontal line But let's use our uniqueness So I will do all the steps properly So that you know how to frame your answer Okay So first site for bijective function That means you are trying to prove that This function is a bijective function Okay Under that you have to first prove for one one So for one one you will say Let X1 and X2 Be Two inputs from the domain of the function Look here this is the domain of the function Okay Yeah And one more thing I forgot to tell you the ground rule Whenever you are answering something You have to answer privately to me Okay So that nobody sees the answer of the next person And changes their answer Okay Alright So let X1 and X2 be two such values from the input That is your domain Such that Such that F of X1 is equal to F of X2 Okay That is We are trying to say two X1 minus one Two X1 minus one By three is two X2 minus one by three In other words Cancel of the factor of three You can do that because Three is a constant which is non-zero So cancelling of non-zero terms is permissible You can cancel of non-zero factors is permissible You can cancel of minus one minus one also So that gives you Two X1 is equal to two X2 And finally it leads to X1 equal to X2 As the only possibility Okay This is the only possibility Okay That means F of X is a one-one function Now let us prove it is onto So for Onto Now for onto what do we do We first write The function given to us As Y equal to that function Then we make X the subject of the formula And try to find the range somehow Three Y I can directly write it as Three Y plus one by two Now this step will be Overlapping with this step of finding the inverse So here any mistake you do That means that mistake will be there also So be careful over here Okay So this is going to be used again in finding the inverse Now What is the range of this function So I can see that All real numbers of Y are permissible So Y can take any real number Of real numbers Okay And hence I can say therefore F of X is an onto function So Since F of X is One-one and onto Implies F of X is a bijective function Is a bijective function Now Comes to the second part of the question Hence find the inverse of F Okay And finding the range That we had done while finding the range Will repeat the same step So X will be equal to Three Y plus one by two Yes or no This particular expression guys Remember This is what we call it F inverse Y And we need F inverse X So all you need to do is Replace Y with X So you can write this intermediate step Implies F inverse Y is three Y plus one That is F inverse X will be Three X plus one by two Okay It's very good practice that you mention It's domain and range as well So if you have to write a final answer You should write it like this So this is how you should write your final answer So Yes sir So why does X equal to One by two become Inverse function of Y Inverse function of Y Yeah Very good question Let us confirm this So this is now a verification step So why does Three X plus one by two Behave as the inverse of Two X minus one by three Let us say I put any input X one in my function What output will I get Two X one minus one by three Correct Let us put this as input to inverse Now this is my inverse definition So three X That means three into this whole thing Two X one minus one by three Plus one Plus one Whole by two Let us see What does this give out to us Does it give you X one back If not then my process Sadly goes for a toss So let us see this So three X gets cancelled I think one plus one Yeah it will become two X one by two So two to get cancelled Sorry I missed out This is Cancel Getting X one back So your input came back over here again That means this indeed Is the inverse of the function Is that fine Who has that Dhruva Is that clear Dhruva Yes sir yes Thank you Now something very interesting Probably I don't know whether you have noticed this or not Let us see the graph of these functions Let us see the graph of f of X And inverse of it Okay For that I will take you to GeoGebra So I will open it on my system only Of the software which normally I use In order to sketch the graphs Okay Let us look at this question itself Which we had discussed right now So I will plot the two functions Which I had given you One is the function Y is equal to two X minus one Whole divided by Whole divided by three Okay so this is the line you can see All of you can see this line Yeah I will also draw the graph of The inverse of this function What was that Three X Yeah I mean inverse was three X Third plus one by two By two By two Okay Now I will also draw one more line Just to make Just to make things very obvious So whatever I have drawn I am just marking it with my pen over here This graph That you see Is your Y equal to X line Okay This graph that you see Let me check which one is what Yeah This graph that you see Is Y is equal to Two X minus one by three Okay Let me change the color of my pen And this blue thingy that you see Is three X plus one by two What do you conclude from this particular Observation They all have a The symmetric about Y is equal to X Absolutely the conclusion that we draw From here is very very loud and clear That F of X And F inverse X graphs Are Symmetrical About Y equal to X line And this is not the only one example Which I am going to give you I am going to give you many examples Let's talk about e to the power X And LNX itself Remember I had given that as the The initial most example So what do you mean by the symmetry Symmetrical Symmetrical about X axis means They are mirror images They are mirror images of each other About Y equal to X line So if you treat this line to be A mirror Let's say I just make a mirror like this On this Then this would be the reflection of this And vice versa So if this is your object This will become your image Or if this is your object This will become your image So imagine as if there is a mirror Being placed over here And this mirror is shaded Both the sides I went on to different tangent This is the mirror Okay So the mirror has been shaded both the Let's say silvered both the sides Then what will happen is that The reflection of 2X minus 1 by 3 On this mirror will become 3X plus 1 by 2 And vice versa I didn't quite understand the question From the line Y equal to X Both will have the same regions I didn't get that About this line Or mirror images about Y equal to X Okay, let me give you another example Let me give you another example Let's say Let me erase this Let me just clear off this image Y is equal to LNX Y is equal to LNX Okay, all of us know the graph Of LNX like this And Y is equal to LNX Okay And if I draw Y equal to X You'll see how they're mapped All mirror images So whatever is the point over here It gets reflected and comes over here Are you getting it? In other words a point A, B Is becoming B, A over here So see this point It's 1, 0 After reflecting it becomes 0, 1 You see that So if you put in input 1 Output is 0 So if this is your input 0 over here Your output will be 1 over here Getting my point Right? So water, juice Juice, water Now got the analogy Okay, and you can do it for any point So if I take a point here Let's say this point is 7, 2 Okay After reflection It will come to Let me just drag this A little bit down So a point 7, 2 over here After getting mirror imaged After getting mirror imaged It will reach a point here Which is 2, 7 Getting it So basically what is happening Every point is mirror imaged About this mirror And that mirror is Y equal to X mirror And their reflections are getting formed On this particular graph Does it make it clear? Let me quote another example Did we do any other function Which was having inverse Any ready made result that you have I told root X, yes But root X I'll draw it only for This part So Y is equal to X square Okay Now please ignore this part So this part I'm scratching out Okay So I'm only considering This function from R plus to R plus Okay Now it's inverse Let me type that out also Y is equal to under root of X Y is equal to under root of X Y is equal to Under root of X Now all of you please in your mind Ignore this part of the graph Okay I'll also plot for you Y equal to X So that you know it is mirror image Y is equal to X Now all of you please pay attention I will do that Don't worry Now here you see Just follow the red line that I'm drawing This red line Is your function Do you see that This is the mirror This red line is the reflection Of this line about that mirror Okay So this down part is getting reflected up And this up part over here See the motion of my pen Is getting reflected like this Okay Let it intersect No worry It should be just mirror images It may intersect It may not intersect also I'll come to that There is a property on that also Getting my point So now I have a question for all of you Can you tell me a function Which is its own inverse A function which is self inverse Y is equal to X Y is equal to X is a very very good example Okay very very simple example So anything which is It's self symmetrical about Y equal to X That will be image of itself And hence that will be inverse of itself One by X One by X is also a very good example What about Xn minus X Sorry sorry Xn minus X Y equal to minus X Y equal to minus X That is also correct Okay so there are so many functions Which are self invertible also Getting my point So any function Whose graph itself is symmetrical About Y equal to X That is what I said Minus X So if you see This graph itself is symmetrical about this line So The mirror image of this guy Is down That is this one And vice versa So now Krish has a question Could you also show us the graph Of the same function from R minus to R plus Okay sure Let me just erase this X square There is also a function If X is great Less than equal to zero X square Yeah So I have now drawn the function X square but only for R minus Only for R minus You could exclude zero also No worries So this is a function which I have drawn For R minus Okay Let me Exclude the Equal to sign here Is that fine Now The inverse of this function Was actually negative root X I think Shankin gave the answer for this Correct Shankin See this Okay You can see here that The graph Again also draw Y equal to X line So that A lot of clarity is there Y equal to X See how beautifully they are mirror image This part That is the grey part Is been mirror image with this Red part And vice versa Sir can you zoom in That Do you appreciate that beauty Isn't it so beautiful Okay So please remember this as a property itself That if you have a function The graph of its inverse Would be mirror image about Y equal to X line Okay So before we move on to the properties We will take a few more questions Okay on finding the inverse Because that exercise is very important So let's go back to our slides Okay Here I will just write down That The graph of F of X Is the Mirror image Mirror image Of F inverse X And vice versa And vice versa About the line About the line Y equal to X Okay Does that make it clear Anything that you would like to ask Dhruva has a question Is the graph of cube root of X self invertible No No I will show you cube root of X Cube of X right You are asking cube root of X So let's say this is the cube root of X If I make X the subject of the formula It becomes Y cube That means this is your F inverse Y That means F inverse X is X cube Okay So these in this graph are different graphs Let me show you on the back So Y is equal to Cube root of X Cube root of X Okay And Y is equal to X cube Different graphs And the beautiful part of it is The relationship holds good Okay So you can see this part Is reflected as this part Okay This part Is reflected as this part And this part Is reflected as this part Okay And same here also So this part Is reflected as this part In short, they are reflections of each other Is that clear Dhruva Any questions Any questions here Okay Let's go back Let's go back to our Yeah board So now let's take a few questions From the same concept Let's have this one Oh oh oh oh I should have opened it in the next board No worries Next board Yeah hope you can see this question This is An option based question Let me just create a poll quickly over here So that Poll 1 Choose the correct option A B C So after you have Solve this Press on a poll button So I don't know how many of you are answering this correctly So the function is Defined from Where is the poll button Are you accessing it on the phone Or are you accessing it on the laptop So it hasn't come to the poll Okay it hasn't come Okay Okay Now Yeah it's here So 15 of you are there So I would like to see your answers If you want to type your answer Type it privately Very easy question Should not take you more than a minute So where is it The poll You can't see the poll Bottom left Bottom left if you're on the laptop Bottom left Yeah there's a small tab kind I can't find it I don't think I've got it No worries Dhruv You can just type the answer to me on the chat box Okay Yes You can just type A B C or D Whichever you feel Okay I've got one response Yeah I've got one response so far Okay Aditi Tashi I would request you If you can see the poll button Press on that poll button also Done Done okay Tashi saw your answer Let's wait for others to answer It's 45 seconds more I'll close the poll in 45 seconds Fast Press on the poll button As soon as you are done 5 4 3 2 1 Guys everybody should press Press something Because you're going to discuss now Okay 4 of you are still Not press anything Okay End of poll And you can see the result 54% of you have said Option B Okay And there's an equal distribution Between A C and D Okay let's see whether 54 is Option B is right or not Okay so I'm just closing the poll over here So it is already invertible We don't have to work hard to Proving it's invertible or not It's a bijective function Okay Now for inverse what is the Methodology that we have seen First we try to express X in terms of Y So cross multiply So 3XY Plus 4Y is equal to 4X Okay bring the X terms to one side So 4X minus 3XY is equal to 4Y And then you take X common You have 4 minus 3Y is equal to 4Y So X will become 4Y by 4 minus 3Y Okay So this cello is your F inverse Y Okay And actually that is what they wanted They wanted GY actually Okay it's not GX So we have to stop over here So this will be your GY What G is your F inverse actually Okay So option number B here Is the right option over here Well done to those who got this site Very good So can you Sorry Can you just blindly say that it's B Without looking at the domain The ranges of the functions See since they have given See none of these were there Then I would have given it a You know a thought But I know one of them Okay But in case In case you have Be just doubtful about it You should always check for it To be a bijective function first Okay In the exam also If the question carries a lot of marks Let's say there's a six marker question And they're saying Find the inverse of this function Now normally Yeah We mentioned that Prove it is one one And prove it is bijective But let's say the examiner Has forgotten to state that And you see that Oh it's a six marker Yeah Then you should definitely Try to prove it first That it's invertible That means it is bijective And then find the inverse Is that fine Sankyam Yes Yes Thank you Let's have another question So f of x is defined as 4x plus 3 Over 6x minus 4 x is not equal to 2 by 3 Show that f of f is x Okay now this is very interesting This actually means F is inverse Of itself Isn't it Because see what has happened If you feed x to a function You get some output This is your output Okay So let's say this is your input x The entire f of x is your output Okay Now this output You are feeding to the function itself So this output You are feeding itself And it is giving you The input back This fellow back What does it claim It claims f is the inverse of itself Okay Some of the examples that we already saw I think Arushi gave this example It was a very beautiful example If you put this It's self in place of x That means 1 by 1 by x You'll end up getting the input back Which is your x value Okay So here you have to show that This function is its own inverse By the way This doesn't suggest you to do that actually But this is an outcome of it This is the inference which I am drawing So for this You have to directly put In place of the x Sorry for writing in white in white So let's say this is your x In place of x You have to feed what You have to feed what 4x plus 3 by 6x minus 4 And show the examiner That you are getting back x From this last step It will be clear From this last question It will be clear that The f was inverse of itself But as of now Pretend as if you don't know that Okay Pretend as if you don't know the function Is inverse of itself Now do these activities one by one Let's have one minute to do this activity Everybody Find f of f Type done once you are done So that I can discuss that Shankar is done Shivam is also done Akshar where are you Let's wrap this up in 30 seconds more Should not take you more time Aditi Isha Tashi Dhruva Sankya Arushi Yeah, Arushi is done Okay Let's discuss it Boys and girls So when I say f of f Matt means I am trying to feed the function In place of x itself Okay Hope nobody is doubtful about How to find composition of functions We had a very good exercise in the class also For finding that out So what I am going to do is 4x In place of x Write the whole thing down Plus 3 By 6x In place of x again Write the whole thing down Minus 4 Is that fine All I need to do is Simplify this And see whether we get the same We get the value of x or not So let's cross Let's take the LCM over here So 4 times 4x plus 3 Plus 3 times 6x minus 4 By 6 times 4x plus 3 Minus 4 times 6x minus 4 Okay Please note that the denominators of both the numerator And the denominator would get cancelled out Okay So 16x plus 18x Which is 34x Right 12 minus 12 Gone for a toss Yes or no What will I get in the denominator? In the denominator I end up getting 18 from here And 16 from here That is 34 Okay 34 34 gone I am left with x Right So therefore f of f Is your x Now let me tell you something very Important over here When a function gives you the same Output as the input Okay We call such functions as i i means identity functions We call these functions as i Identity functions means The function is giving you same Output as the input Right So remember identity relations A will be mapped to A only Okay So whatever input you give The output will also be the same That means the function is called i Now this certain subscript we use over here I will talk about it little later on But meanwhile we are done with this part of the question Can we have the next part of the question What is the inverse of f Now inside your mind you know That this itself is your answer isn't it This itself is our answer But we will not show that explicitly over here We will work it out As if we don't know the answer By our method So what is the method here Do you want some time to do it Let me give you one and a half minutes for this Time starts now do it I am there Shankir don't worry I have muted myself for some time Yeah Shankir I know Shankir is not a break Just a 2 hour class Are you ready to discuss it guys Done everybody Done okay So let me just erase this Because it is taking up our space So what I will do is Again y is equal to 4x plus 3 By 6x minus 4 Let's cross multiply So 6xy minus 4y Is equal to 4x plus 3 Let's bring out the x together So let me bring the 4x to the other side We get x 6y minus 4 Is equal to let's take the 4y to the other side Okay So x is equal to 4y plus 3 Over 6y minus 4 Now please note This represents f inverse y Correct And you have to find f inverse x So f inverse x Just replace your y with x you are done So this and your given function over here See here f of x And f inverse x Let me bring them to the same screen This and this Aren't they the same thing Aren't they the same thing That means the function is Its own inverse And let me show you on the graph also That the graph of this function Will be the mirror image of itself Let me just Show you the The graph of the function So the graph was The function was 4x plus 3 by 6x minus 4 4x plus 3 By 6x minus 4 See this Isn't it? Is that clear guys? Okay Now Let me tell you one important thing over here Which we probably should have discussed In the beginning of the topic Even your relations are invertible Now Let's spend few seconds On inverse of relations See everything comes from relations only See function itself are relations Isn't it? So there could be inverse of relations And hence there is an inverse of functions Okay How do you define an inverse of a relation So let's say there is a relation defined from Set A to set B Where the relation contains Some A comma B Where your A comes from set A And B comes from set B Then inverse of a relation Is basically written as R inverse Again this is symbolic Don't treat it as if I have written 1 by R There is nothing like 1 by R It's just a symbol only Don't take it as if I am raising it to some power and all Right? So its inverse is defined from B to A And its elements would be The reverse of the elements of the R relation Okay So A still comes from set A And B still comes from set B Okay take an example Let me take an example Let's say there is a relation Defined Defined on set A And that A contains Let's say 1, 2, 3, 4 Okay So there is a relation R Which is defined on A Let us say the Rooster form of R Is like this 1 comma 2 1 comma 3 1 comma 4 2 comma 3 4 comma 2 Okay Now R inverse would be The same thing just switch the position Of these two elements 2 comma 1, 3 comma 1 4 comma 1, 3 comma 2 2 comma 4 Like that Here there is no restriction of 1, 1 and onto etc Because that was only in function case So just remember If you have been given some relation And they are asking you the inverse of that relation All you need to do is Switch the position of A and B Switch the position of A and B in that Is that clear everybody Any questions here Okay we will take more questions Just to instill a bit of Confidence in you all Let's have another one We did this Okay we will do this Question as well See Sankya I wrote it Like A belongs to A Or B belongs It doesn't mean A cross B and B cross A R will always come from A cross B and B cross A It could be any relation over here Let me just change the previous Relationship See what I am trying to say is that Here A is Related to B by some relation Okay don't think like it's just A cross B which I am referring to Okay so some relation is there Between A and B In the case of inverse B is also related to A By a inverse of that relation So these guys that you see They are mirror images about Y equal to X See the switching of the 1 and 2 over A Remember I was explaining on the graph That 1 comma 2 will become 2 comma 1 Is that clear everyone I have few more questions on this And we have already Taken up one of them That is this Consider a function from R plus that is positive real numbers To 4 to infinity Given by X square plus 4 Show that F is invertible Okay so you have to Show it's a bijection With the inverse being defined by this relation Let's try this out I'll give you Two minutes for this Is that sufficient? You don't have to verify it You don't have to verify it Verification is optional It's your call If you want to do it do it As the question has never asked to verify it Just type done once you're done Why inverse function becomes a very Important question or a hot question In the exam is because it will test you On various fronts It will test you on your knowledge of 1 1 It will test you on your knowledge of on 2 And of course it will test you On your knowledge of inverse That's why such questions are normally asked Under this chapter function I would say almost 80% of the time Because it is testing your Everything that you have studied in that function Okay so I can see done from Skanda, Shankin, Aditi, Akshar Can we start the discussion Along with solving So first 1 1 In order to show it's a bijective function That means it is an invertible function I have to show it's a 1 1 And on 2 So let's prove for 1 1 So again Let X1 and X2 B2 inputs taken from R plus Okay such that F of X1 is equal to F of X2 Okay That is we are trying to say X1 square plus 4 is X2 Square plus 4 you can cancel off 4 4 so that means X1 square minus X2 square equal to 0 That is X1 plus X2 times X1 minus X2 is 0 Now there are 2 possibilities arising from this 1 X1 equal to X2 And other X1 is equal to negative X2 But this is not possible This is not possible Because your X1 and X2 Both come from R plus Both come from R plus Okay so this is the Only possibility Only possibility Okay And hence you can say your function Is 1 1 function Now let's talk about On 2 Let me write it over here For on 2 For on 2 For on 2 we make Y the subject of the formula Isn't it Yes or no So Y minus 4 is X2 Now given that Your X belongs to R plus X2 should be Positive that means this should be positive Which means Y should be Greater than equal to 4 That means Y should belong To 4 to infinity That means your range Is equal to Your set B Which is your core domain Okay and hence my function F of X is on 2 So no doubt whatsoever Improving it's 1 1 and on 2 Hence it is a bijection Therefore F of X is a bijection Is a bijective function Okay Now Let's talk about finding the inverse Again the process is very similar To this process over here That we are using So X is equal to Y minus 4 Now we get two things from here Mind you one is Plus under root Y minus 4 Other is minus under root Y minus 4 But remember There could be only one inverse Possible This is a property which I am going to tell you next After this particular problem You cannot have two inverses They can only be one inverse So which of them Plus or a minus Plus or a minus guys Plus Why Because R plus Absolutely So you have to watch out Yes absolutely you have to watch out This guy That is R plus So you can never have The negative values coming over here For X X always will take Positive value so plus is accepted You can write Here because X belongs to R plus Okay So this is your I mean don't write it in the paper In your mind you know it's F inverse Y So F inverse X will be By the way they have not asked F inverse X They just wanted you to show F inverse Y Is this so you can stop it over here And say F inverse Y is under root Of Y minus 4 So whenever you are deciding Whenever you get two possible Values of X You decide looking at the domain Of the function Okay that which of them are you going to Accept Both cannot be accepted Remember Inverse of a function Inverse Of a function is Unique Okay So now I am going to talk about few properties And then we will again Carry on with the problem solving The first property Let me give a Topic Properties Of inverse function First property The inverse of a function The inverse of a function Is unique Is itself bijective Okay That means you cannot have Two inverses of the same function And the last statement Is very obvious That your inverse itself is invertible Isn't it So it should also be a bijective function So F inverse should also be a one one And on two Okay because both are inverses of each other Property number two If F is a function from A to B Okay And let's say F inverse Is the inverse of that function Defined from B to A Then Listen to this particular See the difference between this particular expression F O F inverse Will be an identity Function In B And F inverse O F Will also be an identity function In A Now remember I was talking about identity function A little while ago I will give you that later on I will write some subscript next to I What is the subscript I A and I B mean Or Sankya as a question Sir if it is an inverse function Then one of them is a relation The other one has to be a function No no no both are functions only Both of them are functions only Sankya Inverse relation is a different thing So if R is a relation Then R inverse is also a relation But if F is a function F inverse will also be a function What is this B and A over here What are they doing over here See when you say F O F inverse It means you are feeding F inverse X To F right So the input will actually come from here From this set It will come from the domain of F inverse And what is the domain of F inverse B Okay so this X will come from B Will come from set B Right so that's why the answer that you get This X will actually be a Value which will come from set B That's why I have written I B over F Are you getting my point On the other hand When you are doing F inverse F of X This X Will come from This set the domain of F Domain of F is what A so this will come from A That's why I called it as I A Okay so this X will actually Be coming from the set A Okay There is nothing very important about this Is just for your information No question will be asked That X is from A or B This is just for your knowledge Okay Third property This is a scenario where F O G Will be equal to G of F That means if you are seeing that Any function F and G They meet this particular Criteria F inverse X No no no F inverse X X will come from B only Because F inverse domain is B F inverse machine Will only take input from B Yes Correct So if you realize that There are two functions F and G Which satisfy this criteria It actually implies That G is your F inverse Or F inverse Or G inverse is your F That means there are inverses Of each other Is that fine That means under the Condition That two functions F and G Are inverses of each other Then only F O G will be equal to G of F As you can see from the previous example Each Both of them will be equal to X Each So in general F O G is not equal to G of F That is what we discussed in the class In general F O G Is not equal to G of F Correct But in case they are equal It should automatically give you an indication That F and G Are inverses of each other Have I made myself clear Okay Now Next property This I have already told you Sir we have not finished discussing the composite function That was the last property we did Krish The other property that we did That if F O G is one one Then G has to be one one And if F O G has to be on two F has to be on two, that was the last property Okay Sankya I will repeat this property once again See we normally know that F O G and G of F are not equal From the composite function property In general But there are some cases Where they may become equal And if they may become equal Then It can only happen When F and G are inverses of each other Got it Sankya Okay Now last question Okay last property Graphs of Graphs of I think we have already discussed this Graphs of The function and its inverse Are symmetrical Are symmetrical About Y equal to X line And We also say that If F of X X intersect Each other They will do so On Y equal to X line Now let me explain this property to you The first part of the property is clear That the graph of this And this are symmetrical About Y equal to X line No doubt about it Anybody any doubt about it We had seen a lot of graphs also On GOG graph The property you were talking about This was the property which I was talking about In the initial part I showed you E to the power X LNX all those scenarios About the intersection Intersection I am going to discuss now It says that If F of X and F inverse X intersect There is an if condition here That means they may not intersect also For example LNX LNX graph is like this E to the power X graph They will never intersect If they intersect They will intersect each other On Y equal to X line only This property is debatable actually Y debatable In most of the scenario it works fine Any example that we can take up Let's go to the GOG graph again The best way to look at this is The GOG graph Yeah Can we have a few functions That we have taken up Somebody was saying X cube And that rate Let's take that Y is equal to X cube And Y is equal to X to the power Cube root X to the power 1 by 3 And also draw Y equal to X Y equal to X So here you see the blue graph Is intersecting the red graph Correct And it is intersecting only At those points where it is also Intersecting Y equal to X line See here, see these three points Are you getting my point So F of X Is meeting Let me write it here X at Those points Where F of X is Meeting Y equal to X Isn't it? This is a very important Result for us because Let's say Hypothetically If a question comes like this for you to solve Let's say Then the solution of this Will be the same as The solution of this equation Are you getting my point So if somebody says Solve Let me change the color of my pen If somebody says solve X cube is equal to X to the power 1 by 3 That means you have been given this equation to solve If you realize that These are inverses of each other Then this equation is As good as solving this That means the solution would be 0, 1 and minus 1 Hope you know how to solve this You can factorize this like this You can factorize this like this X X plus 1 X minus 1 equal to 0 So solving this Is as good as solving this Are you getting my point Now this is mentioned in most of the books But I have my own personal opinion on this Since I have been teaching maths for so long I have come across some examples where This is not completely true That means All the solutions of this Is not covered under this I will tell you an example I know it is very surprising Because I am trying to go against What your book has mentioned Let me give you A simple scenario With your permission Can I erase whatever is written on the screen Yes sir Let us take A simple scenario Instead of Instead of X cube Instead of X cube Let's have minus X cube Now the inverse of it will be The negative of this If you are confused about this You can find it yourself Negative X cube inverse is Negative X whole to the power 1 by 3 Now here I realize That The red graph And the blue graph Meet at 3 points Getting my point But my red graph And my black line Which is Y equal to X Only meets at 1 point That means the other 2 solutions Let me call this as Solution A and solution B Are not covered Under solving This equal to this Are you getting my point So many books that you will see in the market They have stated this property But Let's come across such cases where The solution Of f of X is equal To X And the solution of f of X Equal to f inverse X Do not completely overlap I mean This is a subset of this solution So as to say It is still symmetric About the same line Let me show you Somebody was confused Let us see this red line Okay Now this red line Over here This blue line is the mirror image of it Getting my point This line over here That is your yellow line This orange line is the mirror image of it Okay, so Krish Is it clear how they are mirror images So there is no doubt about that Okay, yeah There are also mirror images about Y equal to X then that's a different thing Is that fine? Now, coming to Sankha's question Yeah, that's why I want to know Why we consider Y equal to X is the baseline Uh, because we considered everything No See Y equal to minus X is working fine in this case May not work fine for other case but Y equal to X will always work fine Sankha, does that answer your question Right My concern was to show you that when you solve y is equal to f of x when you solve this equation that is f of x equal to f inverse x and when you solve this equation let me write it as this this all the solutions here will not be covered under this some of them will be covered some of them will be missed out what about yes very good example shibab has a excellent example what x if you see one max equal to one by x that means you're trying to solve this it should have infinitely many solutions but when you do one max equal to x it only gives you two solutions that is what I was trying to claim here shibab that not all the not all the values of x here would be covered under this that's a very excellent example shibab has cited getting my point okay this is some extra knowledge which I am giving you probably this may not be very very useful in solving your school level questions school level questions are pretty straightforward okay so these are the properties that will be dealing with more or less and before we end the session I think 20 minutes we have we can do a lot of questions to make ourselves comfortable okay she be on fire okay let's have more questions yeah let's have this one so we have a function from r plus to minus five to infinity and the function is defined like this show that f is invertible I'll make your life simple you don't have to show it as invertible because I'm sure you have done this kind of question why we were doing one one and on two right so leave this so leave this part so leave this part right now just show that f inverse is equal to this f inverse y is equal to this two minutes for that good enough right two minutes done guys how is it going hope nobody is stuck yeah I think an initial bit of problem you may feel in making x the subject of the formula okay nothing to fear in case you're stuck let me help you out see what was the process the process was we used to make x the subject of the formula isn't it so treat as if you have a quadratic equation like this okay you have a quadratic in x you have a quadratic in x now I'll make use of the quadratic equation formula yeah Krish will discuss that so quadratic equation formula says x is equal to minus b plus minus under root of b square minus 4 a c remember this is playing the role of your C by 2 a okay so minus 6 plus minus by the way 30 this will become 36 plus 36 times 5 plus why okay yes I know and we can take out a 36 out that will become plus minus 6 under root of 1 plus 5 plus y by 18 drop the factor of 6 throughout so it'll become minus 1 plus minus under root y plus 6 by 3 now here comes a very very important question should I take a plus or should I take a minus now the answer is hidden here you want positive values of x to come out isn't it will a positive come out if both where minus is no because negative is already negative and negative of a square root will also be a negative quantity correct so plus is the only option now I can further justify it if you take if you take your y is minus 5 onwards then y plus 6 will be greater than 1 onwards yes or no okay that means minus 1 and a quantity greater than 1 will always give you a positive value that's why minus is dropped but remember it will not happen every time so don't be under the impression that it's always the minus which is dropped no not necessarily always you will have some question where plus may be dropped also that depends upon your that depends upon your domain of the function so finally this is your f inverse y and is it not what we wanted to show yes this is what we wanted to show done and shown is that clear yes let's see the questions can you explain that part to me again okay actually see is this step clear finding x in terms of fight using quadratic equation formula okay so this is just a simplification part now how do you choose whether it should be plus or minus that depends upon what is your domain given over here domain is r plus correct if you would have chosen let us say by mistake let's say if you do this in place of sorry this in place of your plus what will happen this is negative this is negative this is negative that means everything will become a negative on top right that will violate your condition that x belongs to R plus so negative cannot be chosen so this cannot be chosen this would be wrong only positive can be chosen does that answer your question yeah R plus correct in the question is R plus that's why you have to take R plus correct good so let's now move on to few more questions of the similar nature first part of the question you can skip show that it is one one you can leave it skip this part find the inverse of this function easy it is a one-minute question not more than that so again same story yeah same story let's cross multiply bring x on this side so x is equal to 2y by 1 minus 5 okay this is f inverse y but probably the question is asking you inverse of the function so you should always in state f inverse x and that will be 2x by 1 minus x is that clear everyone easy question any question please stop me okay I'm moving on to the next part okay let's have this question interesting one so now it's a discrete type of a function which is given state with reason whether following functions have inverse first one does it have an inverse yes or no okay let me have a quick poll on this a for yes b for no okay this poll will only be for 30 seconds a for yes b for no a for yes b for no okay just present does this function have an inverse first one only first one that is against is over are you able to see the poll no sir no oh yes sir okay you can only see you can't see all four options okay fine just press A and B one of A and B don't press press the A and D okay fast fast fast everybody should answer this everybody should answer this only six of you voted what about others okay end of poll 70% of you say no you are correct you are correct the first function will not have an inverse reason reason you can see it's a many one function one is also mapping to 10 2 is also mapping to 10 3 is also mapping to 10 4 is also mapping to 10 so it's not 1 1 it's not 1 1 means not bijection and hence it is not invertible okay good enough same thing for B poll is on sorry same thing for the second one poll is on answer is end of poll answer is no because 4 is mapping to 5 also 4 is mapping to 7 also so again not one one let's have the last one also quick who's pressing D went 30 seconds for this okay yes it is a function it is it has an inverse because it's a one one and onto function because all 2 3 4 5 they have unique images and all the 7 9 11 13 they become your range so answer is yes it has an inverse yeah somebody's at D also okay so we'll move on to the next question we'll move on to the next question let's have yeah so there are two questions over here you may assume that both these functions are invertible okay we don't have to prove it's one one or onto okay of course we are assuming here that they are invertible find the inverse of these two functions let's target the first one first the challenge here is to make extra subject of the formula that's where most of the people make the six has done this type done so that we can discuss it okay yeah I think he has muted himself okay in the interest of time we'll take this up because just five minutes I left so the first one assuming it's invertible my job here is to make x the subject of the formula okay so first thing let me bring the two on the other side okay it becomes e to the power x minus e to the power minus x by e to the power x plus e to the power minus x okay now multiply both the numerator and denominator with e to the power x cross multiply now take e to the power x e to the power 2x common so e to the power 2x will have y minus 2 coming from here and minus 1 coming from here correct and bring this down to the other side that is minus 1 minus of y minus 2 is that fine so e to the power 2x is nothing but y minus 3 by 1 minus y agreed that means e to the power 2x is this anybody has any doubt till this step so now do we have to take a little bit I'll tell you at this step it is fine yeah see where I cross multiplied I split this up early that step also I should write I wrote it as e to the power 2x y minus 2 plus y minus 2 and here it is e to the power 2x minus 1 so what I did I brought this e to the power 2x on this side and I brought this term to the other side correct so this will come with a 1 extra over here so that is what I've written over here so y minus 2 minus 1 and will become minus 1 minus 5 minus 2 like that now coming to your question shankin I'll take ln on both the sides so it becomes 2 to the power x is ln 1 minus y by y minus 3 okay so your x will become half of ln 1 minus y by y minus 3 at the point this fellow is now your f inverse y correct so your final answer that is f inverse x will be half ln 1 minus x by x minus 3 this is ln means log to the base e everybody knows that okay I would request you all to verify this result graphically on GeoGibra or you can also use it on Desmos over your phone so Desmos is also very good software you can install it on your phone this is good for mobile devices okay and you can use GeoGibra for your laptops your laptop slash desktop don't install GeoGibra on phone that will make your phone hang yeah this application is too heavy okay so with this we can now move on to the next question quickly let's say this function is a invertible function we have to find the inverse of it okay so second part of the question we'll do it on top so again in the interest of time let me take this up so y is equal to 1 minus 2 to the power minus x so 2 to the power minus x is 1 minus y so minus x is log of 1 minus y to the base of 2 so x is negative log 1 minus y to the base 2 okay if you want you can further write it as log 1 by 1 minus y that is also fine so this is your f inverse y correct so f inverse x will be nothing but log 2 1 by 1 minus is that fine so with this will stop here I will send you more questions on finding the inverse that is a very important part of our functions chapter whenever we meet next we'll talk about binary operations that's a part which is left off so relations is over function is over we'll talk about binary operations I'll give you a bit of idea about odd even function also because I have seen some questions in your book like that related to odd even functions okay so next class agenda is I'll write it down on the next page so next class we'll talk about number one odd even functions number two binary operations is that fine okay I'll share this with you over and out from my side bye bye I think you'll have another class in some time yeah sir how often will we have these online classes you'll be informed about all the class timings on the group okay sir and so what so what will be the last day of online classes it's what he asked yeah the last day yeah all these classes will happen till you are being informed on the group okay so even I don't know what is the last day don't worry about it I think Manta will keep you updated on the group okay