A one-one (injective) function is defined as a function where if f(x1) = f(x2), then x1 must equal x2, meaning distinct inputs produce distinct outputs. This can be verified using the horizontal line test (graphically) or by proving that f(x1) = f(x2) implies x1 = x2 algebraically. Exponential functions f(x) = a^x (where a > 0, a ≠ 1) have domain R and range (0, ∞), with behavior depending on the base: if a > 1, the function is strictly increasing; if 0 < a < 1, it is strictly decreasing. The y-intercept is always (0, 1), and there is no x-intercept.
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Week 5 Summary - Jul 21, 2026, 8:00 PM
Added:Am I audible?
Yeah. Okay. So, we'll start the class in a second. Just just give me a minute and then we'll start the class.
So today we'll be discussing the summary for week five.
And the first thing in week five is we start with one one functions. What a one one function is.
So the definition of one function is let's say you have function [laughter] f then it's said to be a one one function if at your image is same let's say there are two points in the domain where your x1 and x2 are in the domain of f >> can you hear me ma'am your voice is breaking Okay. Is it better now?
We don't hear them.
So I hope I'm clearly able to you all now and my voice is not breaking. Is it still breaking? Is >> no ma'am.
>> No ma'am.
>> Very clear.
>> So okay. So we'll start with the definition of 11 one functions. It says that if you have to let's say this is your domain.
So by now you all of you must be familiar with what's a domain a co- doain and range and all of those stuff.
So this is let's say your domain. And if I say that I have two two members here two elements here x1 and x2 and they're mapping to the same element. That means this is this is what fx1 and fx2 mean right? This same point is fx1 and same point is fx2. If this happens then we say that the function is not one. That means if two elements are mapping to two uh if two elements are mapping to the same element then that's a problem then then is when we say that it's not one not one one function. So the definition says what should be the definition then it should be that if ever the image of two elements is same then those two elements should also be same right that means if the image is coming out to be elements should have been same then I could say that it's a one one function so whenever the image of two functions ma'am your voice is breaking >> is it breaking again yes ma'am Not too much >> but the case for everyone or uh >> okay it's the internet issue maybe then there's there's not much the domain uh this is the definition for your one one function in simple terms what we're saying here is that if you have two different elements here then their images should be different here this should be mapped to some different element and this should be mapped to some different element. Now if there are two elements that are mapped to the same element this should mean that both of these elements are same they are not different elements then we say that it's a one if so in simple words if ever you see that two elements are being mapped to the same element it's not one one otherwise it's 1 one that's that's also what this definition tells us about one one functions and then coming on how to identify one one function how to tell if a function is one one or not So first is ways to check one oneness or this is also called injectivity by the way.
Injective functions So first one is your graphical method the horizontal line test.
So horizontal line test says that you just take the graph of that function and so far all the kinds of function that we have seen we have seen how to plot them right we have seen how to pl plot a polomial we have seen how to plot or sketch quadratic functions then linear functions all of those we have seen right and some of the popular ones we already know how they look like like the sine function, the cos function, then the tan function. All of these we already know how they look like. If you don't know, they look like this. They're oscillating.
Some of the cos one looks like this. So they're definitely not one one functions, right?
And then you'll see one or two types of functions more and you'll be able to identify by their graph only that one or not. So what's the test for a graph? It's like this. If your graph is let's say this then horizontal line test says that if you draw any horizontal line parall to by horizontal line we mean a line parall to your x-axis. So if I draw any line parall to x-axis that line should only cross this graph this is the graph let's say this is the graph of fx this should only cut the graph at one point then I'll say that it's a one one function.
So for example in this case if I draw this line this is a horizontal line because it's parallel to x-axis but it cuts your graph fx at one 2 and three places that means that it's not a one one function not one one now let's come to this one about this one and to make things simpler. Let's take this.
>> This is one to one >> and this two. This is a straight line.
So this is also this is let's first come to this one. This is one one because if you draw any horizontal line here parall to x-axis this line this line this line any horizontal line will only cut this graph. If this is your f(x), then these lines, this line only cuts here. This one only cuts here. This one only cuts here. That's why each of these horizontal lines cut this fx graph at only one place. That's why it's a one one graph. It's one one. Same goes for this one.
Any horizontal line is only going to cut this graph once. This is 1 one as well.
Now if you have this quadratic one all these lines this one one is enough for you to see that it's not a one one function it cuts your graph at two places right here and here that's why is not.
And if you look at the cubic one, it's like this.
This is also one one because it only cuts your graph one place. One here, one here and one here. That's your graphical test. The horizontally test that you can use to check whether one or not.
>> Excuse me, ma'am.
Then the second one is using your definition.
Use the definition.
>> Ma'am, your voice is breaking.
Hi. So, just give me a second. I'll share my screen again. There's some internet issue on my end. So, I keep getting disconnected.
Just one second.
Okay. So, yeah, someone was saying something. Someone did someone have a doubt somewhere because >> Yes, ma'am. I have a doubt.
>> Yeah.
>> Ma'am, it's a fun.
>> Yes.
F is X to Y is called one to one function. If F if X1 = to FX2 belongs to Y then X = 1 sorry X1 = X2.
>> Yeah.
>> So if X1 = to X2 so the point are same.
So the function will be same. Is it a one? Is it a one one?
>> What?
>> I couldn't hear your question. The last part of it.
>> Oh, okay. So can I repeat again?
>> Yeah.
So function f x2 y >> here x is doain y is co-ain >> okay that's not how you say it but okay >> then >> is is called one to one if f of x1 = to f of x2 belongs to y then x1 = x2 belongs to why what are you trying I don't get it what's your question >> so is this I am confused about this statement >> this is the definition what's your question >> so is it a correct statement >> you're confused about this is this a statement in some book >> no I question.
>> Uh ma'am, sorry to interrupt, but I just want to confirm if this is getting recorded or not.
>> Do one thing. Just drop the screenshot of this.
>> I'm sorry, but your voice is breaking.
>> Yeah, just drop the screenshot of that question if you have it somewhere. Yeah.
Yeah. This is getting recorded.
Okay. So, I mean this is going to happen because I as I said I have some internet connectivity issue here.
So is is it better now?
So you >> Yes, ma'am. Clear.
>> Am I still not clearly audible >> now? It's okay.
>> Am I audible?
>> Yeah, ma'am. Yeah.
Am I ordering?
Yes, sir.
I have some in in internet connectivity issues at my end. Okay. And I want you to because Okay, can you guys hear me now?
>> Oh, so the def how do we use a definition to prove that a function is 11 one. Let's say you've been given a function f(x) equals to this function and then let's say let's assume or let's assume this thing to be true. Let's say for two two elements x1 and x2 the image of these two elements is same. Let's say if this is true then I have to prove that these two elements are also same.
Then is when we show that it's a one one function right?
>> Okay. So so this is what you have to prove. Let's start from here. If this is true, then this would mean that your x1 + 2 is the same as x2 + 2, right? And this is going to prove that your x1 is the same as x2, right?
So that means we started from here and we proved that x1 is equals to x2, which means this is a 1 one function. So this one x + 2 is a 1.
Now let's come to one more example where we we are going to use the definition.
Let's take this one by x + 5. Now if I'll take this to be the function if I take this to be the function then we'll use the definition again fx1 is equals to f_sub_x2 and we have to show that x1 is equals to x2. So if I start from here I should get x1 = to x2. Let's start from here. If f(x1) is equals to f(x), that means 1x x1 + 5 is the same as 1x x2 + 5."
What does this mean? This means that x1 + 5 is the same as x2 + 5. Right?
And what does this mean? This means x1 is the same as x2. So we have reached our conclusion that x1 is equals to x2.
That means this function is 111.
Now let's come to one more function which is let's take x² this time or x² + 1. If this is your function then you will have x1² + 1 is equ= to x2 + 1. Let's start from here. So you will get x1 squar = x2. Now from here can I conclude that x1 and x2 should be equal?
So if the squares of two this means if the square of two elements or two numbers is same then the numbers should be same which isn't true right we if you really want to solve it negative >> yeah it can be negative or if you just want to do it arithmetically you can do this x1 square - x2 square =0 which will give you x1 + x2 x1 - x2 equals to 0.
Now if this is zero then one of this should happen right either x1 either this factor is zero x1 is equ= to x2 because they're both linear or x1 is equ= to minus x2 right either this one is 0 or this one is zero is this okay that means you have two two ways to get this zero so either you can take x1 to be equal to x2 or x1² and x2 X2 square are going to be equal even if X1 X2 are not equal but X1 is the negative of X2 right so that means not 1 one now if I do the same for X cub + 1 or X cub let's say if I do this for X cub + 1 then what's going to happen so we'll have X1 cub + 1 is = X2 + 1 and this will be x1 cq is equals to x2 cq.
Now if you have two numbers the same concept does not apply here. The negative of if you will take the cube of a negative number you'll get a negative number. If you will take the cube of a positive number you'll get a positive number. So the negative positive thing does not happen here. And further if you want to check it there's identity for x1 cub - x2 cq right which is x1 - x2. But if you don't want to use that's why I say that for this one if you want to use it uh if you want to use the definition here that's fine too but the better approach here is to use the graphical method right the horizontal line test and the graph of this one looks like this so it is x cub and y - 1 so it will be here if this is 0 comma 1. So your graph for xq + 1 looks like this.
And you can clearly see from here that it's a 1 one function. Is that okay?
So you just you just need to know how to plot the graph because that's easier in this particular case.
Otherwise you know that the even if you know that minus this one's going to uh require this one's require you to know the formula for x1 cubus what's the formula for x1 - x2 cq it's x1 - x2 and then >> x1² + x1 x2 + x >> this right this is the formula so if you will find this then there are two ways one is when x1 But now if you equate this to be zero, these two are equal. This implies this, this implies this.
This implies you have two things here.
One is this implies either your x1 is equals to x2 or x1² + x2² + x1 x2 is 0. Now can this happen? If I'll take x1 is equals to let's take x1 is equals to 1. And if I'll take x2 is equals to 0. No, this is not going to become zero. Right?
So you'll have to see if this has a solution or not.
And >> x1 + x2 square.
>> No, this is not x1 + x2 whole square. If you really want to write it as the whole square thing, it should be x1 + x2 whole square minus of x1 x2.
This is this expression and then you will say that this is zero.
So basically what you can do is uh I don't know if it has a solution or not.
You'll have to check if there are two numbers or not whether x1 that's two numbers x1 and x2 for which this expression becomes zero. And I think there there should be a number like there shouldn't be a number like that.
So what we're doing here is see we already know by graphical approach we already know that that's why I said you should use the graphical approach here because it's the easiest and we know what the graph of x cube looks like. That's why I've asked you to use the graphical approach. But if you really want to use the algebra approach here then you will need to show that this does not have any solution. Right?
Do you get the point? Which is true.
This is not going to have any solution.
You're not going to find any real numbers such that this expression becomes zero for those real numbers. Two different real numbers.
So, or even the same ones. Even if they're same, you're not going to get this expression zero. Which is why this is one one. What's happening here is by the definition approach.
If there were a solution for this, if you could find x1 and x2 such that this expression becomes zero, then what would happen?
If this expression is zero, that means you have a relation there are x1 and x2 such that both of them not equal. That means this relation is not necessary.
Even if this relation is if this for this to be zero, you would want this relation to be zero. Right? This entire thing to be zero. It's also zero when this thing is zero.
Only this one forces x1 and x2 to be equal. That's why you want this to have no solution.
So for those who have understood this, it's good. For those who haven't, you don't have to use this approach.
Is that okay? Don't use the definition approach for this one. Use the graphical approach. Okay.
Does anyone have a doubt so far? Can we move to the next uh next method?
>> Personally.
>> Okay. So, the next method is let's say you have to you have to see if it's one one or not, right? So what you can do is if you can see do this thing let's say your function is y = to f_sub_x then if y is = to f_sub_x has only one uh not this thing if fx is equals to c has only one root what does it mean If for all C belonging to real numbers that means no matter what real number I take if f(x) is equals to c has one root what does it mean?
>> How do linear function >> has at most one let me just say has at most one root in terms of one one and not one one. If I say that okay if I have a function f(x) and I'm saying for f(x) equals to 1 I only have one that means there's only one x1 which gives you value one there's only one x2 which gives you value 1.5 there's only one x which gives you value 1.7 what does it mean so either there's just one x which gives you a certain value or there is no x which gives you that value what does it means no two x's can take the same value right if there two x's x1 and x2 such that f(x1) different. If the when do we have a function not not one one not one is when for two different x fx1 is equals to fx2 right >> that's when we say that two functions are not one this function is not one one because for two different values of x you're getting same value of f so if I'll say that fx= to c has only one root or it can at most have one root. That means either there is an x such that f(x) is equals to c or there is no x for some c. Let's say for one there is no x.
For two there is one x for 2.66 there is 1 x. Then what does it mean? That means for all C if you'll take any C no matter what this equation is going to have either one solution or no solution that means no two x's can take the same value because if they if they took same values then this would mean let's say they're taking value r that would mean that fx is equals to r has two solutions right x1 and x2 and they are different is that fine so one way to check is that you have to make sure that f(x) is equals to c has at most one root. It only has one root or no root at all. That's also fine. Is this point clear to all?
>> Okay. So, let's do one example on this one. Let's say we have x cub + 5 x² + 6 x + 16. Let's say this is your function and you have to check whether this is one or not. One way is you can try plotting the graph of it if you can and not >> if it's something like this which is what I suspect that it's going to be something like it's going to have some turning points and the moment your function has a you can also check it using the turning points by the way we have another method if you see that your function has a turning point that means it's not one one right if your function has a turning point it cannot it cannot be one one then but that's not what I I was going to use Let's say I equate this to zero. Let's say I I'll equate this to zero. You can equate it to anything and then check if it can have two solutions or not. But I'm equating it to zero.
Then let's see this thing. XQ + 5 X² + 6 X.
Let's check for this one if this can be zero or not for two different x. If this function can be zero this one if this is zero for two different x that means if x1 cub + 5 x1 2 + 6 x1 is 0 and x2 cub + 5 x2 cub + 6 x2 is also 0.
then this will also be zero for two different values of x right the constant does not change anything if the non-constant part this one is zero or take some value let's say if this part is zero for x1 and x2 for two values of x this part is zero that means this entire part including 16 is 16 for both x1 and x2 Is this clear?
There are two things that I need to make sure. One is that I'm audible because I keep getting this corrected and second is if it's understood or not.
>> Yes ma'am, you are audible.
>> Yes, you are audible.
>> So what we've discussed, do you want me to repeat this part?
I I actually joined in pretty late so I will go for the recording that's why I asked for the recording if it's there or not. Okay. So >> u >> Okay. So for other people is this understood the the thing that we're doing here is we're saying this function is 11 one or it's not 11 one. How do you check it's not 111? That if for different x1 and x2 if there are two different x1 and x2 such that this function takes the same value for these two. So how do you check? Let's check it for zero. Let's say can it take can two different x give you zero for the same function. So what we're doing things let's forget about that first. Let's say this thing if you have f of x and f of x is 1 one then can I say that f of x plus some constant is also 11 one.
Can we do that?
>> Yes.
>> Right.
>> Yes ma'am.
>> So and this this goes both ways. If f(x) is 1 one then f_sub_x and added with some constant that also should be 1 one.
So same we're doing here. So instead of checking whether this is 1 one or not one thing that we can do is check whether this is 1 one or not just this part remove the constant part we don't need that because if this part is 1 one non-constant one then the constant added to it will also be 1 one because of this.
So >> we'll only check for this one. Now what we're doing here is forget about everything else. Just take x common from here. Do you see that x is common here?
>> Yes ma'am.
>> It's x and then you will have x² + 5 x + 6 is equals to 0. That means x=0 gives you 0. And x is equals to roots of this.
What are the roots of this? It's 2 minus 2, right? -2 should be a root and uh >> 2 andus 3 >> is it how's two a root this is all positive right >> both are my >> if you will split it as 2x and 3x you'll have x and x plus >> okay so 2 and three are the roots we don't let's not bother about that you're going to have two roots x= to 2 and 3.
Whatever it is the whatever the signs are, we're not bothered about the signs, but you're going to have two roots from here, right? That means for x= to0, x= to 2 and x= to 3, you're going to get the same value zero. That means zero value is attained at three points. f of 0 is also 0. f of 2 is also 0. f of 3 is also zero. Or I shouldn't call this f, I should call this gx instead because f is this, right? That means g of 0 is 0. g of 2 is also 0 and g of 3 is also zero.
That means f of 0 is 16.
f of 2 is also 16 and f of three is also 16. Right?
>> That means at three different points you're getting the same value. That's one way you can check.
And we have particularly chosen zero here. Sometimes what happens is let's say if you have if instead you have gotten something else here you can choose some other constant too that's fine if you can find the root for this is equals to C easier than zero then just put this equals to C and try to find if you can have more than one root for this equation. If you can have more than one root for this equation that means it's not one one taken this here. Is this fine so far? is good.
>> Okay.
Then there we have another method to do it which is increasing decreasing and it's a theorem. So uh any strictly increasing function or strictly decreasing function strictly increasing or strictly decreasing is 1 one. So you don't have to check for such functions. If you have a function which is strictly increasing or it's strictly decreasing, not both. By this I don't mean that in this interval it's strictly increasing then in some other interval it's strictly decreasing. I mean on the entire domain on entire domain if it's strictly increasing and or strictly decreasing then we say that it's a one one function. It's like this only that if you have something like this it's strictly increasing then the horizontal you can you can see why it's one one right because the horizontal line test would pass here this would pass the horizontal line test that's why so for a strictly increasing or strictly decreasing function it's a one one function and uh should we've already discussed what strictly increasing and decreasing functions are right in previous lectures should I discuss that again >> we have >> is everyone familiar with these two terms strictly increasing is different from increasing >> we don't mean increasing here okay >> so that's one way to check and how do you check strictly increasing one way to do is that if your fx f-ash x is strictly greater than zero I know you've not done derivatives but when you whenever you do derivatives You can use that knowledge here as well that if your function's derivative is always greater than zero that means your function is strictly increasing which should mean that it's a one one function.
>> The slope is perfect.
>> Okay. So uh then we have the exponential functions.
So exponential functions are of this kind where you have f(x) is equals to a to the power x. This is called exponential function or for particular let's take uh this is the general form. This is general form for exponential function.
If you want the part like for for quadratic ones you do this right ax² + b x + c this is the general form for quadratic functions. Similarly this is the general form for exponential functions where your x varies and a is a constant.
A belongs to r is a constant.
The example let's take a particular example like let's take fx is equals to 2 to the^ x.
So this is an exponential function and then we'll discuss the properties of exponential functions. So basically in your exponential functions first is a to the power s and uh a to the power t. If this is it then you can add the powers. This will become a to the power s + t.
This is basically a operated s number of times. But when you write a to the power x or 2 the^ x and let's say I have 2 to the power 1.5 then you can't say that 2 is operated 1.5 number of times or 1.7 number of times right. So but that's the core idea here. So your powers get added here. The second one is a to the power s divided by a to the power t. So similarly when you multiply them the powers get added when you divide them the powers get subtracted here a to the power s minus t. The third one is a to the power 0 is equals to 1. That's a property. So any number raised to the power 0 is 1. Fourth one is uh 1x a s this I should have written this one next. So a 1x a to the power s is a to the power minus s.
So when you have a negative power that means it's 1x 8 to the power s and fifth one is a to the power s and then this entire thing to the power t is the same as a to the power s and then a b to the power s is the same as a to the power s and b to the power Yes, we forgot one more here. I should have written.
Okay, I'll write that here. Seventh one.
It is similar to this one which is a to the power s or a to the power m by n. If I'll take a the power m by n. This means a to the power m and this whole to the power 1 by n and power 1 by n basically means maybe I should write power 1 by n first. a to the power 1 by n basically means the nth root of a.
So nth what does nth root of a mean? It means a when let's say I'm saying a to the power 1 by nth root of a is b. This should mean that b to the power n is a.
So a number which when raised to power n gives you a is the nth root of a. Is this okay?
Like when you do power half this basic like 4 to the power half. It means you're trying to find the number which when multiplied two times gives you four. Similarly when you are doing 4 to the^ 1 by 3, it means you're trying to find the number a which when multiplied three number of times gives you four.
Right? So similarly here a to the^ 1 by n means a number which when multiplied n number of times gives you a and this is written in this manner or this manner both of them are same just notations and then a is a to the power if you have a the power m by n this means a to the power m and this entire thing in the nth root that means a is raised to the power m and then you have taken the nth root of it. Is this okay?
These are the basic properties of your exponential function a to the power x.
How do you add expon how do you multiply exponents and all? That's it.
>> U ma'am can you hold on for hold on over it for for us for a minute. I need to read it out.
>> Uh ma'am good evening. Uh just now I was able to join. Can you hear me?
>> Yeah.
>> Okay. uh I because uh we were actually told I don't know which session it was the just before the revision we were told the deadline will be postponed uh not on date >> I forgot to ask ask them to change the deadline so I'll ask them today >> okay >> today is not the deadline right >> tomorrow >> I'll ask them to change it >> and uh second part is like I think it's not you but in in one of the instructor session I requested Because still there is an ambiguity because in the calendar it just says instructor session and we are not clear.
>> Yeah. Okay.
>> We'll change that. We'll name the sessions there.
>> Okay. And when can we expect the deadline the postpone deadline?
>> Tomorrow.
>> I mean to say that when it will be I mean how many?
>> Okay. You mean how? Okay. So today what?
>> It's Tuesday.
>> It's Tuesday, right? So Friday is fine right?
Yeah, I mean because today only we are having a summary. So we'll get if we get a couple of days it'll be good to that we can actually again because just now I was able to join your pro session also.
>> Okay.
>> So let's have it on Saturday. Is Saturday fine for everyone?
>> Yes.
>> Okay. That will be good.
>> Okay.
>> Thank you.
>> So is this one fine?
>> Yeah. No problem.
>> Thank you.
>> Okay. So these are the basic properties that you need to know. Do we need to do the examples on these properties too?
These are just properties and I'm sure you must have done it in your schools as well, right?
>> Um earlier earlier session in during the like TA session we did it already.
>> Okay.
So then let's come to the uh proper function where we'll talk about a to the power x.
So a to the power x as a function is let's say you have a to the power x right and we say that a is a constant number that means your function could be 1.5 to the power x 7 ^x or minus 2.3 ^x this could be your function or let's not take negative values but because we're not taking this is called the base a is called the base x is called the exponent so we're not taking the bases to be negative you can take this to be 0.6 six. You can take all of these things, right? This is a different function.
This is a different function. This is a different function. So, we need to see how these functions behave.
And it depends on a. So, how a to the power x is going to behave depends on a.
Let's see how it how it depends on that.
Let's say if a is between 0 and 1.
Let's take this one. If a is greater than one here and a is between 0 and 1 here.
So for example here you can take a to the power 1.5 or sorry 1.5 to the power x.
So you can take 1.5 to the power x because 1.5 is greater greater than 1 and then here you can take 0.5 to the power x.
This is just very part these are just very particular examples here and here.
You can have a lot of other examples too. Right?
So first is let's talk about the domain of this. What's the domain when your a is less than? What's the domain of this one?
When a is greater than 1, your entire a to the power x function behaves the same way. That means your function 1.5 to the^x behaves the same way as it's not exactly the same but it behaves the same way. 1.7 to the^ x 10 to the power x this they all behave the same way because they are in this category.
Right?
>> Do we have a doubt? Does anyone have a doubt?
And these ones 0.5 to the power x 0.6 6 to the^ x 0.927 to the power x. All of these behave the same way.
Same way as in they're not the same functions but they behave typically the same way. The basic things are same like the domain and all all of these things are same. Let's see what's the domain of a to the power x.
And a to the power x function is by the way this. If you have why do they behave differently for a less than one and for a greater than one.
Let's say you have two here. This is when a is greater than one.
Just one second.
Just give me a minute.
Yeah. So this is when your a is greater than one.
Let's take two.
So if I'll have two two how does 2 ^ x let's see some values of 2 ^ x. 2 ^ 1 2 ^ 2 ^ 3 and so on. Right? 2 ^ 1 is 2 then 4 then 8 then 16 then 32 and it keeps increasing. Right? That's the pattern that you need to observe that this keeps increasing. Now let's have for a less than one let's take 0.2 instance 0.2 to the^ x right now your base is less than one or you can say now your base is less than one and greater than zero.
So here you'll have 0.2 to the power 1 then 0.2 to the power 2 then 0.2 to the power 3 and so on and which is 0.2 then 0.04 4 then 0.008 and then 0.0016.
So these are the values right that means the fun this one keeps decreasing.
So as you increase ma'am I have doubt.
>> Yes ma'am.
0.2 These are two different power 2 power and 2 power and no this confuses you.
What we are studying is we know that a to the power x behaves differently depending on what the value of a is. If a is greater than one then this function behaves differently. If a is between 0 and one it behaves differently. So for all a this that's what I said right? If you have a is equals to 0.5 0.6 0.7 0.927 for all these values of a all of these behave the same way more or less the same way. That means are increasing.
>> Yeah. One give me one.
>> Okay >> just one second. All of these are decreasing, right? These ones are decreasing. This is one of the properties. Similarly, a lot of properties for when your A is between 0 and one, the properties remain same.
When your A is greater than one, for those ones, the properties remain the same. That's why we have two columns, right? One for a greater than one. The other one is for a between these two.
So that's what we're seeing here. We're trying to see when we take a greater than one, we have 2 to the^ x as function. Right? my a to the power x the function that I'm discussing becomes 2 ^x now I can't take all the values I can't take 2 ^0.792 you can't find this right >> no ma'am so what we do is >> so that's why what we're doing is we're only taking natural numbers here so what we're doing is what we're trying to see is when x increases what happens to 2 to the^ x >> so if I'll increase it >> also increased >> yeah that's also increases per 2 to the^ x but it decreases here right >> 0.2 When you write decreases when you write whole number then it decreases. No ma'am.
>> Yeah. So when you take a number between 0 and 1 this decreases.
So a so what we have found is a to the power x decreases when a is between 1 and zero. Right?
So if x increases a to the power x decreases and if x increases a to the power x increases when a is greater than one. Right? I just wanted you guys to notice this thing. That's it. That's why we've done this thing.
>> Yes.
>> Okay. So, let's come to the domain of this function. Let's say you have a to the power x function or 1.5 to the power x in particular. This is a particular example. In general, it's a to the power x. What's the domain of this?
when a is yeah ma'am will be 1.5 and >> no no domain is the input values that you can give this function right if your function is 1.5 to the^ x then what all values can x take here >> 0 to infinity >> 0 to infinity and what's wrong with the other values the negative ones Now you take negative it will decrease >> no if I'll take I can take this right 1.5 to the power minus one >> this is one by one reciprocal of >> then you could do reciprocal that fl >> we just need to know >> that if you why why can't we take this if you have fx= to a to the power x or let's take because a is a little a to the power x is confusing let's take 1.5 to the^ x I just need to know what all values can you have for x like if you have x here then I can't take x negative right because it's not valid minus2 in square root is not valid >> no ma'am root root of is not >> right that's why minus2 is not in the domain of square root of x so that's the question here what all values can x take >> can I take >> may I take may I take minus2 value of x >> yeah you can take this is minus uh 1.5 to the power minus 2 is 1 by 1.5 squared.5 >> which is 1 by 2.25 25 I guess you can you can find the value right it's a real number this is a real number what's the problem >> okay ma'am means all real number can be taken taken as a power >> yeah you can in doain >> yes you can take all the powers uh any real number can be power that's why the domain is real numbers so a domain is real here and it's it's real here as well even if you'll take 0.5 I can take any power here right that's not an issue So your domain is r here as well.
Then the next one is the graph. Let's look at the graph of this. We already know before that we can just write that this is an increasing function right increasing.
>> I'm sorry disturbing decreasing. Yes >> ma'am. Uh if you take the power power ma'am you are taking one. Okay sorry taking two and you are taking the power of x by by means 1.5. Then how can we calculate this?
>> You can't calculate it. Doesn't mean it doesn't exist. Right? You're saying if I'll take 0.5 to the power let's say 1.3.
This is a valid power. But you can't find the value. You'll have to use a calculator for that. But it's valid.
>> Is that okay?
>> Or calculator only.
>> Sorry. What?
>> Means ma'am by calculator we can only find this value. you can find that value. That means you just need to know that that value exists and that value is real. So if I'll take 1.5 to the power 1.327, I want this value to be real and it's a real number. You'll get a real number when you'll find this.
>> Okay ma'am thank you.
>> Okay so this is we already saw that for a greater than one your function is increasing and for a less than one your function is decreasing. Right?
We just checked it here for this one.
So that's one thing. And then let's come to the graph or even before coming to the graph, let's first find the range. What's the range of this range of 1.5 to the power x? What's the range?
>> 0 to infinity >> 0 to infinity.
>> And is 0 included? No zero is not included because >> 1.5 raised to any power cannot give you zero. No matter what power you take here, you are never going to get zero.
That means your range is 0 to infinity.
So this goes from 0 to infinity and it cannot have negative numbers as well because if you have why why why do we not have negative number numbers here because a is also positive. So when a positive number is raised to some power that is always positive. If you have positive here and it's raised to some power x this is always positive.
That's why your range is from 0 to infinity. Same goes here. The range is from 0 to infinity here as well.
Or should I just give you the graph first? I thought okay this is the graph.
This is what the graph looks like. Graph for a to the power x when x is when a is greater than 1 looks like this.
It's uh this.
And this graph looks like this.
When a is less than one or when a is between 0 and one, your graph looks like this.
That means it's decreasing here. It's coming close to x-axis here and it's increasing on this side. It's going to uh to infinity.
And same is with this one. On this side, on the positive side, it's going to plus infinity. And on the negative side, it's side it's going to zero.
Now you can see by the range to infinity right? Yeah.
>> I mean first graph then if it is going in negative so it is going to zero.
>> Yeah. This one's on the negative side it's going to zero. Yeah.
>> And on positive >> on positive it's you tell me. We have discussed this in polinomials right?
>> Yes ma'am.
>> On the positive side it's going to it's going to plus infinity.
Okay.
Okay. So then we have this thing and this point is this is your what is this point called?
y intercept right so the y intercept and how do you find the y intercept for a function let's say your function the function was 1.5 to the^ x right or a to the power x for a less than a greater than 1.
So if this is your function, what's your x y intercept?
>> 1.5 >> 1.5 >> one >> for for y intercept, you put x = to0 and find the value of y.
>> So it's y = to 1.5 to the power x. Put x = to0 in this equation. You'll get y = 1.5 to the power 0. And any number raised to the power 0 is 1. That means your y should be one.
put x = to 0.
So your y intercept is 1 and the coordinate will be 0a 1. Right?
That means this coordinate is 0 comma 1.
Now let's do the same here. If I'll do this for y is equals to >> what did we take on the other side?
>> What?
For every value of a it will be one, right?
>> Yeah, it will be one. So if you'll do it here, if you'll put x = to0, this is still y is equals to 1. So the y intercept here is 1. Again, this is the y intercept and here this is the y intercept.
So y intercepts for both of these are same. That means this point is also 0a 1. This point is also 0a 1. Y intercepts are same.
Then we have x intercept. Although you don't need to solve the equation for this one. Can you just look at the graph and tell me what's the x intercept?
>> No x intercept. Right?
Minus infinity is not a number.
Minus infinity is a limit. So it cannot be x intercept.
Do you see this graph cutting x-axis anywhere?
If you move to minus infinity, it'll keep coming closer to this x-axis. It keeps coming closer to x-axis, but it never touches the x-axis. That means you don't have an x intercept here. And same goes for this one.
So this one is it goes like this keeps coming closer to the x-axis, but it never touches the x-axis.
So this is it. Here also your x intercept is no x intercept and same goes here.
I'm not writing it again. Then let's come to end term behavior.
So for your endterm behavior when x tends to infinity where does your y go here look at the graph when x tends to infinity this is extending to infinity on x-axis your y is also going to infinity right you can say when x goes to infinity a to the power x goes to infinity when x goes to minus infinity a to the power x goes to yeah when x is going to minus infinity. Your a to the power x the function is moving closer to zero, right? Because your height is get is decreasing each time your function is moving closer to the zero x= to0 line y = to0 line sorry is that okay?
So that means when your x ts to minus infinity your ax tends to zero. And what happens here?
When x tends to infinity where does your a power x go this time?
It's going to zero. Right?
And when x ts to minus infinity a to the power x is going to infinity.
So this is it. We've discussed what a to the power x and all the properties of a to the power x are for when a is greater than one and for when a is less than one and greater than zero here.
Okay. Then coming to the next topic it's e and e to the power x is e is 2 uh 7 and something I don't know the values later it's an irrational number though but it's something like this then what should be e to the power x what category should e to the power x fall under this one or this one >> increasing >> the increasing one the first one right >> yes >> so this is going to be this is called natural exponential function and uh if you'll yeah that's all you need to know about this one. One other thing is that the area under this curve is the curve obviously looks like this increasing function. This is the point 0 comma 1. And the area under this curve this one is E. That's it.
And you'll know why the area is E when you do integrals later. Then you'll see why the area is E. Okay. And coming to the next part.
Okay, then coming to compos. Now let's come to compositions.
Do we have onto functions too in our syllabus in this week?
The composition of functions is basically fogg.
This is composition of functions fx and gx let's say. So how do you compose two functions? Composition is basically this f of g. These kind of functions are called composition functions where first you substitute first you apply g on x. So what's happening here is you have a value x.
This is let's say domain some domain then you are applying G to this and you're getting G of D right g applied to this set D and then you're applying F to this thing that means F of G of So for example, let's say you have >> X.
This is how it works. So what you're doing is it's basically this. You have X, then you apply G to it and you'll get GX, right? and then you apply f to this entire thing. So you will get f g of x.
For example, let's say you have a number one and then you apply the function g x = x + 2. And let's say your f is fx= to x². So if I'll apply on this one, if I'll apply g first, what will I get?
>> Three.
>> I'll get three. And then if I'll apply f to g of 1 then what will I get? I'll get f of three right because g of 1 is three and f of three is 9.
So this is what it means that first to this x first you apply g and then to this gx you apply f.
Okay.
So now let's come to let's see some composition functions first. Let's do this one and let's let's talk about the domain of composition functions first. Domain of and this is also written as this you can write as fogg in short we write this as fog g of x.
So your fog g of x is nothing but g applied to x first and then f is applied to gx. That's what fo g of x means.
And g of f of x means f applied to x first and then this to to this entire fx you apply g.
So what is the domain of let's say fo g.
How do you find the domain of this function? Then let's say you have a function fo g of x which is the same as gx. So first you're applying g to x and then f to this entire thing. So what should be your domain? What's the domain of fog g?
So for you to find the domain of fog, you have two things. One is that x should be in the domain of g because you'll apply g to x. Right?
If x is not in the domain of g, then can I apply g to it?
>> No.
>> Right? So you need x to be first condition is for x to be in domain of fo g.
First condition is that x should be in the domain of g because you're going to apply g first and then you will apply f.
And what's the second condition? That your gx should be in the domain of f.
Right?
Is this okay? So x should be in domain of g and gx should be in domain of f.
These two conditions need to be. So if there is any x in the domain of fo g then that x should be in the should be in the domain of g because you're going to apply g to it like and then that gx the value that you'll get that should be in the domain of f. Let's do one example first. Let's say you have x², you have roo<unk>x and x + 1. Let's say you have these two functions. This is your f, this is your g.
Or let's take this as f and this as g.
What's your fo then? Your fo is >> rate of x + one.
>> If you do it one by one, you will get this. fo g of x means gx and then f applied to it. So if you will apply g to x you will get x + 1 of f right this and then if you will apply f2x + 1 you will get square root of x + 1 so that means now for this to happen what we've done here is first thing is for you to be able to apply g to x your x should be in the domain of g right and then x + 1. Now let's take one for example. Let's take x = 1. Is x = 1 in the domain of g.
This belongs to domain of g, right?
>> It belongs to domain of g. And then x + 1 at x = 1 is what?
>> Two, right?
And is 2 in the domain of f? You have to check that x should be in domain of g.
If you want to check for x= to 1, one should be in domain of g and then g of 1 should be in domain of f.
So if you're taking this particular example here, for one to be in domain of fo g, you would want one to be in domain of g and f uh g of 1 to be in domain of f, right?
So your one is in domain of g right you can put one here and that's fine your input can be one for gx right so this is satisfied and g of 1 is g of 1 is two should be in domain of f is two in domain of f it's fine right that means two should be this condition is also satisfied by one this implies is now I can say because these two conditions are satisfied simultaneously that means one is in domain of f now let's come to some other number let's say I'm taking minus2 here if I'll take x= to minus2 what will I get >> x = minus2 I need to check two things first is minus2 should be in the domain of D and second is G of minus2 should be in the domain of F. So your minus2 is in the domain of G. Right? G is X + 1. -2 is in the domain of G. So this is satisfied.
Now let's come to this one. -2 g of -2 is -2 + 1 which is minus1.
That is -1 should be in the domain of f.
Is minus one in the domain of f?
>> No, >> it's not there. That's why minus2.
Now, what does this mean? This is incorrect. So, one condition has failed.
That means minus2 is not in the domain of fog. This is one element which is not in the domain of fog. And we have seen one element which is in the domain of fog. Now, let's try to find the entire domain. So, your domain should be you have two conditions here, right? First is this one. Second is this one.
What what does it mean then? Doain of fo g that means should be equal to domain of g.
So I want both of these to be met to be simultaneously true. So we have to take domain of g and then we have to take the collection of x belonging to real numbers such that gx is in domain of f. Right? That means the inner function when applied to x should be in the domain of outer function and I should take the intersection of both these sets. Right?
This would mean now if x is in this domain that's the definition. If x is here in the domain of fo it should be in the domain of g and intersection means and.
So it should be your x should be in this set and your x should be in this set as well. Right? So that's your domain of fog.
Now how do you find this? Let's take a recent example only. It's let's find the domain of this. f of x was rootx and g of x was x + 1.
What's the domain of g?
>> All real numbers.
>> All real numbers. And then let's come to this set x belonging to real numbers such that g(x) is in the domain of f.
Let's first find domain of f. Then only we know what for what x is gx in the domain of f. Right? What's domain of f?
>> 0 to infinity.
>> 0 to infinity and 0 is also included.
So that means this is the same as x belonging to real numbers such that and what is g of x? g of x is x + 1 right should be in the set 0 to infinity.
This means x such that x + 1 is greater than equals to 0. Right? That's it.
I've written this in equation form in equality form. That's it. This means x such that x is greater than equals to minus1.
Is this fine?
So this set is equal to this set.
This is the same as this set. This set is the same as they're all the same sets. We have simplified the forms.
That's it.
So what should this give me? How can I write this thing in interval? X belonging to real numbers such that X is greater than minus1 or equals to minus1. So it is X greater than equals to minus1 means this thing right.
So this set is the same as this interval. Is this fine so far?
>> Yes ma'am.
>> Okay. So that means I wanted two sets.
One is D of G. The domain of FOG is what?
domain of G intersected with X belonging to real numbers such that G of X is in domain of F.
Okay. So what is D of G? D of G was real numbers and this set is 0 to infinity. Right? What's the intersection of both these sets?
It's like this 0 to infinity is here.
This is zero.
Starting from zero going till infinity.
So this is this set 0 to infinity. And your real line is from here minus infinity to infinity. Right?
So the intersection of these two sets is something that's in this set two and something that's in this set two. Right?
So basically this one I want it to be in this set two and I want it to be in this set two.
Is this example clear to all?
Then we can move to the next example if this one's clear.
>> Okay. So let's try to do this one. f(x) = 1x and g(x) = x² >> statement in the intersection statement.
Uh like uh what about minus one?
We had we had okay sorry I thought this is what we have found thanks for correcting it we had minus1 here right this set was equal to this is this set it was equal to -1 to infinity so this is from minus1 to infinity and here you'll have minus1 And this will also change. Then we'll have minus1 to infinity. Is this okay?
>> Yes.
>> Okay. So this is the next example. Let's do fo g first. So your fog g is 1x x² - 4. Right? But we need to check the domain. What's the domain of this?
Domain of FOG is let's first find for domain of FOG we first need the domain of G which is your G is X squareus 4 right so what's the domain for X squareus 4 this function >> it's the entire real line right all real numbers you can put any real number here no problem so the domain of g is r now let's come to domain of f because we also need that what's domain of f >> 0 to infinity z >> 0 to infinity why >> all real numbers except zero >> yeah so you'll have all real numbers except set zero that means this set minus the zero set. Okay. So this is your domain of f and then you have to find this set x belonging to real numbers such that g of x is in the domain of f right domain of f which is equal to r minus 0.
This set is the same as x belonging to real numbers. What is g of x? G of X is X square - 4 such that X² - 4 does it belongs to this set. When I say that this should belong to R minus 0, what does it mean? It means that this value should not be zero. Right?
This can take all values other than zero. So I can simplify it and write X² - 4 should not be zero.
Right?
When is x² - 4 0? It's when x² is equals to 4.
So this is the same set as x.
>> What do I want? I want x² to be not equals to 4.
So this would mean when do I have x²= to 4? It's when x= -2 and x= to +2. Right?
That means I don't have a problem other than at minus2 and 2 because I don't want the square to be four. That means I don't want x to be minus2 or +2. Other than that all values of x are fine. That means I only have a problem at minus2 and +2. That's it.
That means this other set of rs for the intersection is this.
Now you have two sets for domain of fo g domain of g was real numbers and it's intersected with r - - 2 and 2.
So the intersection is going to be r when intersected with any set gives you that set only provided that set is a subset. This is a subset of r right?
This is a subset of r.
So that means you should get r -2 and 2. That's the domain for fo g.
And if you don't want to go through this entire process, right? You can just have a look at this this thing. You can just have a look at the function fo and then you can see that at for this function the only thing that's going to create a problem is minus2 and plus2. Other than that we're fine with everything, right?
Okay, let's see one thing here which is let's say you have one function f(x)= to x² and then the other function is g(x) is equ= to square root of x then what is uh f4 g what is fo g of x >> x so you'll [snorts] have square root of x and then the square of this right and then this will be x and what about g of f let me just clarify this here we have the domain x belonging to r and here the domain is x belonging to r positive union zero r positive means this is positive real numbers and this is zero what is goof Yes.
Yeah. What's GF?
So GF will be let's see how we can write this. G is square root and then X² right? This is your GF of X. And what is this value?
So this value is mod x. Why is it mod x and not x? This is not equals to x by the way.
This is not x. This is mod x.
And why why is it so? Because if you have let's say if you have minus1 here if your x is equals to minus1 then -1² and then square root of this is 1 and x = 1. you will get 1 square is equ= to 1.
So this means when I put minus 1 I get 1 and when I put one I get one or in simple words in more generalized words it's when I put a negative number I get the I get it posit I get I get the solution as positive of it that means if I put minus 10 here I'll get 10 but if I put 10 I'll get 10. Yeah, >> I'm sorry. But is this week syllabus is completed?
>> What >> is this week syllabus completed until now?
>> No, the syllabus is not completed but left with inverses.
>> Okay ma'am.
>> Okay. So because of this reason this turns out to be mod x and not x. So what's the domain of let's come to the domain of fog in this one. What's the domain for fog?
So if you'll have fog, you just need to check the values of x.
If you quickly check it, the domain of g is r, right? So it will be real numbers.
And then intersection with x such that g of x that means square root of x is in the domain of this. And because the domain is real the entire real line. So this is going to be the intersection with you're only okay the domain of g is we've written this wrong. This is do we do we need to do the domain of this one domain of fo g and all because we've done a sim similar example where we need x² and uh or maybe mod x root square root of x and x - 1. We've done a similar example. Do we need to do this here?
Can you find the domain of fog and go for this one or should we do it here?
Okay, let's move to the other example.
So we are doing a last example on how to find the domain. Let's say you have to find the domain of the function square root of x -1 and x + 2. So if you have to find the domain of this, how do you find the domain?
It's the composition of two functions.
If you see it's the composition of square root of x and g is x -1 and x + 2.
And this entire function that's written here is fogg.
So the domain of FOG This.
So your domain of fog is The domain of g intersected with x belonging to real numbers such that g of x is in domain of f. What's the domain of g? g is this. You have to find the domain of this. And this is what's the domain? We know the domain of f, right?
So let's let's solve this set first.
This will be x belonging to real numbers such that g(x) which is x - 1 x + 2 is in the domain of f and domain of f is r Thank you.
Yeah, the connectivity issue at my end. So this sessions okay.
So this one is your domain of f is greater than domain of f is this right? 0 to infinity. That means you need x such that x -1 and x + 2 is greater than zero.
Okay. So, you need this equation to be true, right?
>> Yes.
>> Okay. So, this is This thing is if you want this if you want x -1 x - let's say x + 2 to be greater than 0 or greater than equals to 0. Then what do you want? Either you want both of these to be positive, right? Or you want both of them to be negative.
It's like this. If you multiply two positive numbers, you get a positive numbers. or when you multiply two negatives you get a positive number. If they are of opposite signs then you'll get a negative number. Right? So for this entire thing to be greater than zero same goes for this one. If you have let's say x + 1 x + 2 x + 3 then what what what would you want? Either you'll want these two to be positive all three to be positive or you would want these two to be negative this to be positive or you would want do we have other choices? these two to be negative, this one positive or these two to be negative and this one positive. Right? So you'll have to see signs as per this.
But for now because we only have two factors, two linear factors.
So we'll only go by these two. So your two cases here are if you want the product of these two to be greater than zero, then there are two cases. Either both of them are greater than zero. X -1 and X + 2 both are greater than 0 or greater than equals to 0 or you have X -2 and X + 2 both of them are less than 0 right so if these both of them are greater than 0 X -1 greater than equals to 0 means X greater than equals to 1 and X greater than equals to -2 right this means now if X is greater than 1 1 is here -2 is here and they have because we have an and here that means Both of the conditions need to be satisfied simultaneously. That means it's from 1 to infinity and minus. So your x should be in this interval 1 to infinity and in this interval -2 to infinity it should be both greater than 1 and greater than minus2. A number greater than minus greater than 1 and minus2 both. So I can simply write this as greater than equals to 1. Right? Any numbers which which is greater than one is also greater than minus2. That means this is it for both of these to be greater than zero. I want x to be greater than equals to 1. That means x should belong to this interval 1 to infinity. Right? And let's come to this one then this one is you want both of them to be negative. That means you want x - 2 to be less than equals to zero. And x >> this was one, right?
>> Yeah. So this was -1. So you want x - 1 to be greater less than equals to 0 and x + 2 to be less than equals to 0. Both of them to be less than 0 simultaneously. That means x should be less than equals to 1 and x should be less than equals to minus2. Now if if you have x less than equals to 1 and x less than equals to minus2 then I can simply write this as x less than equals to minus2. Right? If it's less than minus2 it's obviously less than equals to 1 as well.
So this means interval here is x belonging to minus infinity to minus2. Right? This is the interval here.
And now what does this tell us? This tells us that if you want this thing to be greater than zero, you want either this or this. Right? either I want both the factors to be greater than zero or I want want both the factors to be less than zero. So from here I get this interval and from here I get this interval. That means the set that I'm looking for x belonging to r such that g of x belongs to domain of f.
What should it be equal to? It should be 1 to infinity. And should I use intersection here or union here? I know it has to do with these two sets.
Right? So you'll have to use union here because it's or here it means even if x satisfy this condition then also it's fine. If x satisfy this condition then also it's fine. It doesn't have to satisfy both conditions.
It can satisfy either of these or both of these if it can. Right? It was minus two here.
So this is the this is this other set and then let's come to this one. We have to find domain of G as well. Right? So domain of G is your G was X -1 and X + 2. Right? What's the domain of this?
X -1 X + 2. What's the domain of this function?
>> All real.
>> All real numbers. We don't have a problem here. For any x you will have an output, right? So all real numbers. That means your intersection will be r intersected with this set. Where is it?
This one.
R intersection with 1 to infinity union minus infinity to minus2.
Who was included? Right? Yeah.
That means it's just 1 to infinity union minus infinity to minus2. That's it.
Is this okay?
Okay. Then we are only left with inverses of functions.
So let's do that. Inverse of a function is let's say u fx then gx such that f o g of x and g o f of x both is equal to x. That means if I'll apply fo g to x I should get x and if I'll apply gf to x I should get x again.
If this is happening then I say that g is inverse of gx is inverse of f.
And inverse of f is denoted by this f inverse that is f inverse. So we can say that f inverse is gx because g when operated with f from both sides from the right side from the left side it gives you identity function. This is fogg is equals to goof is equals to identity function. Right?
Okay. Then let's see how you can find the inverses. Let's do one example on how to find an inverse.
And before that let's see when inverse exists. So f has inverse.
f is a function. It has an inverse function if and only if f is both 1 one and on two 1 and on2. So when it's both 1 one and on two then it has an inverse and if it has an inverse it has to be 1 one and onto both.
Let's do one example how to find inverse. Let's say a function fx is = 2x + 1 and x - 4. Then to find the function g of x such that f g of x is equals to x and g f of x is also x. You have to find this function, right? And how do you find this g?
Just take fx = to y. You will have y = 2x + 1 x - 4. And then solve for x from here. By solve for x, I mean write x in terms of y. For now you have y in terms of x. y is in terms of x. Right? I want x in terms of y. So I can do yx - 4 = 2x + 1.
This is going to give me yx - 4 y = 2x + 1. And then you can take x on the same side. You'll have yx - 2x = 4 y + 1.
Take x common from here you'll have y - 2 and this will be 4 y + 1. So that means your x is x in terms of y is 4 y + 1 and y - 2. So this is x in terms of y. You have to get this expression from y in terms of x. You want the expression x in terms of y. And then your gx is simply replace y with an x. So you will get 4x + 1 and x - 2.
This is the inverse function is inverse of fx = 2x + 1 and x - 4.
Is this okay?
That's how you find the inverse. And then there are some things about inver invert in inverse functions which is let's say you have f and then you have inverse function of f inverse of x.
So one thing is that they're mirror images of each other when the mirror is placed along the y= to x y= to x line.
So if you have a line here y = x line looks like this right inclination at 45° with x-axis. This is your y= to x line.
If this is your curve f(x) then the inverse of this is going to be the mirror image of this function when this is the mirror. So if you'll consider this to be the mirror then the mirror image here will be the inverse of this.
This will be your inverse function.
That's one thing about the graphs of in f and f inverse. And then if AB lies on AB lies on f(x). If I say a point lies on the curve y is equals to fx. What does it mean?
This says that if there is a point AB, let's say this point is AB. If there's a there's one point A which lies on FX, then BA the opposite of it. Let's say 1 2 lies on FX.
That means 21 is going to lie on f inverse of x.
And what do we mean when we say that a point lies on f f of x or on the curve?
If I say ab is here, what do we mean by this? We mean that a satisfies this function. Right? That means if I'll put y equals to b, this is ycoordinate and x= to a, then this should be true.
That means the image of this function at a should be equal to b. That's when I say that a is going to lie on this function. Right?
This is the same as y is a here and this is f inverse of b and do you see why these two these two statements are true because if you have b is equals to fa and then if I'll just uh apply f inverse of on both sides of this equation I'll get f inverse of b and here I'll get f inverse f of a and we've discussed this That a function when is composed with itself gives you identity function. f inverse of f a is the same as a right.
So this means f inverse of b is a. So this gives you this. This implies this.
That means if a lies on your curve fx then ba is going to lie on the curve f inverse of x.
Okay.
And then one last thing that if you have intersection of f and f inverse and f inverse function. How do you find the inver the intersection of intersection point of two functions?
That means f(x) should be equal to f inverse of x.
This is only true.
So if there is a point x which lies on both f intersection of f and f inverse means there's a point which lies on f and it lies on f inverse 2. Right?
That's what you call the point of intersection of two curves. So that's only going to happen.
So if x comm y is intersection is point of intersection of f and f inverse. This implies xy is a comma a.
That means only these kind of points can be the point of intersection of f and f inverse. That that means if you have a function say f(x) and the inverse of this function then can 1 comma 2 be the point of intersection of these two functions?
>> Yeah. Because it does not look like this. So if you have to have a point of intersection of these two functions f(x) and f inverse of x it should be of this form a comma a that means it can be 1 comma 1 can be 2 comma 2 that you'll have to have to check depending on the function but it cannot be 1 comma 2.
Now if I say that the point of intersection of f_sub_x and f inverse x lies is of the form a comma a that means your y is also a ycoordinate of the point of intersection is also a x coordinate of that point of intersection is also a that means your point of intersection always lies on the line y is equals to x right >> so that's it that's what you need to Know point of intersection of f and f inverse lies on the line y = x this line.
So if you have f as this then the inverse of this looks like this right? If you have this inverse is going to look like this.
This will be f inverse of x and the point of intersection. This is point of intersection and this is another point of intersection here. This one these two lie on y= to x line. This is y = x.
So with this we're done with the summary for week five. Does anyone have a doubt so far in the summary?
Do we have any doubts from the summary?
>> Excuse me ma'am.
>> Yes, >> ma'am. I want to know about the something about injective function. In this summary, >> what do you want to know about >> ma'am? You said >> function. You said that the definition I know about the that definition which you told but actually professor had told us a different definition also it is said if x1 is not equal to x2 then your fx1 should not be equal to fx >> the same thing right >> it's equivalently contraositive thing right >> but our but our TA session the TA said that you cannot if this definition cannot deduce anywhere so it's misleading you cannot use the other definition >> which one >> uh >> which one cannot >> uh x1 is not equal to x21 >> okay this definition you're saying x1 is not equals to x2 this should imply it's the same thing as what we discussed fx1 should not be equal to fx2 right this is the definition that you're talking about this is also correct are you saying that this definition will not get you anywhere >> yes that's a t headers >> yeah because if you If you have to prove this thing, you will say how will you prove things? Let's say you have a function 2x + 5. I already know that this is a 1 one function. It's a linear function. So it's 1 one. But if I use this definition, I'll take x1 not equals to x2. Then I try to find I'll say if x1 is not equal, you can do that too. If x1 is not equals to x2 then 2 x1 then you'll have to move in the backward direction that 2x1 should not be equal to 2x2 right.
If these two are not equal then the two multiplied with it but it's not the right correct way. It's actually better for you to say the easier thing could have been if 2x1 is equal to 2x2 then x2 and x1 should be same right.
So you guys know that but I was just about to confirm that this definition is correct or not.
>> This definition is correct but for you to use the definition I suggest you to use this one.
>> If you using the definition to prove use this thing but this definition is correct.
These two are equivalent statements.
They're the same thing. So if you're saying f(x1)= to f(x2) should imply x1 is equals to x2." This is the same as saying when if I'm saying if these two are equal, these two should be equal.
Now if these two are not equal, then these two should also not be equal.
Right? That's the contraositive part.
That's correct.
But in while you're using the definition to prove things, use this thing.
>> So >> is that okay?
>> Yes. So yeah >> just I wanted to know that that the definition is correct right.
>> Yeah this is both of them are correct.
>> It is not misleading but only for the proving part we can use the second one.
>> Yeah use this one because it's this one's for proving you you don't actually use it.
Even if you want to use this thing you will indirectly be using this only.
That's why for proving stuff use this one. But this is equivalent to this one.
Both of them are equally correct.
They are essentially the same statements just written in different manner. That's it.
Ma'am, >> is that okay?
>> Yes, ma'am. I want second out. I want to ask you.
>> Yeah.
>> There is there any main difference in a strictly increasing function and increasing function?
>> Yeah, there is a huge difference between a strictly increasing function and an increasing function. Strictly increasing and increasing, right? strictly incre increasing means if x1 is less than equals if you have two different numbers let's say if I say if x1 is less than x2 then my fx1 should be less than equals to fx2 now here you have equals to this says if x1 is here on the real line and x2 is here then they can have same values if they have same values let's say at x1 this is same value at x2 we have same value then also we can say that it's a increasing function but then it's not a strictly increasing function.
So basically this can be this is a constant function right let's say this is fx= to 2 function then this is a strict this is an increasing function why because whenever you will take x1 less than x2 your fx1 which is 2 fx1 is also two fx2 is also two this is going to be less than equals to so the statement holds right so this is a strictly this is an increasing function but the strictly increasing definition says that if x1 is less than x2 Your f(x1) should be strictly this should not be equal. It prohibits you to have these values equal. That means these two values should not be equal. This should be this value should be strictly less than this one. That's what strictly increasing means. That means now if you have something like this f(x) is equals to two function. This is not a strictly increasing function because let's say at one and at two one is less than two but f of one is can I say this this isn't true right f of one is also two and this is also two and two is not strictly less than two strictly less means they should not be equal this has to be strictly less than this that's the difference.
So a strictly increasing function looks like this. It keeps increasing. Your increasing function could look like this and then it becomes constant and then it goes up and then it again becomes constant.
So and actually professor told us that those function which are not strictly increasing they can be they can we can classify into nondereasing function which are not strictly increasing. They can be nondereasing function or >> okay >> that is similar to like if we see a increasing function we can say it is a non-deasreasing function right?
If you see an increasing function, see nondereasing means you're saying if it's strictly increasing, then you can say because it's not becoming constant here. Decreasing also means strictly increasing means it has to go up only. It cannot get conant anywhere or in fact it can at at two different points it cannot have same value. Right?
At two different points you cannot have same value for a strictly increasing function. But for an increasing function at two different points you can have the same value. Now coming to not decreasing.
Not what does decreasing mean?
Decreasing means that it has to be you're not decreasing is this the same as strictly.
If it's not strictly increasing that means uh it can be increasing right it can still be increasing.
If you have a function and I say it's not strictly if I say it's strictly increasing this means it's increasing.
If I say there's a function and it's not not strictly increasing then I don't know don't know about decreasing and increasing don't know about increasing whether it's increasing or not we don't know a strictly not strictly increasing function can be increasing or it can't be increasing we don't know about it but a strictly increasing function is always increasing similarly if you have an increasing function I cannot say that it's a strictly increasing function. And about the not decreasing part, if you have a strictly increasing mean, it's keep keep it keeps increasing. And nondereasing means it should not decrease. That's it.
What is what definition are they using? Because different books use different terminology. So what definition have you read?
Uh I think it was the lesson told by Vishal and he said uh in fact I think he said is strictly functioning strictly increasing function is different and not decreasing function is different because >> that's basically the terminology that's why I said >> so in your course lectures you can check what what terminology in by terminology I mean different courses use different terminology right So you need to check what terminology is used in this course.
Is that okay?
>> Okay.
Does anyone else has anything?
Can we end the session then if no one has any doubts?
Okay. So, we end the session here then.
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