Voiceover:So we have three different function definitions here. This is F of X in blue, here

we map between different values of T and what G of T would be. So you could use this as

a definition of G of T. And here we map from X to H of X. So for example, when X is equal to three, H of X is equal to zero. When X is equal to one,

H of X is equal to two. And actually let me number this one, two, three, just like that. Now what I want to do in

this video is introduce you to the idea of composing functions. Now what does it mean

to compose functions? Well that means to build

up a function by composing one function of other functions or I guess you could think of nesting them. What do I mean by that? Well, let’s think about

what it means to evaluate F of, not X, but we’re

going to evaluate F of, actually let’s just start

with a little warm-up. Let’s evaluate F of G of two. Now what do you think this is going to be and I encourage you to pause this video and think about it on your own. Well it seems kind of daunting at first, if you’re not very

familiar with the notation, but we just have to

remember what a function is. A function is just a mapping from one set of numbers to another. So for example, when

we’re saying G of two, that means take the number two, input it into the function G and

then you’re going to get an output which we are

going to call G of two. Now we’re going to use

that output, G of two, and then input it into the function F. So we’re going to input

it into the function F, and what we’re going to get is F of the thing that we

inputted, F of G of two. So let’s just take it step by step. What is G of two? Well when T is equal to two,

G of two is negative three. So when I put negative three

into F, what am I going to get? Well, I’m going to get negative

three squared minus one, which is nine minus one which

is going to be equal to eight. So this right over here is equal to eight. F of G of two is going

to be equal to eight. Now, what would, using

this same exact logic, what would F of H of two be? And once again, I encourage

you to pause the video and think about it on your own. Well let’s think about

it this way, instead of doing it using this little

diagram, here everywhere you see the input is X,

whatever the input is you square it and minus one. Here the input is H of

two, and so we’re going to take the input, which is H

of two, and we’re going to square it and we’re going to subtract one. So F of H of two is H of

two squared minus one. Now what is H of two? When X is equal to two, H of two is one. So H of two is one, so since

H of two is equal to one, this simplifies two one squared minus one, well that’s just going to be one minus one which is equal to zero. We could have done it

with the diagram way, we could have said, hey

we’re going to input two into H, if you input

two into H you get one, so that is H of two right over here. So that is H of two, and then

we’re going to input that into F, which is going

to give us F of one. F of one is one squared

minus one, which is zero. So this right over here is F of H of two. H of two is the input

into F, so the output is going to be F of our

input, F of H of two. Now we can go even further,

let’s do a composite. Let’s compose three of

these functions together. So let’s take, and I’m doing

this on the fly a little bit, so I hope it’s a good

result, G of F of two, and let me just think

about this for one second. So that’s going to be G of F of two, and let’s take H of G of

F of two, just for fun. Now we’re really doing

a triple composition. So there’s a bunch of

ways we could do this. One way is to just try to

evaluate what is F of two. Well F of two is going to be

equal to two squared minus one. It’s going to be four minus one or three. So this is going to be equal to three. Now what is G of three? G of three is when T is equal

to three, G of three is four. So G of three, this whole thing, is four. F of two is three, three of G is four. What is H of four? Well we can just look back

to our original graph here. When X is four, H of four is negative one. So H of G of F of two, is

just equal to negative one. So hopefully this you

somewhat familiar with how to evaluate the composition of functions.

where you been in my Schooling time? I would be another person… This math killed my dream…

Your explanation is so great and simple. I should've watched your videos earlier. Now I only got 6 hours left to the exam and know nothing except functions and it's graphics

saves my life everytime

headache

Thanks for the help. You sound like a less intimidating version of Bane that helps people with math! lol

very enlightning thank you very much!!!!!!!!!!!!!!!!

People will be coming to this for a long time, Thanks again.

Man i hate my class, My teacher makes this topic so hard when it is so easy. The logic behind this is so simple but my teacher gives us problems that are insane to simplify. I have tried so hard to get an A so far in her class and i have an 82.

😒😒😒😒😒😒 teachers now I say you watch them and learn how to teach

What if x and the f(x) formula are not given?

Oh my gosh thank you for everything you do Mr. Khan. I was about to throw the notebook against the wall and just accept defeat. Aleks does not tell me that those parentheses were for functions instead of multiplication.

thanks bro

i wish i could explain concepts as clearly as you do….

thanks alot!

Sal makes me nervous when he says .. 'Oh I dunno' when hes tryna think of examples off the top of his head lol. Such a wizard tho, biggest fan of KA.

What is this?