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# Intro to composing functions

Sal explains what it means to compose two functions. He gives examples for finding the values of composite functions given the equations, the graphs, or tables of values of the two composed functions. Created by Sal Khan.

Video transcript

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.