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# Finding composite functions

Video transcript

Voiceover:When we first got introduced
to function composition, we looked at actually
evaluating functions at a point, or compositions of functions at a point. What I wanna do in this video is come up with expressions that define
a function composition. So, for example, I wanna figure out, what is, f of, g of x? f of, g of x. And I encourage you to pause the video, and try to think about it on your own. Well, g of x in this case,
is the input to f of x. So, wherever we see the x in this definition, that's the input. So we're going to replace
the input with g of x. We're going to replace the x with g of x. So, f of g of x is going to be
equal to the square root of- Well instead of an x,
we would write a g of x. g of x, g of x squared. g of x squared, minus one. Now what is g of x equal to? Well, g of x is this
thing right over here. So this is going to be
equal to the square root of, g of x, is x over 1 plus x. We're going to square that. We're going to square that, minus 1. So f of g of x, is also a function of x. So f of g of x is a square root of, and we could write this as x
squared over 1 plus x squared, but we could just leave it like this. It's equal to the square
root of this whole thing, x over 1 plus x, squared, minus one. Now let's go the other way round. What is g of f of x? What is g of f of x? And once again, I encourage
you to pause the video, and try to think about it on your own. Well, f of x is now the input into g of x. So everywhere we see the x here, we'll replace it with f of x. So this is going to be equal to, this is going to be
equal to, f of x, over- Let me do it in the same color, so you can appreciate it better. f of x over, one plus f of x. One plus f of x. And what's that equal to? Well, f of x is equal to the square root, of x squared minus one. x squared minus one. So it's gonna be that over
1, plus the square root. One plus the square root
of x squared minus one. So this is a composition f of
g of x, you get this thing. This is g of f of x,
where you get this thing. And to be clear, these are
very different expressions. So typically, you want
the composition one way. This isn't gonna be the
same as the composition the other way, unless the functions are designed in a fairly special way.