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# Finding composite functions

CCSS.Math:

## Video transcript

when we first got introduced a function composition we looked at of actually evaluating functions at a point or compositions of functions at a point what I want to do in this video is come up with expressions that define a function composition so for example I want to 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 one stead 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 for me to 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 1 now let's go the other way around what is G of f of X what is G of f of X and once again 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 where 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 1 plus f of X 1 plus f of X and what's that equal to well f of X is equal to the square root of x squared minus 1 x squared minus 1 so it's going to be that over 1 plus the square root 1 plus the square root of x squared minus 1 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 what the composition one way is it's going to be the same as the composition the other way unless the functions are designed in a fairly special way