Solving Compositions of Functions Algebraically Doing a little practice exercise, and came across the following question:
There are three functions: $f(x)$, $g(x)$, and $h(x)$. 
Find $f(x)$ and $g(x)$, such that $h(x) = f \circ g(x)$ and $g(x) = \sqrt{4} + 8$. 
The value of $h(x)$ is defined as $h(x) = (\sqrt{4} + 8) ^ 4$.
I'm not really sure how to approach the problem. I know I'm going to have to solve this algebraically, but I just can't seem to wrap my mind around the manipulation of functions as opposed to variables and numbers.
 A: The question, as currently posted (even after I edited it) is a mess. Are $g(x)$ and $h(x)$ meant to be constant functions? Was that supposed to be $g(4) = \sqrt{4} + 8$ and $h(4)=(\sqrt{4}+8)^4$, or was it supposed to be $g(x)=\sqrt{x}+8$ and $h(x)=(\sqrt{x}+8)^4$? Something else?
In any case: a composition is a function that is obtained by first applying one function, and then taking the output of that function and plugging it into the next function.
Take the expression $(\sqrt{4}+8)^4$. Suppose you were trying to actually compute this number. How would you go about it? First, we would take the number $4$ and plug it into the square root to obtain $\sqrt{4}$. The we would take the ouput of that, and add $8$. Finally, we would take the result we got from doing all that, and raise it to the fourth power.
So you can think of this as the process of doing three things, one after another, where the input to each of the steps is the output of the previous step:


*

*Take the square root of what you have;

*Add $8$ to what you have;

*Raise what you have to the fourth power.


Now, you are told to try to express this as the result of doing two things in succession: the first one is $g(x)$, and the second one will be $f(x)$. And you are told that the first thing you are doing is $g(x)=\sqrt{4}+8$. So, what else do you need to do to the output of this process in order to get to the final result? You need to raise what you are given to the fourth power. So, whatever $f(x)$ is, it should take the output of $g(x)$ and raise it to the fourth power.
So, what is $f(x)$?
