<s>[INST] If $f(g(x)) = 4x^2-8x$ and $f(x)=x^2-4$, then what's the value of $g(x)$? I'm a little stuck with this simple function:
If ${f(g(x)) = 4x^2-8x}$ and ${f(x)=x^2-4}$, then what's the value of ${g(x)?}$
Any tips?
 [/INST] If $g$ is a polynomial, then note that the degree can't be higher than $1$. Suppose $g(x)=ax+b$. Then
$$f(g(x))=f(ax+b)=a^2x^2+2abx+b^2-4$$
Now we want $a^2=4$, $2ab=-8$, and $b^2-4=0$. Two such solutions are $a=2,b=-2$, or $a=-2,b=2$.
Pluggin those back in, we have
$$f(2x-2)=4x^2-8x+4-4=4x^2-8x$$
and
$$f(-2x+2)=4x^2-8x+4-4=4x^2-8x.$$
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