How would I analytically solve functional equations like this?
$$f(g(x)) = h(x), x\in \mathcal S$$
if $g$ and $h$ are known, $f$ is what we solve for.
I have tried solving special cases, for example $$\cases{g(x) = x^2\\h(x)=(x+0.2)^2}$$
For this one I get multiple solutions when going from $x^2 \to \pm x$ ("undoing" $g$). This could be avoided by splitting the set $\mathcal S$ into two: $\{[-\infty,0],[0,\infty]\}$. But is there some mathematical method / algorithm that can account for the whole set of solutions simultaneously?
Being an engineer I have built something of a "machinery" that can solve it for me, I am just curious of how it may or may not relate to the typical analytical methods.