Figuring out function composition

Question

If I know $f(x)$ and $f(g(x))$,

how do I figure out what is $g(x)$?

I can solve already some simple cases, but I'm looking for a general strategy.

Particularly in harder cases when just trying to figure it out is hard.

Answer 1

Set $h$ so that $f(g(x))=h(x)$.

To get $g(x)$, $f$ is disturbing. $g$ is composited with $f$ from the left. You need the inverse operation. The inverse operation of composition from the left is the left inverse.

Let $f^{-1}$ denote the left inverse of $f$:

$$g(x)=f^{-1}(h(x))$$ $\ $

Let's presuppose that $f$ and $g$ are functions.

If $f$ is bijective, you can set the inverse function (compositional inverse / inverse) of $f$ for $f^{-1}$.

If $f$ is not bijective, you have to choose suitable partial inverses of $f$ for $f^{-1}$. The equation above can give you $g$ as a piecewise given function then.