$f(x)$ from $f(g(x))$

Question

Is it always possible to find $f(x)$ if the composite function $h(x) = f(g(x))$ and $g(x)$ are given?

In other words, can there be any cases where, for given $h(x)$, we can not express it in an explicit function of $g(x)$?

Answer 1

Note that if $g(x)$ is invertible then $$f(x) = h(g^{-1}(x)).$$ What happens if $g(x)$ is not invertible. Consider, e.g., $f(x)=x$ and $g(x) = 1$.