Consider the implicit equation $ f^{-1} (x)=g(x)$. The function $g(x)$ is known and at least can be computed numerically. It may be piecewise continous or oscillating but it is always positive $ g(x) \ge 0 $. Here $ f(x) $ is not known.
Could it be that is there a function $ g(x) $ so it is never invertible and hence we cannot get $ f(x) $?