# Solve for function in a composition

I'm interested in some sort of algebraic system to solve for functions in function equations. What I mean is a way to solve questions of the form:

Given a function $f$, find a function $g$ such that $(f\circ g)(x)=h(x)$.

Which I don't understand how to solve. Why? Because algebra is useful in situations where the symbolic structure is known. However, the symbolic structure of $g$ is unknown. We can't just set $y=f(x)$, $u=g(x)$ and solve for u because the symbolic structure of $u$ relating to $y$ is not implicit in the problem.

Here's a simple example:

Given the function $f\colon x\mapsto n^x$, find a function $g$ such that $(f\circ g)(x)=x$.

The answer to this particular question is relatively trivial. We can set $g(x)=log_n(x)\implies (f\circ g)(x)=x$. But I don't see a larger system for dealing with all kinds of questions.

Either

• I am missing something really obvious
• or to solve questions of this form we need some apriori knowledge of some rules which dictate how to deal with all the situations that arise. **This would equate to being some sort of algebra to solve for functions$. •$x^{1/x}$is not the inverse of$x^x$. And finding the inverse of a function doesn't necessarily have an explicit easy-to-write-down answer in the general case. As for$f \circ g = h$(assuming that$f$has an inverse)$g = f^{-1} \circ h\$.
– dxiv
Oct 27 '16 at 1:47