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.


  • 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$.
  • $\begingroup$ $x^{1/x}$ 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) $g = f^{-1} \circ h$. $\endgroup$
    – dxiv
    Oct 27 '16 at 1:47

Yes there is such a theory – called functional equations. Specifically, see the section on solving function equations.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.