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