I was just deadly curious about a solution of a certain functional equation, or better, a question.
Given $f(x)$ and $g(x)$ as two functions domained in $A$ and $B$, and given the equation for f(x)
$$f(x)=g(f(x))$$
I found out, with some weird reasoning that
$f(x)=g^{-1}(g^{-1}(g^{-1}...(x))..)$
Now I want to ask, with which conditions is this conclusion correct? Because withour any doubt some conditions are to be met and satisfied, and certainly I have not satisfied them or mentioned them because....I do not know them!!! Any insight?
The statement $$ f(x) = g(f(x))$$ is simply the demand that each $y = f(x)$ is a fixed point of $g$, i.e., $y=g(y)$. If $S$ is the set of fixed points of $g$, then the set of solutions consists of all functions $f$ for which $f(x) \in S$ for all $x$. With no restrictions placed on $g$ there is no reason to assume that $g$ has any fixed points. In this case, there are no solutions $f$.
The existence of fixed points is a separate field of study with several theorems ensuring the existence of at least one fixed point. The fixed point theorems of Banach, Brouwer and Schauder are three distinct examples.