Most general solution to the equation $f(x) = f(1/x)$

Question

What is the most general solution of the functional equation $$f(x) = f(1/x)$$ for $x>0$?

Answer 1

Let $A$ be any set, let $g\colon(0,1]\to A$ be any function and define $f\colon (0,\infty)\to A$ by $$f(x)=\begin{cases}g(x)&\text{if }x\le 1\\g(\frac1x)&\text{if }x>1\end{cases} $$