This is kind of a reverse question.
A few years back I was presented with a functional equation problem, I don't remember it completely, and now I would appreciate the help of the math.SE hivemind to recreate it.
It concerned a function $f:\Bbb Q^+\to \Bbb Q$, with $\Bbb Q^+$ being the strictly positive rationals. The problem gave two identities for $f$, one of them being $f(x) = f(x^{-1})$. I can't for the life of me recall what the second one was. However, I believe it related $f(x)$ and $f(x+1)$. There is a possibility that $f(1) = 1$ was also included as a restraint.
What I do remember, though, is the solution. With $a, b\in \Bbb Z$ coprime, $f(\frac{a}{b}) = ab$. Also, that the two identities together is in some way related to the Euclidian algorithm.
Long story short, does anyone know what the second identity is / could be?