Some context, I was trying to derive something and stumbled upon $f(x,y)f(y,x)=1$, when $x , y \ne 0$. Could someone suggest some tricks or books that I could apply to these types of questions? By the way, I'm trying to solve for $f(x,y)$ and I'm aware that one solution could be $f(x,y)=\frac xy$, but I would like to know how to derive it.
Thank you.
There are infinitely many functions satisfying this property. You can start with any function $f(x,y)$ defined for $\{(x,y): x \leq y\}$ satisfying $f(x,x)=\pm 1$ which does not vanish at any point and define $f(y,x)=\frac 1 {f(x,y)}$ for $x >y$.