# What can be said about $f$ in $\frac{f(x,y)}{f(y,x)}=\frac{g(x,y)}{g(y,x)}$ if $g$ is known?

Suppose you have the equailty $\frac{f(x,y)}{f(y,x)}=\frac{g(x,y)}{g(y,x)}$, $x,y\in\mathbb{R}$, and $g$ is known. What non-trivial facts about $f$ can be deduced from this? (Assume $f(x,y)\neq f(y,x)$ and $g(x,y)\neq g(y,x)$.)

• Nothing can be said. $f=(x+y)$ and $g=(x^2+y^2)$ – 1 0 May 4 '16 at 11:51
• One can say that $f=g\cdot h$ where the function $h$ is symmetric. – Did May 4 '16 at 12:11
• Good point @MithleshUpadhyay. I should add to the question that neither $f$ nor $g$ are symmetric in their arguments to exclude constant ratios. – epsilone May 4 '16 at 12:14
• Nothing can be said. $f=\frac{x}{x+y}$ and $g=\frac{x}{x^2+y^2}$ – 1 0 May 4 '16 at 12:20