Let $f : \mathbb{R}\setminus\{0\} \to \mathbb{R}$ be a non-constant function such that $f(y) + f\left(\frac{1}{y}\right) = 0$. I found that $f(y) = h(\log|y|)$ will be a solution , where $h$ is an odd function. Does any other solution exist ?
The need of this functional equation arose in order to get non-polynomial solutions of $g(x)g\left(\frac{1}{x}\right) = g(x) + g\left(\frac{1}{x}\right)$, assuming $g(x)$ to be nonzero I saw that $\frac{1}{g(x)} = h(\log|y|) + \frac{1}{1\pm y^n}$ would be a non-polynomial solution and I think any solution for the function $f$, as conditioned, would yield a non-polynomial solution for $g(x)$.