Question 58
Let $\mathbb{R}^*$ be the set of non-zero real numbers. Find all functions $f: \mathbb{R}^* \rightarrow \mathbb{R}^*$ such that
$$
f\left(x^{2}+y\right)=f^{2}(x)+\frac{f(x y)}{f(x)}
$$
for all $x, y \in \mathbb{R}^*,\ y \neq -x^{2}$.
$$
f\left(x^{2}+y\right)=f^{2}(x)+\frac{f(x y)}{f(x)}
$$
for all $x, y \in \mathbb{R}^*,\ y \neq -x^{2}$.