MathNet 2021 — Part 01 Question 90

Find all functions $f : \mathbb{R}^+ \to \mathbb{R}$ and $g : \mathbb{R}^+ \to \mathbb{R}$ such that
$$
f(x^2 + y^2) = g(xy)
$$
holds for all $x, y \in \mathbb{R}^+$.