Question 54
Let $\mathbb{R}_{\gt 0}$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}_{\gt 0} \to \mathbb{R}_{\gt 0}$ such that for all positive real numbers $x$ and $y$,
$$
f(x^{2023} + f(x)f(y)) = x^{2023} + yf(x).
$$
$$
f(x^{2023} + f(x)f(y)) = x^{2023} + yf(x).
$$