Question 20
Let $\mathbb{R}^+$ be the set of positive real numbers. Find all non-negative real numbers $\alpha$ for which there exists a function $f : \mathbb{R}^+ \to \mathbb{R}^+$ such that
$$
f(x^{\alpha} + y) = (f(x + y))^{\alpha} + f(y)
$$
for any $x, y \in \mathbb{R}^+$.
$$
f(x^{\alpha} + y) = (f(x + y))^{\alpha} + f(y)
$$
for any $x, y \in \mathbb{R}^+$.