Question 52
Let $\alpha \neq 0$ be a real number.
Find all functions $f: \mathbb{R}_{\gt 0} \to \mathbb{R}_{\gt 0}$ with
$$
f(f(x) + y) = \alpha x + \frac{1}{f\left(\frac{1}{y}\right)}
$$
for all $x, y \in \mathbb{R}_{\gt 0}$.
(Walther Janous)
Find all functions $f: \mathbb{R}_{\gt 0} \to \mathbb{R}_{\gt 0}$ with
$$
f(f(x) + y) = \alpha x + \frac{1}{f\left(\frac{1}{y}\right)}
$$
for all $x, y \in \mathbb{R}_{\gt 0}$.
(Walther Janous)