ÖMO 2018 Problem 1

Let $a \ne 0$ be a real number. Find all functions $f : R_{>0}\to R_{>0}$ with $$f(f(x) + y) = ax + \frac{1}{f\left(\frac{1}{y}\right)}$$ for all $x, y \in R_{>0}$ .

(Proposed by Walther Janous)