Školjka

%V0 Determine the functions $f : \mathbb{Q} \to \mathbb{Z}$ satisfying $$ f \left( \frac{f(x) + a}{b} \right ) = f \left( \frac{x + a}{b} \right ) $$ for all $x \in \mathbb{Q}$, $a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{>0}$. (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.)