Question 143
Let $\mathbb{Q}$ be the set of rational numbers. Let $f: \mathbb{Q} \rightarrow \mathbb{Q}$ be a function such that the following property holds: for all $x, y \in \mathbb{Q}$,
$$
f(x+f(y))=f(x)+y \quad \text{or} \quad f(f(x)+y)=x+f(y).
$$
Determine the maximum possible number of elements of $\{f(x)+f(-x) \mid x \in \mathbb{Q}\}$.
$$
f(x+f(y))=f(x)+y \quad \text{or} \quad f(f(x)+y)=x+f(y).
$$
Determine the maximum possible number of elements of $\{f(x)+f(-x) \mid x \in \mathbb{Q}\}$.