Question 7
Let Q be the set of rational numbers. Let $f: \mathbb{Q} \to \mathbb{Q}$ be a function such that the following property holds: for all $x, y \in \mathbb{Q}$,
$$ f(x+f(y))=f(x)+y\qquad \text{or}\qquad f(f(x)+y)=x+f(y). $$
Determine the maximum possible number of elements of $ \{f(x) + f(-x) \mid x \in \mathbb{Q}\} $.
$$ f(x+f(y))=f(x)+y\qquad \text{or}\qquad f(f(x)+y)=x+f(y). $$
Determine the maximum possible number of elements of $ \{f(x) + f(-x) \mid x \in \mathbb{Q}\} $.