Question 51
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that:
$$
f(xy + x + y) + f(xy - x - y) = 2(f(x) + f(y)), \forall x, y \in \mathbb{R}. \quad (1)
$$
Prove that $f$ fulfills the relation:
$$
f(x + y) = f(x) + f(y), \forall x, y \in \mathbb{R}. \quad (2)
$$
$$
f(xy + x + y) + f(xy - x - y) = 2(f(x) + f(y)), \forall x, y \in \mathbb{R}. \quad (1)
$$
Prove that $f$ fulfills the relation:
$$
f(x + y) = f(x) + f(y), \forall x, y \in \mathbb{R}. \quad (2)
$$