Question 63
Let $\alpha$ be a real number.
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying
$$
f(f(x) + y) = f(x^2 - y) + \alpha f(x)y
$$
for all $x, y \in \mathbb{R}$.
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying
$$
f(f(x) + y) = f(x^2 - y) + \alpha f(x)y
$$
for all $x, y \in \mathbb{R}$.