Question 185
Consider functions $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ satisfying $f(0) = 2022$ and
$$
f(x + g(y)) = x f(y) + (2023 - y) f(x) + g(x), \quad \forall x, y \in \mathbb{R}.
$$
a) Prove that $f$ is surjective and $g$ is injective.
b) Find all functions $f, g$ satisfying the given conditions.
$$
f(x + g(y)) = x f(y) + (2023 - y) f(x) + g(x), \quad \forall x, y \in \mathbb{R}.
$$
a) Prove that $f$ is surjective and $g$ is injective.
b) Find all functions $f, g$ satisfying the given conditions.