2. (28 pts) Consider the function $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}$ defined by:
$f(a, b) = (a + b, a - b)$
Prove that $f$ is a bijection and give the inverse function $f^{-1}$, remember to specify the domain and codomain.