Let $\Omega = \{ z = x + iy \in \mathbb{C} : |y| \leq 1 \}$. If $f(z) = z^{2} + 2$, then draw a sketch of $$f(\Omega) = \{ f(z) : z \in \Omega \}.$$ Justify your answer.
Let $\Omega = \{ z = x + iy \in \mathbb{C} : |y| \leq 1 \}$. If $f(z) = z^{2} + 2$, then draw a sketch of
$$f(\Omega) = \{ f(z) : z \in \Omega \}.$$
Justify your answer.