Let C be the complex numbers. Consider the map $f: C \rightarrow C \times \mathbb{R}$ given by
$f(z) = (z^2, \text{Im}(z))$
($\text{Im}(z)$ is the imaginary part of $z$)
a. Is $f$ an immersion?
b. Is the restriction of $f$ to ${z : 1\le |z| \le 2}$ an immersion?
I'm confused about the f here because how would i take the derivative of this map?