How to describe the graph of the complex exponential function $f(z)=e^z$?

Question

This is homework

(a)Describe the domain of definition and the range
(b)Describe the image of the vertical line Re $z=1$
(c)Describe the image of the infinite strip $0 \le \text{Im } z \le \frac{\pi}{4}$

I know $e^z=e^xe^{iy}=e^x(\text{cos } y + i\text{sin }y)$
and this is a circle with radius $e^x$

For (a) since the radius of the circle is, $0 \lt r$, does the domain and range include all real numbers except zero?

For (b) I don't understand what it wants me to say about the line, this is just a vertical line that goes through the real axis at $1$?

For (c) I think this just corresponds to the area in the first quadrant where the argument is from $0$ to $\frac{\pi}{4}$

Answer 1

Re (a): $$ e^0 = 1 $$

Re (b): Yes.

Re (c): No. It is the area between $\text{Im } z = 0$ (the real axis) and $\text{Im } z = \pi/4$