Curves Bounding Complex Function So I have the following complex function: $$f(z) = z^2$$.
This function maps the following region:
$$ D = \{z=x+iy : 1\leq x \leq 3 \hspace{3mm} \& \hspace{3mm} 1 \leq y \leq 2\}$$
onto $G \subset \mathbb{C}$ 
What I want to do is find the equations of the curve bounding $G$ - and then to subsequently plot $G$. However, being a total complex beginner, the only thing I can do with any confidence is to determine the real and imaginary parts of $f(z)$, with anything further merely being guesswork. 
Any help or guidance would be greatly appreciated. 

The first step is to plug $z=x+iy$ into $f(z)=z^2$ and expand to get
$$f(x+iy) = (x+iy)^2 = (x^2-y^2) + (2xy)i.$$
To get one boundary, fix $x=1$ and plot 
$$z(t) = (1-t^2)+2t\,i$$
for $1<t<2$. This should be a parabola. Then do the same thing for $x=3$. You can also plot several curves for choices of $x$ between $1$ and $3$. Finally switch the roles of $x$ and $y$ to examine
$$z(t) = (t^2 - y^2) + 2t \, y \, i$$
for the appropriate choices of fixed $y$.  I guess the result ought to look something like so:

That particular image was produced by Desmos and you can see how it was assembled here.
