In the image below (part (b)), Since $z < |3|$ before the transformation, does that simply imply that the region to be shaded after the transformation is definitely the inside of the circle and not its outside? It seems so to me but I don't want to jump to the conclusion.
Also is the convention to use dotted lines for the locus of the circle, since $z < |3|$ as opposed to $z \leq |3|$ ?
On
I'm not sure if you still want information on this question but I was recently stuck on a similar problem. I presume you were fine with part a), and if my working is correct I get the following equation for the circle in the $w$-plane:
$$ (u-\frac{9}{8})^2 + v^2 = (\frac{3}{8})^2 $$
This describes a circle with centre ($\frac{9}{8}$, 0) and radius $\frac{3}{8}$.
To get to this answer part of my working involved rearranging the given equation to make $z$ the subject and then taking the modulus of either side:
$$ |z| = \frac{|w|}{|1-w|} $$
From this step it is simple to replace the $|z|$ with $3$ as we are told this is what the modulus of $z$ is. But in part b) we are told that $|z|<3$. So the equation becomes:
$$ 3 > \frac{|w|}{|1-w|} $$
Following my working for a), I arrived at the conclusion that:
$$ (u-\frac{9}{8})^2 + v^2 > (\frac{3}{8})^2 $$
Hence it would seem that the new region to shade is outside the cirlce $C$ because the coordinates of any point in this region must lie a distance greater than $\frac{3}{8}$ from the circle centre.
Hope this is useful!
(With regards to dotted lines, I agree with your thinking!).
If $|z - (-i)|$ is very small, then $|w|$ will be very large. In fact, within a certain neighborhood of $-i$, the closer $z$ is to $-i$, the larger $|w|$ is. This implies that wherever the circle $C$ is, there will be points in a neighborhood of $-i$ that are mapped to points outside the circle $C$. But $|z| < 3$ for at least some of those points.
So it seems not to be a good assumption that the image of the points inside a circle will be inside the image of the circle. In this case, in fact, they are not.