Find the image of D on Möbius transformation. I am not sure how to solve the following exercise on Möbius transformations:
Let $D=\{z:|z-1|\le\sqrt{2}\wedge|z+1|\le\sqrt{2}\}$ and $f(z)=\frac{-2}{z+i}$. Find the image of the set D through the function f.
I drew the set D.
Then I computed some points in it and they went to something like parabola. I would like to draw that new set and I am not sure how should I do it. Any help?

Hint: Möbius transformations take lines and circles to lines or circles, right? And they're continuous. This means that you can see what happens to the boundary components $|z-1| = \sqrt{2}$ and $|z+1| = \sqrt{2}$, and the image of $D$ will be one of the regions bounded by the images of those two arcs. Desmos (try it) gives that those images will be two lines with slopes approximately $\pm 1$ so the image of $D$ will look like what's above the graph of $|x|$ after some vertical shift. To do it analytically, note that $$f(z) = \frac{-2z-2i}{|z+i|^2}$$and set $z = 1+\sqrt{2}e^{it}$, $z = -1+\sqrt{2}e^{it}$ to see what happens.
