An exercise on Möbius transformations

Question

I had an exam on Complex Analysis and I could not solve the following exercise on Möbius transformations:

Let $f(z)=\frac{z+1}{z-1}$ and $A=\{z : \operatorname{Im}(z) >0\}\setminus \{|z|<1\}$. Find $f(A)$

I know this probably is not that difficult but I don't know how to solve these kind of exercises and I want to learn in case I want to contest my grade. Any help?

Answer 1

Moebius transformations map circles* (meaning circles or lines) to circles*. The problem tells about two circles* in the $z$-plane. Compute $f(-1)$, $f(0)$, $f(1)$, and $f(i)$ in order to obtain three points of each of the two image circles*. It turns out that $f(A)$ is one of the quadrants in the $w$-plane. In order to determine which one compute $f(2i)$ as well.