From my research, I have figured out that this is a Möbius transformation. The respective wiki page helped me understand a bit more, however I can't figure out how to obtain the image.
So lets talk about what I do know. Well we are describing the set of all values $f(z)$ where $|z| < 1$. I also know that any three points.
I also know that
Given a set of three distinct points z1, z2, z3 on the Riemann sphere and a second set of distinct points w1, w2, w3, there exists precisely one Möbius transformation f(z) which maps the zs to the ws
and from help from another forum I got the following:
To decide which part is the image of the interior |z| < 1 of the disc, figure out which point f sends to infinity.