TITLE: fractional linear transformations
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: If you plug in $z = e^{i\theta}$ into your linear fractional transformation you get
$${3e^{i\theta} + i \over -ie^{i\theta} + 3}$$
This is the same as 
$$e^{i\theta} {3 + ie^{-i\theta} \over 3 - ie^{i\theta}}$$
The numerator and denominators of the fraction are complex conjugates of each other, so it has magnitude $1$, as does $e^{i\theta}$. So their product also has norm $1$ for all $\theta$ and therefore the linear fractional transformation takes the unit circle to itself.
In general when it takes the unit circle to itself you can use factorizations this way to show it.