Any Möbius transformation is injective, since it has a right inverse given by:

(How is this transformation described as a matrix?)
:?:
$R(f^{-1})=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right](ad-bc)$