We want to make a Möbius transformation such that $f(3)=0, \ f(2)=1, \ f(1)=\infty$, so we use the cross-ratio:
\begin{equation} \frac{(w-w_1)(w_2-w_3)}{(w-w_3)(w_2-w_1}=\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1} \end{equation}
and insert for $w_1=0,\ w_2=1, \ w_3=\infty$ and $z_1=3, \ z_2=2, \ z_3=1$.
We get, where the infinity terms cancel out:
\begin{equation} \frac{(w-0)(1-\infty)}{(w-\infty)(1-0)}=\frac{(z-3)(2-1)}{(z-1)(2-3)}\rightarrow w=\frac{(3-z)}{(z-1)} \end{equation}
This looks like this:
Would this be OK?
Thanks