Finding a Mobius Transformation using two points

Question

In lectures I have seen how to find a Mobius Transformation from 3 points using the ratio of $z_1, z_2, z_3$ and $w_1, w_2, w_3$ but not when only given two. The question word for word is:'Is there exactly one Mobius Transformation, $f$, such that $f(3)=4$ and $f(4)=3$. Justify your answer.' This is a new topic for me so I know I must be missing something and cant just make $z_3=0$ and $w_3=0$. Thanks

Answer 1

No, there is not a unique Mobius transformation such that $f(3)=4$ and $f(4)=3$. Take $f(z)=(12/z)$ and $f(z)=(12z/(7z-12))$.