Mobius transformation satisfying certain properties

I'm having some trouble showing that a Mobius transformation F maps 0 to$\mathrm{\infty}\text{}\text{and}\text{}\mathrm{\infty}$ to 0 iff $F\left(z\right)={dz}^{-1}$ for some $d\in \mathbb{C}$ . Mainly with the "only if" part. Do I need to use pictures?

I'm having some trouble showing that a Mobius transformation F maps 0 to