I'm having some trouble showing that a Mobius transformation $F$ maps $0$ to $\infty$ and $\infty$ to $0$ iff $F(z)=dz^{-1}$ for some $d \in \mathbb{C}.$ Mainly with the "only if" part. Do I need to use pictures?
This is Exercise $23$ in Section $3.3$ of Conway's Functions of One Complex Variable.
Write $f(z)=\frac{az+b}{cz+d}$. What are $f(0)$ and $f(\infty)$ in terms of $a,b,c,d$? Deduce further.
(It's a bad idea for the exercise to use the letter $d$ in my opinion; just know you're trying to conclude the function is a constant multiple of the reciprocal function $1/z$.)