2011 AMC 12A Problem 23

Let f(z)= \frac{z+a}{z+b} and g(z)=f(f(z)), where a and b are complex numbers. Suppose that \left| a \right| = 1 and g(g(z))=z for all z for which g(g(z)) is defined. What is the difference between the largest and smallest possible values of \left| b \right|?

\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \sqrt{2}-1 \qquad \textbf{(C)}\ \sqrt{3}-1 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.