According to the Remarks in Section 2.5, since $\displaystyle f(z)=\frac{1}{z}$ is its own inverse function, the mapping $\displaystyle w=\frac{1}{z}$ on the extended complex plane maps the circle $\displaystyle |z-\frac{1}{2}|=\frac{1}{2}$ to the line Re(w) = 1. Verify this.

I am not sure how to do this without working backwards which isn't an easy method. Does anyone know how this can be done in a more efficient manner?

Thank you for your time,

Dustin.