Describe in words what g^-1 (z)= -1/z, f(z)= z+1, and g(z)= -1/z in succession do to the half plane, and hence explain geometrically why gfg^-1 has fixed point at the origin.

For this one, I was thinking that g and g^-1 are reflections about the y axis and the unit circle, and f reflects the plane over the y axis. I'm just not sure if that's correct, and if so why would the origin remain fixed even though f translated it?

Thanks in advance!