Find the image of the unit circle under the transformation $f(z)=\frac{z+1}{2z+1}$. How Do I approach these questions? Find the image of the unit circle under the transformation $f(z)=\frac{z+1}{2z+1}$. How Do I approach these questions?
I tried writing $z$ as $e^{i \phi}$, but I didn't know how to continue from there.
Thank you for any assistance! 

Let $C$ be the unit circle. Since $f$ is a Moebius transformation with no singularity on $C,$ $f(C)$ is a circle. Now $f$ is symmetric about the real axis, hence so is $f(C).$ It follows that $[f(-1),f(1)] = [0,2/3]$ is a diameter of $f(C).$ Therefore $f(C)$ is the circle $(x-1/3)^2 +y^2 = 1/9.$
