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! 

HINT: This is an example of Möbius transformation. Try to decompose the transformation $z\mapsto\frac{z+1}{2z+1}$ into simpler transformations as indicated in the linked page and follow the unit circle through.

For a nice depiction of Möbius transformations, see here.
