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! 

You know $|z| = 1$. Let $A = \frac{z+1}{2z+1}$.
$$(2z+1)A = z + 1 \\
z(2A-1) = 1 - A\\
z = \frac{1-A}{2A-1}\\
1 = \frac{|1-A|}{|2A-1|}\\
|1-A| = |1 - 2A|\\
|1-A| = 2 \left|\frac{1}{2} - A \right|$$
The set of all such $A$ is always a circle.
Next, find two polar opposite points on the circle; that gives you the centre and radius.
