Consider \[ f(z)=\frac12\left(z+\frac1z\right),\qquad z\in\mathbb C\setminus\{0\}. \]
Show that \(f\) is univalent on the punctured disk \(\mathbb D\setminus\{0\}\). What is the image of \(|z|=r<1\) under this map?
Show that \(f\) is univalent on \(\{z\in\mathbb C:|z|>1\}\). What is the image of this domain?
What is the inverse mapping \[ f^{-1}:\mathbb C\setminus[-1,1]\longrightarrow \mathbb D\setminus\{0\}? \]