Conformal map from unit disc $D\setminus\lbrace0\rbrace$ to $\mathbb{C}\setminus\bar D$

Question

Question: If $D$ is the unit disc, find a conformal equivalence from $D\setminus\lbrace0\rbrace$ to $\mathbb{C}\setminus \bar D$

My Attempt: I have no idea how to start this problem... I don't think we are allowed to use linear fractional transformations because that chapter comes after this problem in this textbook. All I know how to do are conformal maps to/from unit discs to parts of the plane (UHP, RHP, quadrants, etc.), but I can't figure out how to apply that to mapping to something like $\mathbb{C}\setminus \bar D$.

Any tips? Thank you so much for any help! :)

Answer 1

What about $f(z)=1/z?$ Doesn't this do the trick?