How do I calculate the possible branch cuts of $f(z)=\frac{1}{\sqrt{z}}$? I know the branch cuts of $g(z)={\sqrt{z}}$ is the same as those of $\log(z)$, so they are halflines originating at the origin. How can I use this to find the branch cuts of $f(z)=\frac{1}{\sqrt{z}}$?
I know that $\frac{1}{\sqrt{z}}=e^{-\frac12\log(z)}$ which is continuous on the same domain on which $\log(z)$ is continuous by the composition rule.  So the branch cuts of $f$ must be the same as those of $\log(z)$.  is this true?

Yes.  Notice that once you have chosen a branch for $\sqrt{z}$, you have chosen a branch for $\frac{1}{\sqrt{z}}$.  That is, the simple algebraic operations, addition, subtraction, multiplication, and division, do not introduce branching.
