What is the domain of $\frac{1}{f(x)}$ in this case? Suppose $f(x) = \frac{x-1}{x-2}$ ,
Then $\frac{1}{f(x)} = \frac{1}{\frac{x-1}{x-2}}$
So would the domain be all real numbers excluding 1 and 2 or would the domain include 2? Since
$\frac{1}{\frac{x-1}{x-2}} = \frac{x-2}{x-1}$ ,
I guess I should also ask whether the denominator this case is $\frac{x-1}{x-2}$ or just $x-1$.

One typically understands "$\frac{1}{f(x)}$" as the composition of the two functions $x \mapsto f(x)$ and $x \mapsto \frac{1}{x}$. So we have to apply $f(x)$ first, hence 2 is not in the domain.
