Answer
See the details below.
Work Step by Step
To find the inverse of $f(x)=\frac{x-2}{x-1}$, we have
$$y=\frac{x-2}{x-1}\Longrightarrow yx-y=x-2\Longrightarrow x=\frac{y-2}{y-1}.$$
So the inverse function is given by
$$f^{-1}(x)=\frac{x-2}{x-1}.$$
Its domian is $R-\{1\}$ and the range is also $R-\{1\}$.
