Calculus (3rd Edition)

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 $R-\{1\}$.