as you should already know to get the inverse of any expression, we start off by doing a quick switcheroo on the variables, and then solve for "y".
[tex]\bf \stackrel{f(x)}{y} = \cfrac{x}{x-1}\implies \stackrel{switcheroo}{x=\cfrac{y}{y-1}}\implies x(y-1)=y\implies xy-x=y \\\\\\ -x=y-xy\implies -x=\stackrel{\textit{common factor}}{y(1-x)}\implies \cfrac{-x}{1-x}=y \\\\\\ \cfrac{x}{-(1-x)}=y\implies \cfrac{x}{x-1}=\stackrel{f^{-1}(x)}{y}[/tex]
