Answer
$f^{-1}(x)=\frac{-2x}{1-x}$
Work Step by Step
To get the inverse function, we must "swap" $x$ and $f^{-1}(x)$ and then rearrange the equation so that $f^{-1}(x)$ is "alone" on one side. Hence here: $x=\frac{f^{-1}(x)}{f^{-1}(x)-2}\\x(f^{-1}(x)-2)=f^{-1}(x)\\-2x=f^{-1}(x)-xf^{-1}(x)\\-2x=(1-x)f^{-1}(x)\\f^{-1}(x)=\frac{-2x}{1-x}$
