Chapter 4 - Exponential and Logarithmic Functions - Section 4.2 One-to-One Functions; Inverse Functions - 4.2 Assess Your Understanding - Page 292: 64

$f^{-1}(x)=\frac{x}{x+2}$

Step 1. $f(x)=-\frac{2x}{x-1} \Longrightarrow y=\frac{2x}{x-1} \Longrightarrow x=-\frac{2y}{y-1} \Longrightarrow y=\frac{x}{x+2} \Longrightarrow f^{-1}(x)=\frac{x}{x+2}$ Step 2. Check $f(f^{-1}(x))=-\frac{2(\frac{x}{x+2})}{(\frac{x}{x+2})-1}=x$ and $f^{-1}(f(x))=\frac{-\frac{2x}{x-1}}{(-\frac{2x}{x-1})+2}=x$