Chapter 4 - Exponential and Logarithmic Functions - Section 4.2 One-to-One Functions; Inverse Functions - 4.2 Assess Your Understanding - Page 292: 65

$f^{-1}(x)=\frac{x}{3x-2}$

Step 1. $f(x)=\frac{2x}{3x-1} \Longrightarrow y=\frac{2x}{3x-1} \Longrightarrow x=\frac{2y}{3y-1} \Longrightarrow y=\frac{x}{3x-2} \Longrightarrow f^{-1}(x)=\frac{x}{3x-2}$ Step 2. Check $f(f^{-1}(x))=\frac{2(\frac{x}{3x-2})}{3(\frac{x}{3x-2})-1}=x$ and $f^{-1}(f(x))=\frac{\frac{2x}{3x-1}}{3(\frac{2x}{3x-1})-2}=x$