Chapter 4 - Exponential and Logarithmic Functions - Section 4.2 One-to-One Functions; Inverse Functions - 4.2 Assess Your Understanding - Page 292: 63

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

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