Chapter 4 - Exponential and Logarithmic Functions - Section 4.2 One-to-One Functions; Inverse Functions - 4.2 Assess Your Understanding - Page 292: 66

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

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