Chapter 4 - Exponential and Logarithmic Functions - Section 4.2 One-to-One Functions; Inverse Functions - 4.2 Assess Your Understanding - Page 292: 72

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

Step 1. $f(x)=\frac{x^2+3}{3x^2}, x\gt0 \Longrightarrow y=\frac{x^2+3}{3x^2} \Longrightarrow x=\frac{y^2+3}{3y^2} \Longrightarrow y=\sqrt {\frac{3}{3x-1}}\Longrightarrow f^{-1}(x)=\sqrt {\frac{3}{3x-1}}$ Step 2. Check $f(f^{-1}(x))=\frac{(\sqrt {\frac{3}{3x-1}})^2+3}{3(\sqrt {\frac{3}{3x-1}})^2}=x$ and $f^{-1}(f(x))=\sqrt {\frac{3}{3(\frac{x^2+3}{3x^2})-1}}=x$