Answer
$f^{-1}(x)=\frac{5x+2}{3x}$.
domain and range of $f(x)$: $\{x|x\ne\frac{5}{3} \}$ and $\{y|y\ne0\}$.
domain and range of $f^{-1}(x)$: $\{x|x\ne0 \}$ and $\{y|y\ne\frac{5}{3}\}$.
Work Step by Step
1. $f(x)=\frac{2}{3x-5} \Longrightarrow y=\frac{2}{3x-5} \Longrightarrow x=\frac{2}{3y-5} \Longrightarrow y=\frac{5x+2}{3x} \Longrightarrow f^{-1}(x)=\frac{5x+2}{3x}$.
2. Check $(f\circ f^{-1})(x)=\frac{2}{3(\frac{5x+2}{3x})-5}=x$ and $(f^{-1}\circ f)(x)=\frac{5(\frac{2}{3x-5})+2}{3(\frac{2}{3x-5})}=x$.
3. The domain and range of $f(x)$: $\{x|x\ne\frac{5}{3} \}$ and $\{y|y\ne0\}$.
The domain and range of $f^{-1}(x)$: $\{x|x\ne0 \}$ and $\{y|y\ne\frac{5}{3}\}$.
