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