If $f\left(x\right)=\frac{3x+2}{5x-3},$ then

 $$\begin{gathered} let f^{ }\left(x\right)=y \Rightarrow \frac{3x+2}{5x-3}=y \Rightarrow 3x+2=5xy-3y \\ \Rightarrow x\left(5y-3\right)=3y+2 \end{gathered}$$ 

$$\begin{gathered} \Rightarrow x =\frac{3y+2}{5y-3} \\ \Rightarrow f^{-1}\left(x\right)=\frac{3x+2}{5x-3} \\ \Rightarrow f^{-1}\left(x\right)=f\left(x\right) \end{gathered}$$