\[\begin{eqnarray*} x&=&\dfrac{y+2}{y-3}\\ xy-3x&=&y+2\\ xy-y&=&3x+2\\ (x-1)y&=&3x+2\\ y&=&\dfrac{3x+2}{x-1}\\ f^{-1}(x)&=&\dfrac{3x+2}{x-1} \end{eqnarray*}\]