as you already know, to get the inverse of any expression we start off by doing a quick switcheroo on the variables and then solving for "y", let's do so.
[tex]\stackrel{f(x)}{y}~~ = ~~\cfrac{2x}{x+3}\implies \stackrel{\textit{quick switcheroo}}{x~~ = ~~\cfrac{2y}{y+3}}\implies xy+3x=2y\implies xy-2y+3x=0 \\\\\\ xy-2y=-3x\implies \stackrel{\textit{common factoring}}{y(x-2)=-3x}\implies y=\cfrac{-3x}{x-2}\implies \stackrel{f^{-1}(x)}{y}=\cfrac{3x}{2-x}[/tex]
