since you know that to get the inverse of any expression we start off by doing a quick switcheroo on the variables and then solve for "y".
[tex]\stackrel{f(x)}{y}=\sqrt[3]{\cfrac{x+6}{2}}\implies \stackrel{\textit{quick switcheroo}}{x=\sqrt[3]{\cfrac{y+6}{2}}}\implies \stackrel{\textit{cubing both sides}}{x^3=\cfrac{y+6}{2}} \\\\\\ 2x^3=y+6\implies 2x^3-6=\stackrel{f^{-1}(x)}{y}[/tex]
