Which function is the inverse of $f(x)=\frac{4}{3} x^3+1$?

A$g(x)=\frac{\sqrt[3]{12 x+12}}{2}$

B$g(x)=\frac{\sqrt[3]{6 x-6}}{2}$

C$g(x)=\frac{\sqrt[3]{12 x-12}}{2}$

D$g(x)=\frac{\sqrt[3]{6 x+6}}{2}$