Source TeX:

\begin{array}{ll} (f^{-1} \circ f)(x) & = f^{-1}(f(x))\\ &=\sqrt[3]{\pmb{\Big({(x-1)^3+2}\Big)}-2}+1\\ &=\sqrt[3]{(x-1)^3}+1\\ &=x-1+1\\f^{-1}(f(x))&=x \end{array}