[tex]f^{-1}(x)[/tex] is supposed to be a function such that
[tex]f^{-1}(f(x))=x[/tex]
In this case, we need
[tex]f^{-1}(\sqrt[3]{x-2})=x[/tex]
To recover [tex]x[/tex] from [tex]\sqrt[3]{x-2}[/tex], we would first need to raise [tex]\sqrt[3]{x-2}[/tex] to the third power:
[tex](\sqrt[3]{x-2})^3=x-2[/tex]
Then add 2:
[tex](x-2)+2=x[/tex]
To recap, we carried out
[tex]f^{-1}(\sqrt[3]{x-2})=(\sqrt[3]{x-2})^3+2=x[/tex]
which implies that the inverse function is
[tex]f^{-1}(x)=x^3+2[/tex]
To verify: we should also have that [tex]f(f^{-1}(x))=x[/tex]. We get
[tex]f(x^3+2)=\sqrt[3]{(x^3+2)-2}=\sqrt[3]{x^3}=x[/tex]
