Prove that f(x) = x/k can be defined on [0, 0) by the require- ment that it be the inverse function of g(x) = x* on (0, ∞), where k is any positive integer. Use the inverse function theorem to derive...

Can we explicity "prove that f(x)=x^{1/k} can be defined on [0,\infty) by the requirement that it be the inverse function of g(x)=x^k on [0,\infty), where k is any positive integer"?

Extracted text: Prove that f(x) = x/k can be defined on [0, 0) by the require- ment that it be the inverse function of g(x) = x* on (0, ∞), where k is any positive integer. Use the inverse function theorem to derive the usual formula for f'.