Quote:
Originally Posted by S485122
IMHO opinion your definition of the inverse function is not correct.
If y=f(x) the inverse function y=f^{ 1}(x) will be such that f(g(x))=y this is obviously not the case your calculations. (For instance it is obvious that x=y^{3} is not the inverse function of y=x^{3}, since substituting x for your inverse function will give y=(y^{3})^{3}=x^{9}.)

Your substitution is inconsistent with the definition you give. Using the example y = x^3, the inverse function implicitly defined by x = y^3 is not y^3, but rather g(x) = y. And, by its formulation (g(x))
^{3} = x.
The only real difficulty with the implicit formulation of the inverse function is that it may not be welldefined. And even then, the problem only really manifests itself at points where two or more possible inverses meet.