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2019-01-25, 19:02   #169
Dr Sardonicus

Feb 2017
Nowhere

2×5×13×37 Posts

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=y3 is not the inverse function of y=x3, since substituting x for your inverse function will give y=(y3)3=x9.)
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 well-defined. And even then, the problem only really manifests itself at points where two or more possible inverses meet.