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Old 2019-01-25, 19:02   #169
Dr Sardonicus
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Feb 2017

2·33·5·19 Posts

Originally Posted by S485122 View Post
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.
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