Question about inverse functions

[Spirochete]

If you've already found that F(g(X))=X, is it necessarry to also prove that g(f(X))=X to know that you have inverse functions? Would there be a case where the first statement is true but the second is false?

 

[MeJennifer]

If you've already found that F(g(X))=X, is it necessarry to also prove that g(f(X))=X to know that you have inverse functions? Would there be a case where the first statement is true but the second is false?

The inverse of many functions are not functions themselves.

 

[HallsofIvy]

In particular, if f(x)= x^2] and g(x)= \sqrt{x}, f(g(x))= (\sqrt(x))^2= x but g(f(x))= \sqrt{x^2}= |x|. Of course, if the domain of f is restricted to the non-negative numbers, then they are inverses.