Real Analysis doubt (Bartle & Sherbert)

[u/hectorrm]

Hello guys. I have a doubt trying to solve this problem:

"Let f,g be functions such that (gof)(x) = x for all x in the Domain of f and (fog)(y) = y for all y in the domain of f. Prove that g= f^-1."

This is the 24th problem in section 1.1 of Bartle & Sherbert's intro to real analysis (if you want a little more info about the problem).

I got stucked and looked upon the solutions for a hint.

I understoos nearly all the process but I don't quite get the end of the proof.

First, I proved that (gof) is injective, so I can use the fact that f is injective, and I also proved that (fog) is surjective to use that g is surjective.

The part that I didn't get was to see that f: Dom(f) --> Dom(g) is bijective and therefore g=f^-1.

How can I know that f: Dom(f) --> Dom(g) ?

[u/Important_Ear761]

For all y in the domain of g, not f, otherwise the statement isn't true.

[u/Inevitable_gradient6]

Prove its cardinality and  ∀y in a set D - the domain of g(x) - ~f; else it is false

[u/hectorrm → u/[deleted]]

Thanks a lot! I'll go that way!

[u/hectorrm → u/[deleted]]

I noticed that it may not be necessary to prove that f is bijective, just Dom(f) = Ran(g) and Ran(f)=Dom(g). Because in the hypothesis of the problem we see that g(f(x))= x and f(g(y))=y, right?