An intro to real analysis question. eazy?

[eibon]

Homework Statement

Let f : A -> B be a bijection. Show that if a function g is such that f(g(x)) = x for
all x ϵ B and g(f(x)) = x for all x ϵ A, then g = f^-1. Use only the definition of a
function and the definition of the inverse of a function.

Homework Equations

The Attempt at a Solution

well since f is a bijection then there exist an f^-1 that remaps f back to A and since g does that then g is the unique inverse of f, or something like that please help I am not very good that this stuff

 

[losiu99]

Yes, you only need to prove uniqueness of an inverse function. If f(g(x))=x, then g(x) must be the unique argument a for which f(a) = x. And so, this condition completely defines g for all the arguments. g is then obviously equal to f^-1.

 

[eibon]

thanks losiu for your response.
but how do you prove that it is the unique inverse?