Proving Equivalence and Function Equality in Real Analysis

[SomeRandomGuy]

Hey guys, wasn't sure what forum to post this in. So if this is the wrong forum, I apologize. Anyway, I have a problem in Real Analysis that I can't quite get. Here it is:

Let f:A->B and R is a relation on A such that xRy iff f(x) = f(y).
a.) Prove R is an equivalence relation
b.) Show g:A->E is surjective
c.) Show h:E->B is injective
d.) Prove f(x) = h(g(x)).

I solved parts a, b, and c. My problem is part d... I don't even know where to begin. It just doesn't make sense to me when I think about it. Thanks for any help.

EDIT: I just realized I didn't put what E is. E is the equivalence classes on any particular element. So, it's the set of all equivalence classes for this function.

 

[Galileo]

For two functions to be equal, they have to send the same element to the same image.

 

[SomeRandomGuy]

For two functions to be equal, they have to send the same element to the same image.

So are we showing that if f(x) = h(g(x)), then g(x) = x? Here is exactly what I have written so far:

"Proof: In order to show that two functions are equal, we must show that for any x in the domain, we will get the same output y in the codomain. So, if f(x) = x, then h(g(x)) = x as well. By the definition if being injective, x = g(x)."

I'm lost from there.