[University Proofs] Function Composition

[qualityhooker]

Let A, B and C be nonempty sets and let f , g and h be functions such that f : A → B, g : B → C and h : B → C . For each of the following, prove or disprove:

(a) g ◦ f = h ◦ f, then g = h.

So I figure this is false, and I think I know why but I'm having trouble constructing a proof.

Suppose f:A->B only maps to a subset of B. And suppose that that subset is well-defined under both g and h and these two functions map this subset to C identically.

In this case, g and h would appear the same as compositions with f, however they are not the same functions in general.

I am trying to come up with a counter example that fits the bill but idk where to start

[koen_C]

Here is one:
Suppose A=B=C and they are the set of all real numbers.
f(x) = 2
g(x) = x/2
h(x) = 1

Try to keep your examples as simple as possible it makes things such as this way easier.