How do you prove this math problem?

Question

How do you prove something like this?

If f is one-to-one and g: A-->B and h: A-->B are functions such that fog = foh (composition), then g = h?

Answer 1

If f is 1-1, then the inverse of f, f^(-1), exists.

Thus, if fog = foh, apply the inverse of f on both sides of the composition.

f^(-1)o(fog) = f^(-1)o(foh)
(f^(-1)of)og = (f^(-1)of)oh
Id_B o g = Id_B o h
g = h

Where the Id_B is the identity function on its domain B.

Answer 2

Since f is one to one, f has an inverse f⁻¹ from the range of f to B. Therefore, we have:

f∘g=f∘h
f⁻¹∘f∘g = f⁻¹∘f∘h
g=h (since f⁻¹∘f is the identity function).