Prove that if f A B is an injective function and h A C is

Prove that if f : A B is an injective function and h : A C is any function, then there always exists a function g : B C with h = g f.

Solution

Let, b be in f(A)

So, f(a)=b for some a in A

Let, h(a)=c

Define:

g(b)=c

So,

g(f(a))=c

the function gof is well defined because f is injective

Hence, g(f(a))=c=h(a)

So, such a function g exists