To summarize the injectivity, surjectivity and bijectivity of the composition of two functions and its components, we have the following results.
Suppose f: A\to B, g: B\to C and h = g\circ f: A\to C.
- If f and g are injective, then so is h.
- If f and g are surjective, then so is h.
- If h is injective, then so is f.
- If h is surjective, then so is g.
- It’s possible that h is injective but g is not.
- It’s possible that h is surjective but f is not.