# Recitation 15

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.