This question is from the book Abstract Algebra: Theory and Applications by Thomas W. Judson.
Chapter 1 – Question 22
Let $f : A \to B$ and $g : B \to C$ be maps.
(a) If $f$ and $g$ are both one-to-one functions, show that $g \circ f$ is one-to-one.
(b) If $g \circ f$ is onto, show that $g$ is onto.
(c) If $g \circ f$ is one-to-one, show that $f$ is one-to-one.
(d) If $g \circ f$ is one-to-one and $f$ is onto, show that $g$ is one-to-one.
(e) If $g \circ f$ is onto and $g$ is one-to-one, show that $f$ is onto.