9.72. $A,B$ are nonempty sets. $f: A \rightarrow B$ and $g: B \rightarrow A$ are functions.
Suppose $g \circ f = $ the identity function on $A$. (♦) Are the following true or false?
$1.$ $f$ has a left inverse (ie: $g \circ f = id_A)$ $\iff$ $f$ is injective.
$2.$ $g$ has a right inverse $\iff$ $g$ is surjective. (Question on its proof)
$3.$ $f$ need not be onto.
$4.$ $g$ need not be one-to-one.
$5.$ $f$ is onto $\iff$ $g$ is one-to-one.
How would I determine truth or untruth for each, before proving or finding a counterexample? Moreover, what are the intuitions? I tried sketching possibilities from the given info, but it became desultory. I'm not asking about formal arguments.
Sources: Chartrand 3rd Ed P239 9.72 = 2nd Ed 9.48 and D Velleman P248 Thm 5.3.3