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> 9.72. $A,B$ are nonempty sets. $f: A \rightarrow B$ and $g: B \rightarrow A$ are functions.  
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>  Suppose $g \circ f = $ the identity function on $A$. (♦) Are the following true or false? 
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$1.$ $f$ 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](https://math.stackexchange.com/questions/553186/intuition-for-1-left-inverse-iff-injective-2-right-inverse-iff-surjecti))  

$3.$ $f$ 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