Suppose that $f:A \mapsto B$ is injective.
Note that means $x \in A,\,y \in A\;\& \,x \ne y\, \Rightarrow \,f(x) \ne f(y)$
Using that property; $\left( {\forall a \in A} \right)\left\{ {\{ f(a)\} } \right\}$ is collection of pair-wise disjoint subsets of $B$.
If $B$ is finite then what can you say about $A$?