 Solution

Enumerate $A$ as $a_1,\dots ,a_a$. A function is specified by choosing the image of each of these elements independently, with $b$ possibilities each. The multiplication principle gives $\boxed{b^a\text{ functions}}$.

For an injective function, successive images must be distinct. If $a\le b$, there are $b$ choices for the first, $b-1$ for the second, and so on. If $a>b$, the pigeonhole principle makes injectivity impossible. Thus the number of injective functions isThe product in the first case is a falling factorial.