1. f1 and f4 are functions. f2 and f3 are not.
2. Letting A and B be the left and right sets. \\(\forall a \in A, f1(a) \in B \wedge f4(a) \in B \\). However, for both f3 and f2 \\(\exists a \in A \text{ s.t. } f(a) \notin B\\). For f2 it's \\(\circ\\), for f3 it's \\(\bullet\\).
3. To have an injective, but not surjective, function \\(|A| \lt |B|\\). An example of this would be \\(A = \\{1,2,3\\}\\) and \\(B = \\{1,2,3,4,5,6\\}\\) with \\(f (a) = 2a\\). for each b in B either there is a (singular) a in A which maps to it or no a maps to it (at most one).
4. To have a surjective, but not injective, function we require the opposite: \\(|A| \gt |B|\\). Using the same two sets from part 3, we can define \\(g(b) = \lceil \frac{b}{2} \rceil\\). Each of the 3 elements of A will be mapped to, but each by 2 elements of B.

If the sizes of A and B are equal, then we can find functions which are both surjective and injective or functions which are neither. However we cannot find functions which are one or the other.