Define a function as a correspondence between two sets.

A function is a correspondence between two sets of elements, such that for each element in the first set there is only one corresponding element in the second set.

The first set is called the domain and the set of all corresponding elements in the second set is called the range.

Let A = {1, 2, 3} and  B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Let f : A → B be the correspondence which assigns to each element in A its square.

Hence,

f (1) = 12 = 1

f (2) = 22 = 4

f (3) = 32 = 9

Since for each element (1 or 2 or 3) of A, there is exactly one element of B, so f is a function.

In this case, every element of B is not an image of some element of A.