Functions

A function f from A to B is a relation from A to B in which every element of A has one and only one image in B. Equivalently, the domain of f is the whole of A and no two different pairs of f share a first entry.

We write f : A → B and f(x) = y when (x, y) ∈ f. y is the image of x under f, and x is a preimage of y.

A is the domain of f and B its codomain. The range of f is the set of all images, and range ⊂ codomain.

The test has two parts: no element of A is left without an arrow, and no element of A sends two arrows. Several elements of A may share an image, and some elements of B may receive no arrow at all.

Every function is a relation, but a relation is a function only when it passes both parts of the test. {(1, 3), (2, 3), (3, 5)} is a function on {1, 2, 3}; {(1, 4), (1, 6), (2, 6)} is not, because 1 has two images.

A function whose range is R or a subset of R is a real valued function. If its domain is also R or a subset of R, it is a real function.

Evaluating a real function means substituting into its rule: for f(x) = 3x − 4, f(0) = −4, f(2) = 2 and f(−1) = −7.

Counting result: when n(A) = p and n(B) = q, each of the p elements of A picks one of q images independently, so there are qᵖ functions from A to B.