Functions

Let D, F ⊂ R be some sets of real numbers.

Definition 1.1. A function f : D → F is a rule which assigns to every x ∈ D exactly one real number from the set F which we denote by f(x). We often write that x → f(x) which is taken to mean that “x maps to f (x)”. The notation f : D → F tells us that f takes an element of D and maps it to an element of F.

Definition 1.2. For a function f : D → F, the set D is called its domain. The set F is called its co-domain.

 We can define a function f by,

f : R→R, x → x^2         (R equals real numbers)

Clearly this satisfies the definition of a function with domain D = R and co-domain F=R. It is important to note that for a function f : D → F, not every element of the co-domain F need be a possible output of the function. For instance in Example 1.3 there is no x ∈ D = R such that x2=-4, and yet -4 ∈ F=R. Thus we have an example of a function for which there are elements of the co-domain which do not have a corresponding elements in the domain. This naturally leads us to the concept of the “range” of a function