The notion of a function is that of something which provides a distinct output for a given input.

**Definition**

Think about two sets, *D* and *R* along with a principle which appropriates a unique element of *R* to each and every element of D. This rule is termed a function and it is represented by a letter such as *f*. Given n *x ∈ D, f (x) *is the name of the thing in *R* which comes from doing f to x. D is called the domain of *f*. In order to establish that D refers to f, the representation D (f) may be used. The set *R* is sometimes described as the range of *f*. Nowadays it.

is known as the ** codomain**. The set of all elements of

*R*which are of the form

*f (x)*for some

*x ∈ D*is consequently, a subset of R. This is sometimes referred to as the image of

*f*. When this set equals

*R*, the function

*f*is said to be

**onto**, also

**surjective**, if whenever

*x ̸= y*it followss

*f (x) ̸= f (y)*, the function is called

**one-to-one**, also

**injective**.

It is typical representation to write *f : D → R* to denote the condition just described within this definition where *f* is a function characterized on a domain *D* which has values in a codomain *R*.