Function

Any relation on A x B in which (i) no two second elements have a common first element and (ii) every first element has a corresponding second element is called a function.Any relation on A x B in which (i) no two second elements have a common first element and (ii) every first element has a corresponding second element is called a function.

It is denoted by where y is a function of x.

(a) First element is also called

(i) abscissa or

(ii) first component or

(iii) pre-image

(b) Second element is also called

(i) ordinate or

(ii) second component or

(iii) image

We give below some examples of relations which are functions:

{(1, 2), (2, 3), (3, 4)}

{(1, 1), (2, 4), (3, 7)}

One-one function

Every one element of A corresponds to one element of B. Every first element has a corresponding second element. One-one relations in examples 1 and 2 are called one-one functions.

Every element of set A has one image in set B.

{(1, 4), (2, 4), (3, 4), (4, 5)}

Here, one image has three pre-images.

{(3, 4), (5, 4), (7, 4)}

Here, one image has three pre-images.

Many-one function

Every one element of A corresponds to more than one element of B. In example 3, one image has three pre-images. In example 4, one image has four pre-images. Therefore, many-one relations in examples 3 and 4 are many-one functions.

Consider the following examples and note why these relations are NOT functions.

{(a, 1), (b, 2), (c, 3)} one - one relation

First element d has no image.

The relation is not a function.

{(1, 2), (1, 4), (2, 6), (3, 8)}; one - many relation.

Two pre-images have a common image.

The relation is not a function.

{(1, 1), (2, 1), (2, 2), (3, 3)}; many - many relation.

Two images have a common pre-image.

The relation is not a function.

The relations given in examples 6 and 7 are not functions. These examples given above should make clear to the students the definition of a function. There should be no doubt left in their mind to know when a relation is a function and when it is not a function.

Domain

The set of all the first elements of the ordered pairs of a function is called the domain.

Range

The set of all the second elements of the ordered pairs of a function is called the range.

Co-Domain

If (a, b) is an ordered pair of the function then the set B is called the Co-Domain. The range is a subset of the co-domain.

Domain and range have same definitions in relations and functions.