Function or Mapping

Let be two non-empty sets. Then, a function or a mapping from is a rule which associates to each element , a unique , called the image of . If is a function from , then we write

For to be a function from :

(i) Every element in must have its image in .

(ii) No element in must have more than one image

Example 1:

Let

Consider the rule,

Clearly, each has a unique image in . Hence, is a function from .

Representation of a Function

You can represent the function in three different ways:

__Arrow Diagram: __The function in the above Example can be represented as follows.

__Roster Method:__ Let be a function between . The first thing is to form ordered pairs of all elements in that have image in . Then the function f is represented as the set of all such ordered pairs.

The function in the above example can be written as follows:

__Equation Form:__ Let be a function between . If f can be represented as a rule of association, then it would take equation for. For example, in the above example,

If *. *Hence, equation represents the function .

Let’s do one example for more clarification.

Example 2:

Let

Define

Represent this function by the above three methods.

Solution:

First find out the following:

__Arrow Method__: Now draw the diagram

__Roster Method__: In Roster form the function can be represented as:

__Equation Form__: In Equation form the function can be represented as

Domain, Co-Domain and Range of a Function

Let f be a function from . Then, we define:

Domain

Co-Domain

Range = Set of all images of

Function as a Relation

Let A and B be two non-empty sets and R be a relation from . Then is called a function from

, if (i) domain and (ii) no two ordered pairs in have the same first components.

The following example will make it more clear:

Example 3:

Let

Let

Justify, which of the above relations is a function from

Solution

- Domain Hence is not a function of
- Two different ordered pairs, namely have the same first co-ordinates. Hence is not a function of
- Domain . Also, no two different ordered pairs in have the same first co-ordinates. Hence is a function of

Real Valued Functions

A rule which associates to each real number , a unique real number , is called a real valued function. Here, is an expression in .

Let’s do an example.

Example 4:

Let Find the value of

Solution:

Substitute corresponding values of in the function. We get