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