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
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.
Represent this function by the above three methods.
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:
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:
Justify, which of the above relations is a function from
- 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.
Substitute corresponding values of in the function. We get