Function:
            Let A and B be any two non–empty sets. Then a function '' is a rule or law which associates each element of 'A' to a unique element of set 'B'.

Notation:
            (i) Function is usually denoted by small letters i.e. etc and the Greek letter i.e. etc.
            (ii) If '' is a function from 'A' to 'B' then we write.
            (iii)
            
Let 'y' be the element of set 'B' which associate with the element, then we write
                                                               
Read as 'y' is the image of 'x' under ''.
Examples:
            Let,                     
            

It is a function.
            

It is not a function because '2' has no image.
            

It is not a function because 1 has two images.
            

It is a function.

Ordered Pair:
            Let 'a' and 'b' be any two elements then an elementis called ordered pair.
Cartesian product:
            e.g.           ,                      
               
            Let 'A' and 'B' be any two non–empty sets, then the set of all those elements of the form, where, is called Cartesian product.
It is denoted by
Binary Relation:
            Let 'A' and 'B' be any two non–empty sets, then every sub–set of is called binary relation from A to B.
            It is denoted by ''
                i.e.          
e.g.           ,                      
               
               
                '' is a binary relation from A to B.
Function as a Binary Relation:
            Let 'A' and 'B' be any two non–empty sets, then a binary relation '' from 'A' to 'B' is called a function iff it satisfied the following two conditions.
            (i) Domain of ,           i.e.
            (ii) For each element 'x' of 'A' there exist a unique element such that
                               
Domain of a Relation:
            The set of the first elements of all ordered pair in a relation is called, domain of a relation.
                                e.g.
                                Dom
Range of a Relation:
            The set of the second elements of the ordered pair in a relation is called, range of a relation.
                                e.g.
                                Range
Interval:
(i) Closed Interval:
            
Let 'a' and 'b' be any two real numbers, such that , then the set of numbers satisfying inequality.
                                                                i.e.
Read as: 'x' is less than or equal to 'b' and greater than or equal to 'a' is called closed interval.
It is denoted by.
(ii) Open Interval:
            
Let 'a' and 'b' be any two real numbers, such that then the set of all numbers satisfying inequality.
                                                                i.e.
Read as: 'x' is less than 'b' and greater than 'a' is called open interval.
It is denoted by.
Similarly, the semi–closed interval is defined to be , and is denoted by, and the semi–open interval is defined to be, and denoted by.
Domain of the Function:
            e.g.          
            For
            For

            For
            Domain
The set of all those values of 'x' for which 'y' is finite, definite real number is called domain of the function.

(Introduction to Functions)

Functions and Limits

(Examples of Functions)