A function is defined as a relation in which to every element in domain there exist corresponding one and only one image in the co- domain.
We write a function as f: A->B and this is denoted by y=f(x) such that A and B sets are non empty sets i.e to every element of A, there exist one unique element in B.

Note that it may happen that there may exist some elements in B which might not have a corresponding element in A. But there should not be any element in set A left which has no element in B. Now if the function is y = f(x), elements in set A are called x and elements in B are called y.
A mapping that is multi valued i.e. if there aye more than one y's for one value of x then it is called a relation..

1. SET A                SET B .            2. SET A               SET B                       3. SET A               SETB
a                         1                         a                      1,2                                 a                       1
b                         2                         b                       1                                  b                       1
c                         3                         c                        0                                  c                       1

Example 1 and 3 are functions but 2 is not a function. In the first example there is only one y for every x but in the second for x = a there exist two values of y and hence it is a relation. Similarly in third every x has one  value of y, no matter the value  of y are same. Hence, it is also a function.

Here is one more pictorial representation of a function. This is a function since all the elements  in A have images in set B and also no A element has two images.

How to test whether a relation is a function or not?

Before asking this  question you must be aware that all functions are relations but all relations are not functions i.e. functions can be thought as a subset of a relation.

Now, a relation f: A->B  is a function or not can be checked  by its graph. If a vertical line drawn on domain i.e on x values cuts the  curve at two points then it does not satisfy the criteria of  being a function. If the vertical line cuts at only one point then it is a function.

Important terms: Domain, Range, Co-domain

If a function f is defined  from a set A to set B, then set A is called the  domain of function and B is called the co - domain. Now  the  set of all values that corresponds to each value of x (set A) is called the range of  the function.

Please don't confuse between co- domain and  range.  Range is  the  set of values of y which necessarily correspond  to the value of x. But co- domain contain those values which may or may not correspond to each x. Hence, the  range is a subset of co-domain.

Set of all possible values for which f(x) exist is  called domain.

Type of Functions

1. One - one mapping (injection):
A function f is said to be injective if all x values have different values of y. Thus no two elements in A can have the same image in B,

• How to decide whether a function is injective or not?

a. Theoretically / Analytically
Take two arbitrary elements x1 and x2 in the domain. Operate f(x1) = f(x2). If f(x1) = f(x2) gives x1= x2, then only the function is  injective or one- one or many one.

b. Graphically
Draw the graph of the function. Draw horizontal lines. If the horizontal  lines cut the function at one point then only it is injective.

c. By calculus approach
Find dy/dx.  If dy/dx > 0 (function is strictly increasing) or dy/dx < (strictly decreasing ), then only injective.

2. Onto mapping (surjection)
A function in which each element of B is paired with at least one element in A i.e. the  range of  the function is equal to the co- domain then only the function is called surjective • How to decide whether mapping is surjective or onto?

a.  Theoretically / Analytically
Find range of the function and then check if range = co-domain or not.

b. Graphically
From the graph, check whether its vertical span is equal to co- domain or not.

3. Bijective functions
A function is  said to be bijective if and only if it is onto as well as one one.