FUNCTION

DEFINITIONS,PROPERTIES AND INCLUDES ITS IMPORTANCE

[USEFUL FOR ALL ENTRANCES LIKE EAMCET,JEE(MAINS&ADVANCE),BITSAT,VIT,NDA..ETC]

Name is from Latin,

Meaning:Operations

Synonims:Mappings&map

Role:Very important Role in Differential and Integral Calculus..

Function as a special kind of Relation:

Definition1:

Let A and B be two non-empty sets,A relation f from A to B that is a subset of AXB ,is called a function from A to B,if

1)for each a∈A there exists b∈B such that (a,b)∈f.

2)(a,b)∈f and (a,c)∈f =>b=c.

Definition2:

let A and B be non-empty sets and f is a relation from A to B.if for each element a∈A there exists a unique element b∈B such that (a,b)∈f then f is called a function (or mapping from A to B)

(or AintoB)

The set A is called domain of f and B is called co-domain of f.

Function as a correspondance:

let A and B be two non-empty sets then a function f from set A to set B is a rule of method (or) correspondance which associates elements of set A to elements of set B such that

1)All elements of set A are associated to elements in set B

2)All elements of set A is associated with unique element in set B

Notation:

if is a function from a set A to a set B then we write f:A->B which is read as f is a function from A to B or f maps A to B.

Note:

if an element a∈A associated to an element b∈B

then b is called the f-image of a (or) image of a under f.

here 'a' is called Pre-image of b under the function f

therefore b=f(a).

Description of a Function:

Let A is a finite set,

let f(x)=2x+1 then f:A->B is a function such that the set A consists of a finite number of elements

for example:A={1,2,3}

if x=1 then f(1)=3,

if x=2 then f(2)=5,

if x=3 then f(3)=7

let A is an infinite set :-

f cannot be described by listing the images at points in its domain,functions are generally described by formula,

for Example:f:z->z given by f(x)=X^2+1 and f:R->R defined by f(x)=e^x....

Examples in list:

Ex1:

let A={1,2,3} B={2,3,4}

F1={(1,2),(2,3),(3,4)}

F2={(1,2),(1,3),(2,3),(3,4)}

F3={(1,3),(2,4)}

Note:

1)F1 is a function from A to B but F2 and F3 are not functions from Ato B

F2 is not a function from A to B because 1∈A has two images 2 1nd 3 in B

F3 is not a Function from A to B because 3∈A has no images in B

Ex2:if x,y∈{1,2,3,4} then which are given set?

F1={(x,y);y=x+1}

F1={(1,4),(2,3),(3,4)} is not a Function from the given set to itself.

2) F4={(x,y);x+y=4}

F4={(1,4),(2,3),(3,2),(4,1)}

Range of F={1,2,3,4}

We observe that each element of the given sets has appered as first components in one and only one ordered pair of F4 so F4 is a function in the given set

hence range of F ={1,2,3,4}.