DISCRETE MATHEMATICS

TOPIC:-FUNCTIONSBY:-ANSHUL GOUR ROLL NO. :-16110

FUNCTIONS

• A function is a relation in which each element of the domain is paired with exactly one element of the range. Another way of saying it is that there is one and only one output (y) with each input (x).

X f(x) Y

FUNCTION NOTATION

Output

InputName of

Function

y f x

• We commonly call functions by letters. Because function starts with f, it is a commonly used letter to refer to functions.

632 2 xxxfThis means

the right hand

side is a

function

called f

This means the

right hand side

has the variable

x in it

The left side DOES NOT MEAN

f times x like brackets usually

do, it simply tells us what is

on the right hand side.

The left hand side of this equation is the function notation.

It tells us two things. We called the function f and the

variable in the function is x.

• Variable x is called independent variable

• Variable y is called dependent variable

• For convenience, we use f(x) instead of y.

• The ordered pair in new notation becomes:

• (x, y) = (x, f(x))

REPRESENTATION OF GRAPH

• Verbally

• Numerically, i.e. by a table

• Visually, i.e. by a graph

• Algebraically, i.e. by an explicit formula

Domain, Codomain and Range of a Function

• Let be the function, then set ‘A’is called the domain of f and set ‘B’ iscalled the codomain of f. The set ofthose elements of B which are relatedby elements of A is called range of f or image of set A under f and isdenoted by f(A), i.e.

Range of f.

f :A B

f A f a |a A

f A B.

For example:

1

2

A BR7

a

b

Dom (R7) = {a, b},Codomain = {1, 2}Range (R7) = {1}

1

2

A BR8

a

b

Domain (R8) = {a, b}Codomain (R8) = {1, 2}Range (R8) = {1, 2}

= Codomain (R8)

Domain, Codomain and Range of a Function

TYPES OF FUNCTION

One-one function (or injective)

A function is said to be one-onefunction or injective if different elementsof A have different images in B, i.e. if

f : A B

a, b A s.t. a b f a f b

Thus if f : A B is 1 1

a b f a f b a, b A

or f a f b a b a, b A

FOR EXAMPLE:-

• Let be thefunction given by

That’s why (i) is not one to one function.

f : a, b 1, 2

1

2

A B

f : A B

a

b(i) (ii)

1

2

A B

f : A B

a

b

Here only (ii) is one to one function

a b but f a f b 1

TYPES OF FUNCTION

Onto function (or surjective)

A function is said to be ontofunction or subjective if all theelements of B have preimage in A,i.e. for each

f : A B

b B some a A st f a b or a, b f

i.e. A function is not onto if s.t. thereis no for which f(a) = b.

b B

a A