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Class 11 Maths Revision Notes Relations and Functions

Maths notes for exam notes for teachers
Course

Mathematics – Iii

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Revision Notes

Class 11 Mathematics

Chapter 2 - Relation & Function-I

1. INTRODUCTION:

● In this chapter, we'll learn how to link pairs of objects from two sets to

form a relation between them.

● We'll see how a relation can be classified as a function.

● Finally, we'll look at several types of functions, as well as some standard

functions.

2. RELATIONS:

2 Cartesian product of sets Definition: ● Given two non-empty sets P and Q.

● The Cartesian product P Q is the set of all ordered pairs of elements

from P and Q that is

● P×Q = {( p, q ); p  P ; qQ}

2 Relation: 2.2 Definition: ● Let A and B be two non-empty sets. ● Then any subset ‘ R ’ of A×B is a relation from A and B.

● If ( a, b ) R , then we can write it as a R b which is read as a is related to

b ‘by the relation R ’, ‘ b ’ is also called image of ‘ a ’ under R.

2.2 Domain and range of a relation:

● If R is a relation from A to B , then the set of first elements in R is

known as domain & the set of second elements in R is called range of R symbolically.

● Domain of R = { x:( x, y )R}

● Range of R = {y: (x, y )R}

● The set B is considered as co-domain of relation R.

● Note that,

range co-domain

● Note :

Total number of relations that can be defined from a set A to a set B is the

number of possible subsets of A×B.

If n A( )= p and n B( )= q, then

n A×B( )= pq and total number of relations is 2 pq .

2.2 Inverse of a Relation:

● Let A, B be two sets and let R be a relation from a set A to set B. Then

the inverse of R denoted as R-1 , is a relation from B to A and is defined by

R –1 = {(b, a : a,)( b ) R}

● Clearly

(a, b )  R (b, a )R –

● Also,

Dom R( )= Rang e(R –1) and

Range R( )= D o m( R–1)

3. FUNCTIONS:

3 Definition:

A relation ‘ f ’ from a set A to set B is said to be a function if every element

of set A has one and only one image in set B.

Rules for finding Domain:

  1. Even roots (square root, fourth root, etc.) should have non–negative

expressions.

  1. Denominator  0

  2. log xa is defined when x > 0, a > 0 and a  1

  3. If domain y = f(x) and y = g(x) are D 1 and D 2 respectively then the domain

of f (x) g(x) or f (x).g(x) is D 1 D 2.

While domain of g(x)

f (x)

is D 1  D 2 − x : g(x) = 0 

Range: The set of all f - images of elements of A is known as the range of f and can be denoted as f (A) .

Range = f (A) = f (x) : x A

f(A) B {Range  Co-domain}

Rule for finding range: First of all find the domain of y =f (x)

i. If domain  finite number of points  range  set of corresponding f (x)

values. ii. If domain R or R −{Some finite points} Put y =f (x)

Then express x in terms of y .From this find y for x to be defined. (i., find

the values of y for which x exists).

iii. If domain  a finite interval, find the least and greater value for range using monotonicity.

Note:

  1. Question of format:
 
 = = = 
 
Q Q L

y ; y ; y

Q L Q

Q →Quadratic

L → Linear

Range is found out by cross-multiplying & creating a quadratic in 'x' &

making D  0 (as xR)

  1. Questions to determine the range of values in which the given expression

y =f (x) can be converted into x (or some function of x = expression in ‘ y

’.

Do this & apply method (ii).

  1. Two functions f & g are said to be equal if

a. Domain of f = Domain of g

b. Co-domain of f =Co-domain of g

c. f (x) =g(x)  x Domain

3 Kinds of functions:

  1. Graphically: A function is one-one, if no line parallel to x −axis meets the graph of function at more than one point.

  2. By Calculus: For checking whether f (x) is One-One, find whether function is only

increasing or only decreasing in their domain. If yes, then function is one-one, that is if f '(x)  0,  x domain or, if f '(x)  0,  x domain, then function is

one-one.

3 Some standard real functions & their graphs: 3.4 Identity Function: The function f : R →R defined by

y = f ( x )= x x R is called identity function.

3.4 Constant function: The function f : R →R defined by

y = f (x )= c, x R

3.4 Modulus function: The function f : R →R defined by

− 
= 

x; x 0

f (x)

x; x 0

is called modulus function. It is denoted by y = f (x) =x

It is also known as “Absolute value function”.

Properties of Modulus Function: The modulus function has the following properties:

  1. For any real number x , we have x 2 =x

  2. xy =x y

Properties of Greatest Integer Function: If n is an integer and x is any real number between n and n + 1 , then the greatest integer function has the following properties:

1. − n = −n

2. x + n = x +n

3. − x = x − 1

4.
 
+ − = −

0, if x I

x x

1, if x I

   

Note:

Fractional part of x , denoted by  xis given by x – x , Hence

 + −  
= − =   
 −  

x 1 1 x 0

x x x x 0 x 1

x 1; 1 x 2

   

3.4 Exponential Function: f (x) =ax , a  0,a  1

Domain: x R

Range: f (x)  (0, )

3.4 Logarithm Function: f (x) =log xa , a  0,a  1

Domain: x  (0, )

Range: y R

  1. If a > 1 then log x > pa x > ap

  2. If 0 < a < 1 then log x < pa x > ap

  3. If 0 < a < 1 then log x > pa 0 < x < ap

Note: ● The logarithm is positive if the exponent and base are on the same side of unity. ● The logarithm is negative if the exponent and base are on opposite sides of unity.

4. ALGEBRA OF REAL FUNCTION:

We'll learn how to add two real functions, remove one from another, multiply a real function by a scalar (a scalar is a real integer), multiply two real functions, and divide one real function by another in this part.

4 Addition of two real functions: Let f : X →R and g : X →R by any two real functions, where x R. Then,

we define ( f + g : X) →R by

( f + g )(x )= f (x )+g x( )for all x X .

4 Subtraction of a real function from another: Let f : X →R be any two any two real functions, where x R.

Then, we define ( f − g : X) →R by

( f − g )(x )= f (x )−g x( )for all x X.

4 Multiplication by a scalar: Let f : X →R be a real valued function and  be a scalar. Here by scalar, we mean a real number. Then the product f is a function from X to R defined as

(  f )(x )=  f (x , x) X.

4 Multiplication of two real functions: The product (or multiplication) of two real functions f : X →R and g : X →R

is a function fg : X →R defined as

( fg )(x ) =f (x g x) ( ) for all x X .

This is also known as pointwise multiplication.

4 of two real functions: Let f and g be two real functions defined from X →R where X R.

The quotient of f by g denoted by g

f a is a function defined as  

  =
 

g g x

x

f f x

( )

( )

( )

Provided g x( ) 0, x X.

4 Even and Odd Functions ● Even function:

o f ( − x )= f (x ,) x Domain

o The graph of an even function y =f ( x)is symmetric about the y − axis,

that is ( x, y)lies on the graph  ( −x, y) lies on the graph.

Some standard results on periodic functions:

Functions Periods

i sin x, cos x, sec x, cosec xn n n n  ; if n is even 2  ;(if n is odd or fraction) ii tan x, cot xn n  ; n is even or odd

iii

cotx , secx , cosecx

sinx , cosx , tanx , 

iv x − x , . represents greatest

integer function

1

v Algebraic functions for example x , x , x +5, ..... 2 3 etc.

Period does not exist

Properties of Periodic Function:

i. If f ( x )is periodic with period T , then

a) c ( x ) is periodic with period T

b) f ( x c)is periodic with period T

c) f ( x )cis periodic with period T

Where c is any constant

i. If f ( x )is periodic with period T , then

kf ( cx +d)has period

c

T

That is Period can be only affected by coefficient of x where k, c, d constant.

ii. If f 1 ( x ,f) 2 (x)are periodic functions with periods T ,T 1 2 respectively,

Then we have,

h x( ) = af 1 (x )bf 2 (x)has period as, LCM of T ,T 1 2 

Note:

a. of  

  =
 

b d f HCF of (b,d,f )

, ,

a c e LCM of (a,c,e)

b. LCM of rational and rational always exists. LCM of irrational and irrational sometime exists. But LCM of rational and irrational never exists.

For example, LCM of ( 2 π, 1, 6π )is not possible because 2 π, 6π

irrational and 1  rational.

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Class 11 Maths Revision Notes Relations and Functions

Course: Mathematics – Iii

39 Documents
Students shared 39 documents in this course

University: MATS University

Was this document helpful?
Class IX Mathematics 1
Revision Notes
Class 11 Mathematics
Chapter 2 - Relation & Function-I
1. INTRODUCTION:
●In this chapter, we'll learn how to link pairs of objects from two sets to
form a relation between them.
●We'll see how a relation can be classified as a function.
●Finally, we'll look at several types of functions, as well as some standard
functions.
2. RELATIONS:
2.1
Cartesian product of sets
Definition:
●Given two non-empty sets
P
and
Q
.
●The Cartesian product
PQ
is the set of all ordered pairs of elements
from
P
and
Q
that is
●
 q{P×Q = p, q ; p P ; Q}
( )
2.2
Relation:
2.2.1
Definition:
●Let
A
and
B
be two non-empty sets.
●Then any subset ‘
R
’ of
A×B
is a relation from
A
and
B
.
●If
a, b R
( )
, then we can write it as a
R b
which is read as
a
is related to
b
‘by the relation
R
’, ‘
b
’ is also called image of ‘
a
’ under
R
.
2.2.2
Domain and range of a relation:
●If
R
is a relation from
A
to
B
, then the set of first elements in
R
is
known as domain & the set of second elements in
R
is called range of
R
symbolically.
●Domain of
R = x:{ x, y R}
( )

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