Let g : R → R be given by g x = 1 + 2 x . If g n x = g o g o … o g ( x ) ⏟ n times , then find g 4 ( 1 ) .

Given: g x = 1 + 2 x Now, g 2 x = g o g x = 1 + 2 1 + 2 x ⇒ g 2 x = 3 + 4 x ⇒ g 3 x = g 3 + 4 x = 1 + 2 3 + 4 x ⇒ g 3 x = 7 + 8 x ⇒ g 4 x = g 7 + 8 x = 1 + 2 7 + 8 x ⇒ g 4 x = 15 + 16 x ∴ g 4 1 = 15 + 16 × 1 = 31

Let f : R - { - 2 } → R - { 1 } be given by f x = x + 1 x + 2 . Find f - 1 ( x ) .

Given: f x = x + 1 x + 2 f x is one-one and onto function, hence it is invertible. Now, y = f x = x + 1 x + 2 ⇒ y x + 2 = x + 1 ⇒ x y + 2 y = x + 1 ⇒ x y - 1 = 1 - 2 y ⇒ x = 1 - 2 y y - 1 Hence, f - 1 x = 1 - 2 x x - 1

Let us consider a real valued function f defined as f x = 1 - x 1 + x , x ≠ - 1 . Find the value of f ( f ( x ) ) + f f 1 x . where x ≠ 0 .

We have, f x = 1 - x 1 + x Now, f f x = 1 - f x 1 + f x ⇒ f f x = 1 - 1 - x 1 + x 1 + 1 - x 1 + x ⇒ f f x = 1 + x - 1 + x 2 ⇒ f f x = x And, f 1 x = 1 - 1 x 1 + 1 x ⇒ f 1 x = x - 1 x + 1 So, f f 1 x = 1 - f 1 x 1 + f 1 x ⇒ f f 1 x = 1 - x - 1 x + 1 1 + x - 1 x + 1 ⇒ f f 1 x = x + 1 - x + 1 x + 1 + x - 1 ⇒ f f 1 x = 1 x Therefore, f f x + f f 1 x = x + 1 x ⇒ f f x + f f 1 x = x 2 + 1 x

Let A be the set of all real numbers except 1 and o be an operation on A defined by a o b = a + b - a b for all a , b ∈ A . Prove that A is closed under the given operation.

We have, a o b = a + b - a b If the binary operation * defined on set A , then " * " is closed if for all a , b ∈ A , a * b ∈ A . Let x , y ∈ R , then x o y = x + y - x y Now, x + y ∈ R , since sum of real numbers is real. x y ∈ R , since product of real numbers is real. Hence, x + y - x y ∈ R ⇔ x o y ∈ R as difference of real numbers is real.

Let A = { 2 , 3 , 4 , 5 , 6 , … … , 9 } . Let R be the relation on A defined by { ( x , y ) : x ∈ A , y ∈ A and x divides y } . Find R .

We have, A = { 2 , 3 , 4 , 5 , 6 , … … , 9 } . R is the relation on A defined by { ( x , y ) : x ∈ A , y ∈ A and x divides y } Therefore, R = 2 , 4 , 2 , 6 , 2 , 2 , 2 , 8 , 3 , 6 , 3 , 3 , 3 , 9 , 4 , 8 , 4 , 4 , 5 , 5 , 6 , 6 , 7 , 7 , 8 , 8 , 9 , 9

Let A = { 2 , 3 , 4 , 5 , 6 , … … , 9 } . Let R be the relation on A defined by { ( x , y ) : x ∈ A , y ∈ A and x divides y } . Find domain of R .

We have, A = 2 , 3 , 4 , 5 , 6 , … … , 9 . R is the relation on A defined by { ( x , y ) : x ∈ A , y ∈ A and x divides y } Therefore, R = 2 , 4 , 2 , 6 , 2 , 2 , 2 , 8 , 3 , 6 , 3 , 3 , 3 , 9 , 4 , 8 , 4 , 4 , 5 , 5 , 6 , 6 , 7 , 7 , 8 , 8 , 9 , 9 Domain of R is 2 , 3 , 4 , 5 , 6 , … … , 9 .

Check whether the relation R defined in the set 1 , 2 , 3 , 4 , 5 , 6 as R = a , b : b = a + 1 is reflexive, symmetric or transitive.

Suppose A = 1 , 2 , 3 , 4 , 5 , 6 . A relation R is defined on set A as: R = { ( a , b ) : b = a + 1 } ∴ R = { ( 1, 2 ) , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 } Reflexive Property : A relation R in a set A is said to be reflexive if x , x ∈ R , for every x ∈ A Now, 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 , 6 ∉ R ∴ R is not reflexive. Symmetric Property : A relation R in a set A is said to be symmetric if x , y ∈ R implies that y , x ∈ R for all x , y ∈ A . Now it can be observed that 1 , 2 ∈ R , but 2 , 1 ∉ R . Hence , R is not symmetric. Transitive Property : A relation R in a set A is said to be transitive if x , y ∈ R and y , z ∈ R implies that x , z ∈ R for all x , y , z ∈ A . Now, 1 , 2 , 2 , 3 ∈ R but, ( 1 , 3 ) ∉ R ∴ R is not transitive. Hence, R is neither reflexive, nor symmetric, nor transitive.

E M B I B E

Gamma Question Bank for Engineering Mathematics>Chapter 18 - Relation and Functions>Assignment>Q 21

MEDIUM

JEE Main/Advance

IMPORTANT

Earn 100

Let $A = {1, 2, 3, 4}$ . Find the number of onto functions from set $A$ to set $A$ .

Important Questions on Relation and Functions

MEDIUM

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 18 - Relation and Functions>Assignment>Q 22

Let

f (x) = 1 - x

and

g (x) = 1 + x^{2}

, then find

(f o g) (x)

and

(g o f) (x)

.

HARD

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 18 - Relation and Functions>Assignment>Q 23

Let

g : R \to R

be given by

g (x) = 1 + 2 x

. If

g^{n} (x) = \underset{n times}{\underset{⏟}{g o g o \dots o g (x)}}

, then find

g^{4} (1)

.

MEDIUM

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 18 - Relation and Functions>Assignment>Q 24

Let

f : R - {- 2} \to R - {1}

be given by

f (x) = \frac{x + 1}{x + 2}

. Find

f^{- 1} (x)

.

MEDIUM

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 18 - Relation and Functions>Assignment>Q 25

Let us consider a real valued function

f

defined as

f (x) = \frac{1 - x}{1 + x}, x \neq - 1

. Find the value of

f (f (x)) + f (f (\frac{1}{x}))

. where

x \neq 0

.

EASY

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 18 - Relation and Functions>Assignment>Q 26

Let

A = {a, b} .

Find the number of binary operations that can be defined on

A

.

EASY

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 18 - Relation and Functions>Assignment>Q 27

Let

A

be the set of all real numbers except

1

and

o

be an operation on

A

defined by

a o b = a + b - a b

for all

a, b \in A

. Prove that

A

is closed under the given operation.

EASY

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 18 - Relation and Functions>Assignment>Q 28

Let $A = {2, 3, 4, 5, 6, \dots \dots, 9}$ . Let $R$ be the relation on $A$ defined by ${(x, y) : x \in A, y \in A$ and $x$ divides $y}$ . Find $R$ .

MEDIUM

JEE Main/Advance

IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 18 - Relation and Functions>Assignment>Q 28

Let $A = {2, 3, 4, 5, 6, \dots \dots, 9}$ . Let $R$ be the relation on $A$ defined by ${(x, y) : x \in A, y \in A$ and $x$ divides $y}$ . Find domain of $R$ .