# RD Sharma Solution CLass 12 Mathematics Chapter 2 Functions

## Chapter 2 – Functions Exercise Ex. 2.1

Question 1

Give an example of function which is one-one but not onto.

Solution 1

Question 2

Give an example of a function which is not one-one but onto.

Solution 2

Question 3

Given an example of a function which is neither one-one nor onto.

Solution 3

Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14

Solution 14

Question 15
Solution 15
Question 16

Solution 16

Question 17

Classify the following functions as injection, surjection or bijection:

f : R → R, defined by f(x) = sin2x + cos2x

Solution 17

Question 18
Solution 18
Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24

Question 25
Solution 25
Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29

If A = { 1, 2, 3}, show that a ono-one function f : A → A must be onto.

Solution 29

Since f is one-one, three elements of {1, 2, 3} must be taken to 3 different elements of the co-domain {1, 2, 3} under f.

Hence, f has to be onto.

Question 30

If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.

Solution 30

Suppose f is not one-one.

Then, there exists two elements, say 1 and 2 in the domain whose image in the co-domain is same.

Also, the image of 3 under f can be only one element.

Therefore, the range set can have at most two elements of the co-domain {1, 2, 3}

i.e f is not an onto function,  a contradiction.

Hence, f must be one-one.

Question 31
Solution 31
Question 32
Solution 32
Question 33
Solution 33
Question 34
Solution 34
Question 35
Solution 35
Question 36

Solution 36

Question 37

Solution 37

## Chapter 2 – Functions Exercise Ex. 2.2

Question 1
Solution 1
Question 2

Solution 2

Question 3

Solution 3

Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15

Consider f : N → N, g : N → N and h : N → R defined as f(x) = 2x, g(y) = 3y + 4 and

h(z)  = sin z for all x, y, z ϵ N.

show tht ho (gof) = (hog) of.

Solution 15

Question 16

Given Examples of two functions f : N → N and g ; N → N such that gof is onto but f is not onto.

Solution 16

Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19

## Chapter 2 – Functions Exercise Ex. 2.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10
Solution 10
Question 11
Solution 11
Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Let f be a real function given by

Find each of the following

fof

Solution 19

Question 20

Let f be a real function given by

Find each of the following

fofof

Solution 20

Question 21

Let f be a real function given by

Find each of the following

(f0f0f) (38)

Solution 21

Question 22

Let f be a real function given by

Find each of the following

f2Also, show that fof ≠ f2 .

Solution 22

Question 23

Solution 23

## Chapter 2 – Functions Exercise Ex. 2.5

Question 1

Solution 1

Thus, h is a bijection and is invertible.

Question 2

Solution 2

Question 3

Solution 3

Question 4

Consider  and  defined as .

Show that  are invertible. Find  and show that

Solution 4

Question 5

Solution 5

f-1 = {(3, 1), (5, 2), (7, 3), (9, 4)}

f-1 0g-1 = {(7, 1), (23, 2), (47, 3), (79, 4)} ……. (A)

Now (A) & (B) we have (g0f)-1 = f-10g-1

Question 6

Show that the function  defined by  is invertible. Also find .

Solution 6

Question 7
Solution 7

Question 8

Solution 8

Question 9
Solution 9
Question 10

Consider f : R+ → [-5,] given by f(x) = 9×2 + 6x – 5. Show that f is invertible with

Solution 10

Question 11
Solution 11
Question 12
Solution 12
Question 13

Solution 13

Question 14

Let A = R – {3} and B = R – {1}. Consider the function  defined by . Show that f is one-one and onto and hence find .

Solution 14

Question 15

Consider the function f ; R+
[-9, ∞]
given by

f(x) = 5x2
+ 6x – 9.

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

## Chapter 2 – Functions Exercise Ex. 2VSAQ

Question 1

Figure (a)

Figure (b)

Solution 1

Question 2

Figure (a)

Figure (b)

Solution 2

Question 3

If A = {1, 2, 3} B = {a, b}, write total number of functions from A to B.

Solution 3

A = {1, 2, 3} B = {a, b}

The total number of functions is 8

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1,4), (2,5), (3,6)} be a function from A to B.

State the whether f is one – one or onto

Solution 36

It is given that

A = {1, 2, 3}, B = {4, 5, 6, 7} and f = {(1,4), (2,5), (3,6)}

The function f is one-one from A to B

## Chapter 2 – Functions Exercise MCQ

Question 1

Let A = {x ε R : – 1 ≤ x ≤ 1} = B and C = {x ε R : x ≥ 0} and let S = {(x, y) ε A × B : x2 + y2 = 1} and s0 = {(x, y) ε A × C : x2 + y2 = 1}. Then

(a) S defines a function from A to B

(b) S0 defines a function from A To C

(c) S0 defines a function from A to B

(d) S defines a function from A to C

Solution 1

Question 2

(a)  injective

(b) surjective

(c) bijective

(d) none of these

Solution 2

Question 3

If f : A → B given by 3f(x) + 2-x = 4 is a bijection, then

(a) A = {x ε R : – 1 < x < ∞], B = {x ε R ; 2 < x < 4]

(b) A = {x ε R : – 3 < x < ∞}, B = {x ε R ; 0 < x < 4}

(c) A = {x ε R : – 2 < x ε R : 0 < x < 4}

(d) none of these

Solution 3

Question 4

The function f : R → R defined by f(x) = 2x + 2|x| is

(a) one-one and onto

(b) many-one and onto

(c) one-one and into

(d) many-one and into

Solution 4

Question 5

(a) f is one-one but not onto

(b) f is onto but one-one

(c) f is both one-one and into

(d) none of these

Solution 5

Question 6

The function f : A → B defined by f(x) = -x2 + 6x – 8 is a bijection, if

(a) A = ( -∞, 3] and B = ( -∞, 1]

(b) A = [-3, ∞) and B = (-∞, 1]

(c) A = ( -∞, 3] and B = [1, ∞)

(d) A = [3, ∞) and B = [1, ∞)

Solution 6

Question 7

Let A = {x ε R : – 1 ≤ x ≤ 1} = B. then, the mapping f : A → B given by f(x) = x|x| is

(a) injective but not surjective

(b) surjective but not injective

(c) bijective

(d) none of these

Solution 7

Question 8

Let f : R  → R be given by f(x) = [x]2 + [x + 1] – 3, where [x] denotes the greatest integer less than or equal to x. Then, f(x) is

(a) many-one and onto

(b) Many-one and into

(c) one-one and into

(d) one-one and onto

Solution 8

Question 9

let m be the set of all 2 × 2 matrices with entries from the set R of real numbers. Then the function f : M → R defined by f(A) = |A| for every A ε M, is

(a)  one-one and onto

(b) neither one-one nor onto

(c) one-one but-not onto

(d) onto but not one-one

Solution 9

Question 10

(a) one-one and onto

(b) one-one but not onto

(c) onto but not one-one

(d) neither one-one nor onto

Solution 10

Question 11

The range of the function f (x) = 7-xPx-3 is

(a) {1, 2, 3, 4, 5}

(b) {1, 2, 3, 4, 5, 6}

(c) {1, 2, 3, 4}

(d) {1, 2, 3}

Solution 11

Question 12

A function f from the set of natural numbers to integers defined by

is

(a) neither one-one nor onto

(b) one-one but not onto

(c) onto but not one-one

(d) one-one and onto both

Solution 12

Question 13

Let f be an injective map with domain {x, y, z} and range {1, 2, 3} such that exactly one of the following statements is correct and the remaining are false.

f(x) = 1, f(y) ≠ 1, f(z) ≠ 2.

The value of f-1 (1) is

(a) x

(b) y

(c) z

(d) none of these

Solution 13

Question 14

Which of the following functions from Z to itself are bijections?

(a) f(x) = x3

(b)  f(x) = x + 2

(c) f(x) = 2x + 1

(d) f(x) = x2 + x

Solution 14

Question 15

Which of the following from A = {x : -1 ≤ x ≤ 1} to itself are bijections?

Solution 15

Question 16

Let A = {x : -1 ≤ x ≤ 1} and f; A → A such that f(x) = x|x|, then f is

(a) a bijection

(b) injective but not surjective

(c) surjective but not injective

(d) neither injective nor surjective

Solution 16

Question 17

(a) R

(b) [0, 1]

(c) (0, 1]

(d) [0, 1)

Solution 17

Question 18

if a function f : [2, ∞) → B defined by f(x) = x2 – 4x + 5 is a bijection, then B =

(a) R

(b) [1, ∞)

(c) [4, ∞)

(d) [5, ∞)

Solution 18

Question 19

The function f : R → R defined by f(x) = (x – 1)(x – 2)(x – 3) is

(a) one-one but not onto

(b) onto but not one-one

(c) both one and onto

(d) neither one-one nor onto

Solution 19

Question 20

(a) bijection

(b) injection but not a surjection

(c) surjection but not an injection

(d) neither an injection nor a surjection

Solution 20

Question 21

(a) f is a bijection

(b) f is an injection only

(c) f is surjection on only

(d) f is neither an injection nor a surjection

Solution 21

Question 22

(a) f is one-one onto

(b) f is one-one into

(c) f is many one onto

(d) f is many one into

Solution 22

Question 23

(a) one-one but not onto

(b) one-one and onto

(c) onto but not one-one

(d) neither one-one nor onto

Solution 23

Question 24

(a) one-one but not onto

(b) many-one but onto

(c) one-one and onto

(d) neither one-one nor onto

Solution 24

Question 25

The function f ; R → R, f(x) = x2 is

(a) injective but not surjective

(b) surjective but not injective

(c) injective as well as surjective

(d) neither injective nor surjective

Solution 25

Question 26

(a) neither one-one nor onto

(b) one-one but not onto

(c) onto but not one-one

(d) one-one and onto both

Solution 26

Question 27

Which of the following functions from A = {x ε R : – 1 ≤ x  ≤ 1} to itself are bijections?

Solution 27

Question 28

(a) onto but not one-one

(b) one-one but not onto

(c) one-one and onto

(d) neither one-one nor onto

Solution 28

Question 29

The function f ; R → R defined by f (x) = 6x + 6|x| is

(a) one-one and onto

(b) many one and onto

(c) one-one and into

(d) many one and into

Solution 29

Question 30

Let f(x) = x2 and g(x) = 2x. Then the solution set of the equation fog (x) = gof (x) is

(a) R

(b) {0}

(c) {0, 2}

(d) none of these

Solution 30

Question 31

if f : R → R is given by f(x) = 3x – 5, then f-1(x)

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Let A = {x ε R : x ≥ 1}. The inverse of the function f ; A → A given by f(x) = 2x(x – 1), is

Solution 34

Question 35

Let A = {x ε R : x ≤ 1} and f : A → 1 A be defined as f (x) = x(2 – x). Then, f-1 (x) is

Solution 35

Question 36

Solution 36

Question 37

If the function f: R→R be such that f(x) = x – [x], where [x] denotes the greatest integer less than or equal to x, then f-1(x) is

Solution 37

Question 38

Solution 38

Question 39

(a) x

(b) 1

(c) f(x)

(d) g(x)

Solution 39

Question 40

Solution 40

Question 41

The distinct linear functions which map [-1, 1] onto [0, 2] are

(a) f(x) = x + 1, g(x) = -x + 1

(b) f(x) = x – 1, g(x) = x + 1

(c) f(x) = – x – 1, g(x) = x – 1

(d) none of these

Solution 41

Question 42

Let f: [2, ∞) → X be defined by f(x) = 4x – x2. Then, f is invertible, if X=

(a) [2, ∞)

(b) ( -∞, 2]

(c) (-∞, 4]

(d) [4, ∞)

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

(a) 2x – 3

(b) 2x + 3

(c) 2x2 + 3x + 1

(d) 2x2 – 3x – 1

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Let f(x) = x3 be a function with domain {0, 1, 2, 3}. Then domain of f-1 is

(a) {3, 2, 1, 0}

(b) {0, -1, -2, -3}

(c) {0, 1, 8, 27}

(d) {0, -1, -8, -27}

Solution 48

Question 49

Let f : R → R be given by f(x) = x2 – 3. Then domain of f-1 is

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Let A = {1, 2, …, n} and B = {a, b}. Then the number of subjections  from A into B is

Solution 52

Question 53

If the set A contains 5 elements and the set B 6 elements, then the number of one-one and onto mappings from A to B is

(a) 720

(b) 120

(c) 0

(d) none of these

Solution 53

Question 54

If the set A contains 7 elements and the set B contains 10 elements, then the number one-one functions from A to B is

Solution 54

Question 55

Solution 55

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