RD SHARMA Solutions for Class 12-science Maths Chapter 2 - Functions

Chapter 2 - Functions Exercise MCQ

Question 55

Let Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions be defined by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions Then,

a. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

b. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

c. fof(x) = -x

d. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Solution 55

Given: Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions be defined by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

For inverse, we first need to check whether f is invertible.

Let Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions such that f(x) = f(y)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

So, f is one-one.

Let y = f(x)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Therefore, for every Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions there exist y R such that f(x) = y.

Thus, f is bijective and so invertible.

Taking y = f(x), we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Hence, Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 56

Let f: R R be defined by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions Then, f is

a. one-one

b. onto

c. bijective 

d. not defined

Solution 56

Given: f: R R be defined by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

As x R, so it can take the value 0 as well.

At x = 0, f(x) is not defined.

Thus, f is not defined.

Question 57

Let f: R R be defined by f(x) = 3x2 - 5 and g: R R by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions Then, (gof)(x) is

a. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

b. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

c. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

d. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 57

f: R R is given by f(x) = 3x2 - 5

g: R R is given by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Question 58

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

a. f(x) = x2

b. f(x) = x + 2

c. f(x) = 2x + 1

d. f(x) = x2 + 1

Solution 58

a. If we take f: Z Z defined by f(x) = x2

Clearly, it is not one-one as considering f(x) = f(y) will give us

x = ±y

Therefore, it is not a bijection.

b. Consider f: Z Z defined by f(x) = x + 2

Let x, y Z such that f(x) = f(y)

x + 2 = y + 2

x = y

So, f is one-one.

For x in the domain Z, f(x) will also take integer values.

So, the range of f is Z.

Therefore, f is onto.

Thus, f is a bijection.

c. Consider f: Z Z defined by f(x) = x + 2

Let x, y Z such that f(x) = f(y)

x = y

So, f is one-one.

For x in the domain Z, f(x) = 2x + 1 will take odd integer values.

So, the range of f is not same as its co-domain.

Therefore, f is not onto.

Thus, f is not a bijection.

d. If we take f: Z Z defined by f(x) = x2 + 1

Clearly, it is not one-one as considering f(x) = f(y) will give us

x = ± y

Therefore, it is not a bijection.

Hence, option (b) is correct.

Question 59

Let f: A B and g: B C be the bijective functions. Then Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

a. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

b. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

c. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

d. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 59

It is given that f: A B and g: B C are two bijective functions.

Consider,

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Thus, Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Question 60

Let f: N R be the function defined by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions and g: Q R be another function defined by g(x) = x + 2. Then (g o f)(3/2) is

a. 1

b. 2

b. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

c. none of these

Solution 60

It is given that f: N R as Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions and g: Q R as g(x) = x + 2.

Now,

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Hence, Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a)  injective

(b) surjective

(c) bijective

(d) none of these

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) f is one-one but not onto

(b) f is onto but not one-one

(c) f is both one-one and into

(d) none of these

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 12

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 15

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

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) R

(b) [0, 1]

(c) (0, 1]

(d) [0, 1)

Solution 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) bijection

(b) injection but not a surjection

(c) surjection but not an injection

(d) neither an injection nor a surjection

Solution 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 24

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) one-one but not onto

(b) many-one but onto

(c) one-one and onto

(d) neither one-one nor onto

Solution 24

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 26

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 27

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

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 27

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 28

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 31

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

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 31

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 32

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 32

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 33

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 33

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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 

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 34

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 35

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsANSWER PENDING

Question 36

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 36

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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 

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 37

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsANSWER PENDING

Question 38

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 38

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 39

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) x

(b) 1

(c) f(x)

(d) g(x)

Solution 39

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 40

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 40

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsANSWER PENDING

Question 43

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 43

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 44

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 44

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 45

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

(a) 2x - 3

(b) 2x + 3

(c) 2x2 + 3x + 1

(d) 2x2 - 3x - 1

Solution 45

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 46

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 46

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 47

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 47

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 49

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

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 49

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 50

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 50

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 51

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 51

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 52

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

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 52

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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 

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 54

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Chapter 2 - Functions Exercise Ex. 2.1

Question 1(i)

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

Solution 1(i)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 1(ii)

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

Solution 1(ii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 1(iii)

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

Solution 1(iii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 5(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 5(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 5(ii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 5(ii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 5(iii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 5(iii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 5(iv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 5(iv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 5(v)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 5(v)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 5(vi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 5(vi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 5(vii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 5(vii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 5(viii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 5(viii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 5(ix)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 5(ix)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 5(x)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 5(x)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 5(xi)

Classify the following functions as injection, surjection or bijection:

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

Solution 5(xi)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 5(xii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 5(xii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 5(xiii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 5(xiii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 5(xiv)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 5(xiv)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 5(xv)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 5(xv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 5(xvi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 5(xvi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 10
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 10
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 12

We have f : R → R given by f(x) = ex

let x, y  R, such that

f(x) = f(y)

⇒ ex = ey

⇒ ex-y = 1 = e°

⇒ x - y = 0

⇒ x = y

∴ f is one-one

clearly range of f = (0, ∞) ≠ R
∴ f is not onto

when co-domain is replaced by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions i.e, (0, ∞) then f becomes an onto function.

Question 13
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 13
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 14

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

Solution 14

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 15

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

Solution 15

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 16
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 16
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 17
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 17
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 18
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 18
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 19
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 19
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 20
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 20
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 5(xvii)

Classify the following function as injection, surjection or bijection:

f: R R, defined by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 5(xvii)

Given: f: R R, defined by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Injective: Let x, y R such that f(x) = f(y)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

This does not imply x = y

Therefore, f is not one-one.

 

Surjective: Let y R be arbitrary, then

f(x) = y

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

This does not give any value for x.

Therefore, f is not onto.

Thus, it is not bijective.

Question 8(i)

Let A = [-1, 1]. Then, discuss whether the following function from A to itself are one-one, onto or bijective.

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Solution 8(i)

Given: f: A A, defined by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Injective: Let x, y A such that f(x) = f(y)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Therefore, f is one-one.

 

Surjective: For f(x), x [-1, 1]

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Therefore, range is the subset of codomain.

Thus, f is not onto.

Hence, f is one-one but not onto.

Question 8(ii)

Let A = [-1, 1]. Then, discuss whether the following function from A to itself are one-one, onto or bijective.

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Solution 8(ii)

Given: g: A A, defined by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Injective: Let x, y A such that g(x) = g(y)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Therefore, f is not one-one.

 

Surjective: For f(x), x [-1, 1]

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Therefore, range is the subset of codomain.

Thus, f is not onto.

Hence, f is neither one-one nor onto.

Question 8(iii)

Let A = [-1, 1]. Then, discuss whether the following function from A to itself are one-one, onto or bijective.

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Solution 8(iii)

Given: h: A A, defined by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Injective: Let x, y A such that h(x) = h(y)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Therefore, f is not one-one.

 

Surjective: For f(x) = x2, x [-1, 1]

x2 [0, 1]

So, the range is subset of co-domain.

Thus, f is not onto.

Hence, f is neither one-one nor onto.

Question 9(i)

Is the following set of ordered pairs function? If so, examine whether the mapping is injective or surjective.

{(x, y): x is a person, y is the mother of x}

Solution 9(i)

Given set of ordered pair is {(x, y): x is a person, y is the mother of x}

Now biologically, each person 'x' has only one mother.

So, the above set of ordered pairs is a function.

Also, more than one person may have same mother.

So the function is many-one and surjective.

Question 9(ii)

Is the following set of ordered pairs function? If so, examine whether the mapping is injective or surjective.

{(a, b): 'a' is a person, 'b' is an ancestor of 'a'}

Solution 9(ii)

Given set of ordered pair is {(a, b): 'a' is a person, 'b' is an ancestor of 'a'}

Clearly, a person can have more than one ancestors.

So, the above set of ordered pairs is not a function.

Question 21

Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:

a. an injective map from A to B

b. a mapping from A to B which is not injective

c. a mapping from A to B

Solution 21

Given: A = {2, 3, 4}, B = {2, 5, 6, 7}

a. Consider f: A B defined by f(x) = x + 3

Clearly, f is injective because each element of A has distinct output in B.

b. Consider g: A B defined by g(x) = 3

Here, g is a constant function.

Therefore, each element in A has the same output in B.

Thus, g is not injective.

c. Consider h: A B such that h = {(2, 2), (3, 5), (4, 7)}

Clearly, h is a mapping from A to B.

Chapter 2 - Functions Exercise Ex. 2.2

Question 1(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 1(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 1(ii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1(ii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions


Question 1(iii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1(iii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 1(iv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 1(iv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 1(v)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 1(v)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 1(vi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 1(vi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 8
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 8
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 10

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 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 11

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

Solution 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 12
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 12
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 14
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 14
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 13

If Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions then find fof(x).

Solution 13

Given: Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Hence, fof(x) = x.

Chapter 2 - Functions Exercise Ex. 2VSAQ

Question 37

If f: [5, 6] [2, 3] and g: [2, 3] [5, 6] are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, find f o g.

Solution 37

We have f: [5, 6] [2, 3] and g: [2, 3] [5, 6] are given by:

f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}

Now, (f o g)(2) = f(g(2)) = f(5) = 2

And, (f o g)(3) = f(g(3)) = f(6) = 3

Therefore, fog = {(2, 2), (3, 3)}.

Question 38

Let f: R R be the function defined by f(x) = 4x - 3 for all x R. Then write f-1.

Solution 38

Let x, y R such that f(x) = f(y)

4x - 3 = 4y - 3

x = y

So, f is one-one.

For x R, 4x - 3 R

Therefore, range and co-domain are same.

So, f is onto.

Thus, f is bijective and so it is invertible.

Let y = f(x)

y = 4x - 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Hence, Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Question 39

Which one of the following relations on A = {1, 2, 3} is a function?

f = {(1, 3), (2, 3), (3, 2)}, g = {(1, 2), (1, 3), (3, 1)}

Solution 39

Consider, f = {(1, 3), (2, 3), (3, 2)}

In f, every element in the domain A has unique image.

So, f is a function.

It is given that g = {(1, 2), (1, 3), (3, 1)}

For g, the element 1 in the domain A has two different images.

So, g is not a function.

Question 40

Write the domain of the real function f defined by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Solution 40

Given: Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

The function f is defined only when Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Hence, the domain is [-5, 5].

Question 41

Let A = {a, b, c, d} and f: A A be given by f = {(a, b), (b, d), (c, a), (d, c)}, write f-1.

Solution 41

Given: A = {a, b, c, d} and f: A A is given by f = {(a, b), (b, d), (c, a), (d, c)}

Here, f-1 will be a function from A to A and given by

f-1 = {(b, a), (d, b), (a, c), (c, d)}

Question 42

Let f, g: R R be given by f(x) = 2x + 1 and g(x) = x2 - 2 for all x R, respectively. Then, find gof.

Solution 42

Given: f, g: R R

Therefore, gof: R R will be

gof(x) = g(f(x))

= g(2x + 1)

= (2x + 1)2 - 2

 = 4x2 + 4x - 1

Hence, gof(x) = 4x2 + 4x - 1.

Question 43

If the mapping f: {1, 3, 4} {1, 2, 5} and g: {1, 2, 5} {1, 3}, given by f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)}, write fog.

Solution 43

Given: f: {1, 3, 4} {1, 2, 5} and g: {1, 2, 5} {1, 3}

fog(1) = f(g(1)) = f(3) = 5

fog(2) = f(g(2)) = f(3) = 5

fog(5) = f(g(5)) = f(1) = 2

Therefore, fog: {1, 2, 5} {5, 2}

Hence, fog = {(1, 5), (2, 5), (5, 2)}.

Question 44

If a function g = {(1, 1), (2, 3), (3, 5), (4, 7)} is described by g(x) = αx + β, find the values of α and β.

Solution 44

Given: g(x) = αx + β and g = {(1, 1), (2, 3), (3, 5), (4, 7)}

Now, g(1) = 1

∴ α + β = 1 … (i)

And, g(2) = 3

2α + β = 3 … (ii)

Solving (i) and (ii), we get

α = 2 and β = -1 

Question 45

If f(x) = 4 - (x - 7)3, write Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 45

Given: f(x) = 4 - (x - 7)3

Let y = 4 - (x - 7)3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Hence, Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Question 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Figure (a)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

Figure (b) 

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Figure (a)  
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

Figure (b) 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 24

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 24

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 25

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 25

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 26

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 26

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 27

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 27

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 28

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 28

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 29

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 29

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 30

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 30

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 31

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 31

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 32

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 32

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 33

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 33

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 34

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 34

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 35

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 35

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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 Ex. 2.3

Question 13

If f, g: R R be two functions defined as f(x) = |x| + x and g(x) = |x| - x for all x R. Then, find fog and gof. Hence, find fog(-3), fog(5) and gof(-2).

Solution 13

Given: f(x) = |x| + x and g(x) = |x| - x for all x R

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

fog(-3) = -4(-3) = 12, fog(5) = 0 and

gof(-2) = 0

Question 1(i)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1(i)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 1(ii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1(ii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 1(iii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1(iii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 1(iv)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1(iv)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 1(v)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1(v)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 1(vi)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1(vi)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 1(vii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1(vii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 1(viii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1(viii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 1(ix)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1(ix)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 11(i)

Let f be a real function given by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Find each of the following

 

fof

Solution 11(i)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 11(ii)

Let f be a real function given by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Find each of the following

 

fofof

Solution 11(ii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 11(iii)

Let f be a real function given by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Find each of the following

 

(f0f0f) (38)

Solution 11(iii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 11(iv)

Let f be a real function given by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Find each of the following

 

f2Also, show that fof ≠ f2 .

Solution 11(iv)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Chapter 2 - Functions Exercise Ex. 2.4

Question 15

Let f: N N be a function defined as f(x) = 9x2 + 6x - 5. Show that f: N S, where S is the range of f, is invertible. Find the inverse of f and hence find f-1(43) and f-1(163).

Solution 15

Given: f(x) = 9x2 + 6x - 5

Let x, y N such that f(x) = f(y)

9x2 + 6x - 5 = 9y2 + 6y - 5

9(x2 - y2) + 6(x - y) = 0

(x - y){9(x + y) + 6} = 0

As x, y N, 9(x + y) + 6 can't be zero.

x - y = 0

x = y

So, f is one-one.

Clearly, f is onto as range and co-domain are same.

Therefore, f is bijective.

Thus, f is invertible.

Let y = f(x) = 9x2 + 6x - 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Question 16

Let Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions be a function defined as Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions Show that Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions is one-one and onto. Hence, find f-1.

Solution 16

Given: Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Let x, y Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions such that f(x) = f(y)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

So, f is one-one.

Clearly, f is onto as the co-domain of Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions is same as its range.

Therefore, f is bijective.

Thus, f is invertible.

Let y = f(x)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Question 17

Let A = R - {2} and B = R - {1}. If f: A B is a function defined by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions show that f is one-one and onto. Find f-1.

Solution 17

Given: Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Let x, y A such that f(x) = f(y)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

So, f is one-one.

Let y B

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

So, for every x A, there is a y B such that f(x) = y.

Therefore, f is onto.

Thus, f is bijective and so it is invertible.

Hence, Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 18

Show that the function f: N N defined by f(x) = x2 + x + 1 is one-one but onto. Find the inverse of f: N S, where S is range of f.

Solution 18

Given: f: N N defined by f(x) = x2 + x + 1

Let x, y N such that f(x) = f(y)

x2 + x + 1 = y2 + y + 1

x2 - y2 + x - y = 0

(x - y)(x + y + 1) = 0

As x, y N, so (x2 + x + 1) can't be zero.

x - y = 0

x = y

So, f is one-one.

Since x N, x2 + x + 1 > 3 as the minimum value x can take is 1.

Therefore, range of f = (3, )

As the co-domain and range does not match, f is not onto.

But, f: N (3, ) is onto as the range matches with co-domain.

Thus, f: N S is a bijective function and so it is invertible.

Let y = f(x)

y = x2 + x + 1

x2 + x + 1 - y = 0

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Hence, Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Question 19

Consider the bijective function f: R+ (7, ) given by f(x) = 16x2 + 24x + 7, where R+ is the set of positive real numbers. Find the inverse function of f.

Solution 19

Given: f: R+ (7, ) given by f(x) = 16x2 + 24x + 7

As f is a bijective function, so it is invertible.

Let y = f(x)

y = 16x2 + 24x + 7

16x2 + 24x + 7 - y = 0

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Hence, Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Question 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Thus, h is a bijection and is invertible.
 

Question 2(i)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 2(i)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 2(ii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 2(ii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 3

Consider Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions and Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions defined as Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions.

Show that Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions are invertible. Find Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions and show that Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions    

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

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

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

Question 5

Show that the function Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions defined by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions is invertible. Also find Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions.

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 9

Consider f : R+ → [-5,] given by f(x) = 9x2 + 6x - 5. Show that f is invertible with

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 12