# Chapter 2 : Functions - Rd Sharma Solutions for Class 12-science Maths CBSE

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## Chapter 2 - Functions Exercise Ex. 2.1

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

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

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

Classify the following functions as injection, surjection or bijection:

f : R → R, defined by f(x) = sin^{2}x + cos^{2}x

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

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.

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

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.

## Chapter 2 - Functions Exercise Ex. 2.2

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.

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

## Chapter 2 - Functions Exercise Ex. 2.3

Let f be a real function given by

Find each of the following

fof

Let f be a real function given by

Find each of the following

fofof

Let f be a real function given by

Find each of the following

(f0f0f) (38)

Let f be a real function given by

Find each of the following

f^{2}^{}Also, show that fof ≠ f^{2 }.

## Chapter 2 - Functions Exercise Ex. 2.5

Thus, h is a bijection and is invertible.

Consider and defined as .

Show that are invertible. Find and show that

∴ 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^{-1}0g^{-1}

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

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

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

Consider the function f ; R^{+}→
[-9, ∞]
given by

f(x) = 5x^{2}
+ 6x - 9.

## Chapter 2 - Functions Exercise MCQ

Let A = {x ε R : - 1 ≤ x ≤ 1} = B and C = {x ε R : x ≥ 0} and let S = {(x, y) ε A × B : x^{2} + y^{2} = 1} and s_{0} = {(x, y) ε A × C : x^{2} + y^{2} = 1}. Then

(a) S defines a function from A to B

(b) S_{0} defines a function from A To C

(c) S_{0} defines a function from A to B

(d) S defines a function from A to C

(a) injective

(b) surjective

(c) bijective

(d) none of these

If f : A → B given by 3^{f(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

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

(a) one-one and onto

(b) many-one and onto

(c) one-one and into

(d) many-one and into

(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

The function f : A → B defined by f(x) = -x^{2} + 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, ∞)

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

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

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

(a) one-one and onto

(b) one-one but not onto

(c) onto but not one-one

(d) neither one-one nor onto

The range of the function f (x) = ^{7-x}P_{x-3} is

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

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

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

(d) {1, 2, 3}

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

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

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

(a) f(x) = x^{3}

(b) f(x) = x + 2

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

(d) f(x) = x^{2} + x

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

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

(a) R

(b) [0, 1]

(c) (0, 1]

(d) [0, 1)

if a function f : [2, ∞) → B defined by f(x) = x^{2} - 4x + 5 is a bijection, then B =

(a) R

(b) [1, ∞)

(c) [4, ∞)

(d) [5, ∞)

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

(a) bijection

(b) injection but not a surjection

(c) surjection but not an injection

(d) neither an injection nor a surjection

(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

(a) f is one-one onto

(b) f is one-one into

(c) f is many one onto

(d) f is many one into

(a) one-one but not onto

(b) one-one and onto

(c) onto but not one-one

(d) neither one-one nor onto

(a) one-one but not onto

(b) many-one but onto

(c) one-one and onto

(d) neither one-one nor onto

The function f ; R → R, f(x) = x^{2} is

(a) injective but not surjective

(b) surjective but not injective

(c) injective as well as surjective

(d) neither injective nor surjective

(a) neither one-one nor onto

(b) one-one but not onto

(c) onto but not one-one

(d) one-one and onto both

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

(a) onto but not one-one

(b) one-one but not onto

(c) one-one and onto

(d) neither one-one nor onto

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

(a) one-one and onto

(b) many one and onto

(c) one-one and into

(d) many one and into

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

(a) R

(b) {0}

(c) {0, 2}

(d) none of these

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

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

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

ANSWER PENDING

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

ANSWER PENDING

(a) x

(b) 1

(c) f(x)

(d) g(x)

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

Let f: [2, ∞) → X be defined by f(x) = 4x - x^{2}. Then, f is invertible, if X=

(a) [2, ∞)

(b) ( -∞, 2]

(c) (-∞, 4]

(d) [4, ∞)

ANSWER PENDING

(a) 2x - 3

(b) 2x + 3

(c) 2x^{2} + 3x + 1

(d) 2x^{2} - 3x - 1

Let f(x) = x^{3} 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}

Let f : R → R be given by f(x) = x^{2} - 3. Then domain of f^{-1} is