Relations and Functions is the second chapter of Class 11 Maths and forms the foundation for important topics such as inverse trigonometric functions, calculus, algebra, probability and higher-level function analysis. A strong understanding of relations, functions, domain, range, composition and inverse functions helps students solve both board-level and entrance-level questions with confidence.

To help students revise this chapter effectively, we have compiled 100+ Relations and Functions MCQs with Answers. These topic-wise objective questions cover relations, types of relations, equivalence relations, functions, domain, co-domain, range, one-one functions, onto functions, bijective functions, composition of functions, inverse functions and binary operations.

What are Relations and Functions?

A relation from set A to set B is any subset of A × B. A function is a special type of relation in which every element of the domain has exactly one image in the co-domain.

For example, if A = {1, 2} and B = {3, 4}, then:

A × B = {(1,3), (1,4), (2,3), (2,4)}

Any subset of A × B is a relation from A to B.

A function from A to B must assign every element of A to exactly one element of B.

Also Check: Class 11 Maths MCQs

Important Definitions for Relations and Functions

Also Check: Class 11 Maths Sets MCQs with PDF

Important Formulas for Relations and Functions MCQs

Difference Between Relation and Function

Difference Between One-One, Onto and Bijective Functions

Relations and Functions MCQs with Answers

Relations MCQs for Class 11

Q1. A relation from set A to set B is defined as:

(a) Any subset of A
(b) Any subset of B
(c) Any subset of A × B
(d) Any subset of B × A

Answer: (c) Any subset of A × B

Solution: A relation from A to B is any subset of the Cartesian product A × B.

Q2. If A = {1, 2} and B = {3, 4}, then A × B is:

(a) {(1,3), (2,4)}
(b) {(1,3), (1,4), (2,3), (2,4)}
(c) {(3,1), (4,2)}
(d) {(1,2), (3,4)}

Answer: (b) {(1,3), (1,4), (2,3), (2,4)}

Solution: Cartesian product A × B contains all ordered pairs (a,b) where a ∈ A and b ∈ B.

Q3. If n(A) = 3 and n(B) = 2, then n(A × B) is:

(a) 3
(b) 5
(c) 6
(d) 8

Answer: (c) 6

Solution: n(A × B) = n(A) × n(B) = 3 × 2 = 6.

Q4. If A has 2 elements and B has 3 elements, then the number of relations from A to B is:

(a) 6
(b) 8
(c) 32
(d) 64

Answer: (d) 64

Solution: n(A × B) = 2 × 3 = 6. Number of relations = 2⁶ = 64.

Q5. If A = {1, 2, 3}, then the number of relations on A is:

(a) 6
(b) 8
(c) 64
(d) 512

Answer: (d) 512

Solution: A relation on A is a subset of A × A. Since n(A × A)=3²=9, number of relations = 2⁹ = 512.

Q6. The domain of the relation R = {(1,2), (2,3), (4,5)} is:

(a) {1, 2, 4}
(b) {2, 3, 5}
(c) {1, 2, 3, 4, 5}
(d) {1, 3, 5}

Answer: (a) {1, 2, 4}

Solution: The domain is the set of all first elements of ordered pairs.

Q7. The range of the relation R = {(1,2), (2,3), (4,5)} is:

(a) {1, 2, 4}
(b) {2, 3, 5}
(c) {1, 2, 3, 4, 5}
(d) {4, 5}

Answer: (b) {2, 3, 5}

Solution: The range is the set of all second elements of ordered pairs.

Q8. If A = {1, 2}, then A × A is:

(a) {(1,1), (2,2)}
(b) {(1,1), (1,2), (2,1), (2,2)}
(c) {(1,2), (2,1)}
(d) {(1,2)}

Answer: (b) {(1,1), (1,2), (2,1), (2,2)}

Solution: A × A contains all ordered pairs formed using elements of A.

Q9. The empty relation on a set A is:

(a) A
(b) A × A
(c) ∅
(d) {A}

Answer: (c) ∅

Solution: The empty relation contains no ordered pair.

Q10. The universal relation on a set A is:

(a) ∅
(b) A × A
(c) A
(d) P(A)

Answer: (b) A × A

Solution: Universal relation on A contains all possible ordered pairs of A × A.

Types of Relations MCQs

Q11. A relation R on A is reflexive if:

(a) (a,b) ∈ R ⇒ (b,a) ∈ R
(b) (a,b), (b,c) ∈ R ⇒ (a,c) ∈ R
(c) (a,a) ∈ R for every a ∈ A
(d) R is empty

Answer: (c) (a,a) ∈ R for every a ∈ A

Solution: Reflexive relation must contain every self-pair (a,a).

Q12. A relation R on A is symmetric if:

(a) (a,a) ∈ R for every a
(b) (a,b) ∈ R ⇒ (b,a) ∈ R
(c) (a,b), (b,c) ∈ R ⇒ (a,c) ∈ R
(d) R = ∅

Answer: (b) (a,b) ∈ R ⇒ (b,a) ∈ R

Solution: Symmetric relation requires the reverse pair to be present whenever a pair is present.

Q13. A relation R on A is transitive if:

(a) (a,a) ∈ R for every a
(b) (a,b) ∈ R ⇒ (b,a) ∈ R
(c) (a,b), (b,c) ∈ R ⇒ (a,c) ∈ R
(d) R contains no pair

Answer: (c) (a,b), (b,c) ∈ R ⇒ (a,c) ∈ R

Solution: This is the definition of transitive relation.

Q14. If A = {1,2} and R = {(1,1), (2,2)}, then R is:

(a) Reflexive
(b) Not reflexive
(c) Empty
(d) Universal

Answer: (a) Reflexive

Solution: Since R contains (1,1) and (2,2), it is reflexive on A.

Q15. If A = {1,2,3} and R = {(1,1), (2,2)}, then R is:

(a) Reflexive
(b) Not reflexive
(c) Universal
(d) Symmetric only

Answer: (b) Not reflexive

Solution: For reflexive relation on A, (3,3) must also be present.

Q16. The relation R = {(1,2), (2,1)} on A = {1,2} is:

(a) Reflexive only
(b) Symmetric
(c) Transitive only
(d) Universal

Answer: (b) Symmetric

Solution: Since (1,2) and (2,1) are both present, R is symmetric.

Q17. The relation R = {(1,2), (2,3), (1,3)} is:

(a) Reflexive
(b) Symmetric
(c) Transitive
(d) Empty

Answer: (c) Transitive

Solution: Since (1,2) and (2,3) imply (1,3), the relation is transitive for the given pairs.

Q18. The universal relation on a non-empty set is always:

(a) Reflexive only
(b) Symmetric only
(c) Transitive only
(d) Reflexive, symmetric and transitive

Answer: (d) Reflexive, symmetric and transitive

Solution: Universal relation contains all ordered pairs, so it satisfies reflexive, symmetric and transitive conditions.

Q19. The empty relation on a non-empty set is:

(a) Reflexive
(b) Not reflexive
(c) Universal
(d) Always equivalence

Answer: (b) Not reflexive

Solution: Empty relation contains no self-pairs, so it is not reflexive on a non-empty set.

Q20. A relation that is reflexive, symmetric and transitive is called:

(a) Empty relation
(b) Universal relation
(c) Equivalence relation
(d) Inverse relation

Answer: (c) Equivalence relation

Solution: An equivalence relation must be reflexive, symmetric and transitive.

Equivalence Relation MCQs

Q21. Which of the following is required for an equivalence relation?

(a) Reflexive only
(b) Symmetric only
(c) Transitive only
(d) Reflexive, symmetric and transitive

Answer: (d) Reflexive, symmetric and transitive

Solution: An equivalence relation must satisfy all three properties.

Q22. The relation “is equal to” on the set of real numbers is:

(a) Reflexive only
(b) Symmetric only
(c) Transitive only
(d) Equivalence relation

Answer: (d) Equivalence relation

Solution: Equality is reflexive, symmetric and transitive.

Q23. The relation R on integers defined by aRb if a − b is divisible by 3 is:

(a) Reflexive only
(b) Symmetric only
(c) Transitive only
(d) Equivalence relation

Answer: (d) Equivalence relation

Solution: Divisibility of a − b by 3 satisfies reflexive, symmetric and transitive properties.

Q24. If R is an equivalence relation, then R must be:

(a) Only reflexive
(b) Only symmetric
(c) Only transitive
(d) Reflexive, symmetric and transitive

Answer: (d) Reflexive, symmetric and transitive

Solution: This is the definition of equivalence relation.

Q25. The smallest equivalence relation on A = {1,2,3} is:

(a) ∅
(b) {(1,1), (2,2), (3,3)}
(c) A × A
(d) {(1,2), (2,3)}

Answer: (b) {(1,1), (2,2), (3,3)}

Solution: The smallest equivalence relation is the identity relation.

Q26. The largest equivalence relation on A = {1,2,3} is:

(a) ∅
(b) Identity relation
(c) A × A
(d) {(1,1)}

Answer: (c) A × A

Solution: The universal relation is the largest equivalence relation.

Q27. If R is an equivalence relation on A, then it partitions A into:

(a) Empty sets
(b) Singleton sets only
(c) Equivalence classes
(d) Ordered pairs only

Answer: (c) Equivalence classes

Solution: Every equivalence relation divides a set into disjoint equivalence classes.

Q28. If relation R on integers is defined by aRb if a ≡ b (mod 2), then the equivalence classes are:

(a) Positive and negative integers
(b) Even and odd integers
(c) Prime and composite numbers
(d) Natural and whole numbers

Answer: (b) Even and odd integers

Solution: Integers with the same remainder when divided by 2 form two classes: even and odd.

Q29. Identity relation on a set A is:

(a) Reflexive only
(b) Symmetric only
(c) Transitive only
(d) Equivalence relation

Answer: (d) Equivalence relation

Solution: Identity relation contains only self-pairs and satisfies reflexive, symmetric and transitive properties.

Q30. Universal relation on a set A is:

(a) Never equivalence
(b) Always equivalence
(c) Not symmetric
(d) Not transitive

Answer: (b) Always equivalence

Solution: Universal relation contains all pairs, so it is reflexive, symmetric and transitive.

Functions MCQs for Class 11

Q31. A function is a relation in which:

(a) Every element of domain has no image
(b) Every element of domain has exactly one image
(c) One element of domain has many images
(d) Every element of co-domain has exactly one pre-image

Answer: (b) Every element of domain has exactly one image

Solution: A function maps each element of the domain to exactly one element of the co-domain.

Q32. Which of the following relations is a function?

(a) {(1,2), (1,3)}
(b) {(2,3), (2,4)}
(c) {(1,2), (2,3), (3,4)}
(d) {(1,2), (1,4), (2,5)}

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

Solution: No first element is repeated with different images.

Q33. Which of the following is not a function?

(a) {(1,2), (2,3)}
(b) {(1,3), (2,3)}
(c) {(1,2), (1,4)}
(d) {(2,5), (3,5)}

Answer: (c) {(1,2), (1,4)}

Solution: The input 1 has two different images, 2 and 4. Hence, it is not a function.

Q34. The set of all input values of a function is called:

(a) Range
(b) Co-domain
(c) Domain
(d) Image

Answer: (c) Domain

Solution: Domain is the set of all values for which the function is defined.

Q35. The set of actual output values of a function is called:

(a) Domain
(b) Range
(c) Co-domain
(d) Pre-image

Answer: (b) Range

Solution: Range contains the actual images of elements of the domain.

Q36. If f(x) = 2x + 3, then f(2) is:

(a) 5
(b) 6
(c) 7
(d) 8

Answer: (c) 7

Solution: f(2) = 2(2) + 3 = 7.

Q37. If f(x) = x², then f(−3) is:

(a) −9
(b) 3
(c) 6
(d) 9

Answer: (d) 9

Solution: f(−3) = (−3)² = 9.

Q38. If f = {(1,2), (2,4), (3,6)}, then f(3) is:

(a) 1
(b) 2
(c) 3
(d) 6

Answer: (d) 6

Solution: In the ordered pair (3,6), the image of 3 is 6.

Q39. If f = {(1,5), (2,6), (3,7)}, then the range of f is:

(a) {1,2,3}
(b) {5,6,7}
(c) {1,5}
(d) {2,6}

Answer: (b) {5,6,7}

Solution: Range is the set of second elements.

Q40. Every function is:

(a) A relation
(b) Not a relation
(c) An equivalence relation
(d) A binary operation

Answer: (a) A relation

Solution: A function is a special type of relation.

Domain, Co-domain and Range MCQs

Q41. The domain of f(x) = 1/(x − 2) is:

(a) All real numbers
(b) All real numbers except 2
(c) All real numbers except 0
(d) Positive real numbers only

Answer: (b) All real numbers except 2

Solution: The denominator cannot be zero. So x − 2 ≠ 0, hence x ≠ 2.

Q42. The domain of f(x) = √(x − 3) is:

(a) x < 3
(b) x ≤ 3
(c) x ≥ 3
(d) All real numbers

Answer: (c) x ≥ 3

Solution: For square root to be real, x − 3 ≥ 0, so x ≥ 3.

Q43. The range of f(x) = x², where x ∈ R, is:

(a) R
(b) Negative real numbers
(c) Non-negative real numbers
(d) Positive integers

Answer: (c) Non-negative real numbers

Solution: Square of a real number is always greater than or equal to zero.

Q44. If f: A → B, then B is called:

(a) Domain
(b) Range
(c) Co-domain
(d) Relation

Answer: (c) Co-domain

Solution: In f: A → B, A is domain and B is co-domain.

Q45. The range of a function is always a subset of:

(a) Domain
(b) Co-domain
(c) Cartesian product
(d) Power set

Answer: (b) Co-domain

Solution: The actual outputs form the range, which is contained in the co-domain.

One-One, Onto and Bijective Function MCQs

Q46. A function f is one-one if:

(a) Different inputs have different images
(b) Every output has two inputs
(c) Range is empty
(d) Domain is empty

Answer: (a) Different inputs have different images

Solution: A one-one function maps distinct domain elements to distinct co-domain elements.

Q47. A function f is onto if:

(a) Domain is equal to range
(b) Range is equal to co-domain
(c) Domain is empty
(d) Function is constant

Answer: (b) Range is equal to co-domain

Solution: Onto function means every element of the co-domain is used as an image.

Q48. A function that is both one-one and onto is called:

(a) Many-one
(b) Into
(c) Bijective
(d) Constant

Answer: (c) Bijective

Solution: Bijective means both injective and surjective.

Q49. The function f: R → R defined by f(x) = x + 5 is:

(a) One-one only
(b) Onto only
(c) Bijective
(d) Neither one-one nor onto

Answer: (c) Bijective

Solution: Every real input gives a unique real output, and every real output has a pre-image.

Q50. The function f: R → R defined by f(x) = x² is:

(a) One-one and onto
(b) One-one but not onto
(c) Onto but not one-one
(d) Neither one-one nor onto

Answer: (d) Neither one-one nor onto

Solution: f(1)=f(−1)=1, so it is not one-one. Negative real numbers are not in its range, so it is not onto R.

Q51. The function f: R → R defined by f(x) = 3x − 2 is:

(a) One-one only
(b) Onto only
(c) Bijective
(d) Neither

Answer: (c) Bijective

Solution: A linear function with non-zero coefficient is one-one and onto from R to R.

Q52. A constant function from a non-empty set A to B is generally:

(a) One-one
(b) Many-one
(c) Onto always
(d) Bijective always

Answer: (b) Many-one

Solution: In a constant function, all elements of the domain have the same image.

Q53. If f: A → B is one-one and n(A) > n(B), then:

(a) f always exists
(b) f cannot exist
(c) f is onto
(d) f is bijective

Answer: (b) f cannot exist

Solution: A one-one function needs distinct images for distinct elements. This is impossible if the domain has more elements than the co-domain.

Q54. If f: A → B is bijective, then:

(a) n(A) < n(B)
(b) n(A) > n(B)
(c) n(A) = n(B) for finite sets
(d) A is empty only

Answer: (c) n(A) = n(B) for finite sets

Solution: A bijection pairs each element of A with exactly one unique element of B.

Q55. If f: R → R is given by f(x)=2x, then f is:

(a) One-one but not onto
(b) Onto but not one-one
(c) Bijective
(d) Neither

Answer: (c) Bijective

Solution: It is linear with non-zero slope, so it is both one-one and onto.

Number of Relations and Functions MCQs

Q56. If n(A)=2 and n(B)=4, then the number of functions from A to B is:

(a) 4
(b) 8
(c) 16
(d) 32

Answer: (c) 16

Solution: Number of functions from A to B = [n(B)]ⁿ⁽ᴬ⁾ = 4² = 16.

Q57. If n(A)=3 and n(B)=2, then the number of functions from A to B is:

(a) 6
(b) 8
(c) 9
(d) 12

Answer: (b) 8

Solution: Number of functions = 2³ = 8.

Q58. If n(A)=2 and n(B)=3, then the number of one-one functions from A to B is:

(a) 3
(b) 6
(c) 8
(d) 9

Answer: (b) 6

Solution: Number of one-one functions = 3P2 = 3 × 2 = 6.

Q59. If n(A)=3 and n(B)=3, then the number of bijections from A to B is:

(a) 3
(b) 6
(c) 9
(d) 27

Answer: (b) 6

Solution: Number of bijections between two 3-element sets = 3! = 6.

Q60. If A has 4 elements, then the number of relations on A is:

(a) 16
(b) 64
(c) 256
(d) 65536

Answer: (d) 65536

Solution: Number of relations on A = 2ⁿ² = 2¹⁶ = 65536.

Composition of Functions MCQs

Q61. If f(x)=2x+1 and g(x)=x², then (f ∘ g)(x) is:

(a) 2x² + 1
(b) (2x+1)²
(c) x² + 1
(d) 2x + x²

Answer: (a) 2x² + 1

Solution: (f ∘ g)(x)=f(g(x))=f(x²)=2x²+1.

Q62. If f(x)=2x+1 and g(x)=x², then (g ∘ f)(x) is:

(a) 2x² + 1
(b) (2x+1)²
(c) x² + 1
(d) 2x + 1

Answer: (b) (2x+1)²

Solution: (g ∘ f)(x)=g(f(x))=g(2x+1)=(2x+1)².

Q63. In general, (f ∘ g)(x) is:

(a) f(x) + g(x)
(b) g(f(x))
(c) f(g(x))
(d) f(x)g(x)

Answer: (c) f(g(x))

Solution: By definition, (f ∘ g)(x)=f(g(x)).

Q64. In general, composition of functions is:

(a) Always commutative
(b) Not always commutative
(c) Always impossible
(d) Same as addition

Answer: (b) Not always commutative

Solution: Usually, f ∘ g ≠ g ∘ f.

Q65. If f(x)=x+3, then (f ∘ f)(x) is:

(a) x+3
(b) x+6
(c) x²+3
(d) 2x+3

Answer: (b) x+6

Solution: (f ∘ f)(x)=f(f(x))=f(x+3)=x+3+3=x+6.

Q66. If f(x)=x−1 and g(x)=3x, then (g ∘ f)(2) is:

(a) 1
(b) 2
(c) 3
(d) 6

Answer: (c) 3

Solution: f(2)=1, so g(f(2))=g(1)=3.

Q67. If f(x)=x² and g(x)=x+1, then (f ∘ g)(2) is:

(a) 3
(b) 4
(c) 8
(d) 9

Answer: (d) 9

Solution: g(2)=3, so f(g(2))=f(3)=9.

Q68. If f(x)=2x and g(x)=x−4, then (f ∘ g)(x) is:

(a) 2x−8
(b) 2x−4
(c) x−8
(d) x²−4

Answer: (a) 2x−8

Solution: f(g(x)) = f(x−4)=2(x−4)=2x−8.

Q69. If f(x)=x² and g(x)=√x, then (f ∘ g)(x) is:

(a) x²
(b) √x
(c) x
(d) 2x

Answer: (c) x

Solution: f(g(x)) = f(√x) = (√x)² = x, for x ≥ 0.

Q70. If (g ∘ f)(x)=g(f(x)), then the function applied first is:

(a) g
(b) f
(c) both together
(d) none

Answer: (b) f

Solution: In g(f(x)), f is applied first and then g.

Inverse and Invertible Function MCQs

Q71. A function is invertible if it is:

(a) One-one only
(b) Onto only
(c) Both one-one and onto
(d) Constant

Answer: (c) Both one-one and onto

Solution: A function has an inverse if and only if it is bijective.

Q72. If f(x)=x+4, then f⁻¹(x) is:

(a) x+4
(b) x−4
(c) 4x
(d) x/4

Answer: (b) x−4

Solution: Let y=x+4. Then x=y−4. Hence, f⁻¹(x)=x−4.

Q73. If f(x)=2x, then f⁻¹(x) is:

(a) 2x
(b) x+2
(c) x/2
(d) x−2

Answer: (c) x/2

Solution: Let y=2x. Then x=y/2. Hence, f⁻¹(x)=x/2.

Q74. If f(x)=3x−5, then f⁻¹(x) is:

(a) (x+5)/3
(b) (x−5)/3
(c) 3x+5
(d) 3x−5

Answer: (a) (x+5)/3

Solution: Let y=3x−5. Then x=(y+5)/3.

Q75. If f is bijective, then f⁻¹ is:

(a) Not a function
(b) A function
(c) Empty relation
(d) Universal relation

Answer: (b) A function

Solution: The inverse of a bijective function is also a function.

Q76. If f = {(1,2), (2,3), (3,4)}, then f⁻¹ is:

(a) {(1,2), (2,3), (3,4)}
(b) {(2,1), (3,2), (4,3)}
(c) {(1,3), (2,4)}
(d) {(2,3), (3,4)}

Answer: (b) {(2,1), (3,2), (4,3)}

Solution: The inverse relation is obtained by interchanging each ordered pair.

Q77. If f and g are invertible functions, then (g ∘ f)⁻¹ is:

(a) g⁻¹ ∘ f⁻¹
(b) f⁻¹ ∘ g⁻¹
(c) f ∘ g
(d) g ∘ f

Answer: (b) f⁻¹ ∘ g⁻¹

Solution: The inverse of a composition reverses the order: (g ∘ f)⁻¹ = f⁻¹ ∘ g⁻¹.

Q78. A constant function from R to R is:

(a) Always invertible
(b) Never invertible
(c) Always one-one
(d) Always onto

Answer: (b) Never invertible

Solution: A constant function is not one-one, so it is not invertible.

Q79. If f: R → R, f(x)=x³, then f is:

(a) One-one only
(b) Onto only
(c) Bijective
(d) Neither

Answer: (c) Bijective

Solution: The cubic function maps R to R, is one-one and covers all real values.

Q80. If f(x)=x² from R to R, then f is not invertible because:

(a) It is not defined
(b) It is not one-one
(c) It is not a relation
(d) It has no domain

Answer: (b) It is not one-one

Solution: Since f(1)=f(−1)=1, f is not one-one and therefore not invertible.

Binary Operation MCQs

Q81. A binary operation on A is a function from:

(a) A to A
(b) A × A to A
(c) A to A × A
(d) A × B to B

Answer: (b) A × A to A

Solution: A binary operation combines two elements of A and gives an element of A.

Q82. Addition on the set of integers is:

(a) Not a binary operation
(b) A binary operation
(c) Not closed
(d) Undefined

Answer: (b) A binary operation

Solution: Sum of two integers is always an integer, so addition is a binary operation on integers.

Q83. Subtraction on natural numbers is:

(a) Always a binary operation
(b) Not a binary operation
(c) Always commutative
(d) Always associative

Answer: (b) Not a binary operation

Solution: Natural numbers are not closed under subtraction. For example, 2−5 = −3, which is not natural.

Q84. A binary operation * is commutative if:

(a) ab = ba
(b) a*(bc) = (ab)c
(c) ae = a
(d) a*b = e

Answer: (a) ab = ba

Solution: Commutativity means changing the order does not change the result.

Q85. A binary operation * is associative if:

(a) ab = ba
(b) a*(bc) = (ab)c
(c) ae = a
(d) a*a = a

Answer: (b) a*(bc) = (ab)*c

Solution: Associativity deals with grouping of elements.

Q86. The identity element for addition on integers is:

(a) 0
(b) 1
(c) −1
(d) 2

Answer: (a) 0

Solution: For every integer a, a+0 = 0+a = a.

Q87. The identity element for multiplication on non-zero rational numbers is:

(a) 0
(b) 1
(c) −1
(d) 2

Answer: (b) 1

Solution: For every non-zero rational number a, a×1 = 1×a = a.

Q88. The inverse of 5 under addition on integers is:

(a) 5
(b) −5
(c) 1/5
(d) 0

Answer: (b) −5

Solution: Additive inverse of 5 is −5 because 5 + (−5) = 0.

Q89. The inverse of 5 under multiplication on non-zero rational numbers is:

(a) −5
(b) 0
(c) 1/5
(d) 5

Answer: (c) 1/5

Solution: Multiplicative inverse of 5 is 1/5 because 5 × 1/5 = 1.

Q90. If a*b = a + b + ab, then 0 is:

(a) Identity element
(b) Inverse element
(c) Not related
(d) Universal element

Answer: (a) Identity element

Solution: a*0 = a + 0 + a(0) = a and 0*a = 0 + a + 0(a) = a. Hence, 0 is identity.

JEE Level Relations and Functions MCQs

Q91. If f: R → R is defined by f(x)=|x|, then f is:

(a) One-one and onto
(b) One-one but not onto
(c) Onto but not one-one
(d) Neither one-one nor onto

Answer: (d) Neither one-one nor onto

Solution: f(1)=f(−1)=1, so not one-one. Also, negative real numbers are not in range, so not onto R.

Q92. If f(x)=x²+1, where x ∈ R, then the range of f is:

(a) R
(b) [0,∞)
(c) [1,∞)
(d) (−∞,1]

Answer: (c) [1,∞)

Solution: Since x² ≥ 0, x²+1 ≥ 1.

Q93. If f(x)=1/x, then the domain is:

(a) R
(b) R − {0}
(c) R − {1}
(d) Positive integers

Answer: (b) R − {0}

Solution: Division by zero is not defined, so x ≠ 0.

Q94. If f(x)=x+1 and g(x)=2x, then (f ∘ g)(3) is:

(a) 6
(b) 7
(c) 8
(d) 9

Answer: (b) 7

Solution: g(3)=6, then f(6)=7.

Q95. If f(x)=2x−1, then f⁻¹(5) is:

(a) 2
(b) 3
(c) 4
(d) 5

Answer: (b) 3

Solution: f⁻¹(x)=(x+1)/2. So f⁻¹(5)=6/2=3.

Q96. Assertion: Every bijective function is invertible.

Reason: A function is invertible if it is both one-one and onto.

(a) Both Assertion and Reason are true, and Reason is the correct explanation
(b) Both Assertion and Reason are true, but Reason is not the correct explanation
(c) Assertion is true, but Reason is false
(d) Assertion is false, but Reason is true

Answer: (a) Both Assertion and Reason are true, and Reason is the correct explanation

Solution: A bijective function is both one-one and onto; hence, it is invertible.

Q97. Assertion: Every relation is a function.

Reason: A function is a special type of relation.

(a) Both Assertion and Reason are true
(b) Assertion is false, but Reason is true
(c) Assertion is true, but Reason is false
(d) Both are false

Answer: (b) Assertion is false, but Reason is true

Solution: Every function is a relation, but every relation is not a function.

Q98. A relation R on A = {1,2,3} is given by R = {(1,1), (2,2), (3,3), (1,2), (2,1)}. This relation is:

(a) Reflexive and symmetric
(b) Reflexive but not symmetric
(c) Symmetric but not reflexive
(d) Neither reflexive nor symmetric

Answer: (a) Reflexive and symmetric

Solution: All self-pairs are present, so it is reflexive. Also (1,2) and (2,1) are both present, so it is symmetric.

Q99. If A = {1,2,3} and B = {a,b}, then the number of functions from A to B is:

(a) 6
(b) 8
(c) 9
(d) 12

Answer: (b) 8

Solution: Number of functions = [n(B)]ⁿ⁽ᴬ⁾ = 2³ = 8.

Q100. If A = {1,2} and B = {a,b,c}, then the number of one-one functions from A to B is:

(a) 3
(b) 6
(c) 8
(d) 9

Answer: (b) 6

Solution: Number of one-one functions = 3P2 = 3 × 2 = 6.

Relations and Functions MCQ Answer Key

Common Mistakes in Relations and Functions MCQs

1. Confusing Relation and Function

Every function is a relation, but every relation is not a function. A relation becomes a function only when each element of the domain has exactly one image.

2. Confusing Co-domain and Range

The co-domain is the target set, while the range is the set of actual outputs. The range is always a subset of the co-domain.

3. Forgetting Reflexive Pairs

For a relation on A to be reflexive, every element of A must have its self-pair. For example, if A = {1,2,3}, then (1,1), (2,2) and (3,3) must be present.

4. Confusing Symmetric and Transitive Relations

Symmetric means if (a,b) is present, then (b,a) must also be present.
Transitive means if (a,b) and (b,c) are present, then (a,c) must also be present.

5. Assuming One-One Means Onto

A function can be one-one without being onto. A function is bijective only when it is both one-one and onto.

6. Assuming f ∘ g = g ∘ f

Composition of functions is not always commutative. In most cases, f ∘ g ≠ g ∘ f.

7. Forgetting the Condition for Invertibility

A function is invertible if and only if it is both one-one and onto.

8. Forgetting Closure in Binary Operations

For an operation to be binary on A, the result of combining any two elements of A must also belong to A.