JEE Main Mathematics - Unit 1: Sets, Relations and Functions - Practice Test (100 MCQs)

JEE Main Mathematics - Unit 1: Sets, Relations and Functions - Practice Test (100 MCQs)

Targeted 100 MCQ Practice Module focusing on: Sets and their representation; Union, intersection and complement of sets and their algebraic properties; Power set; Relations, types of relations, equivalence relations, functions; one-one, into and onto functions; the composition of functions.

Select your answers and click Submit Assessment Answers at the bottom to calculate your total score and review explanations.

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SECTION I: Sets - Basic Concepts and Representation [Questions 1 to 10]

1. A well-defined collection of distinct objects is called a:

Relation Set Function Sequence

Explanation: A set is a well-defined collection of distinct objects, considered as an object in its own right.

2. Which symbol denotes 'is an element of' in set theory?

⊂ ∈ ∪ ∅

Explanation: The symbol ∈ is read as 'belongs to' or 'is an element of'.

3. The set {1, 2, 3, 4, 5} written in set-builder form is:

{x : x ∈ N, x ≤ 5} {x : x ∈ N, x < 5} {x : x ∈ Z, x ≤ 5} {x : x ∈ N, x ≥ 5}

Explanation: Natural numbers from 1 to 5 satisfy x ∈ N and x ≤ 5.

4. A set containing no element at all is called a:

Singleton set Null set Universal set Infinite set

Explanation: A set with no elements is called the null set (or empty set), denoted by ∅ or { }.

5. If A = {x : x² = 9, x ∈ N}, then A is:

{3} {-3, 3} {9} {3, 9}

Explanation: x² = 9 gives x = 3 or x = -3, but since x ∈ N, only x = 3 qualifies, so A = {3}.

6. Which of the following is an example of an infinite set?

Set of days in a week Set of vowels in English alphabet Set of natural numbers Set of prime factors of 30

Explanation: The set of natural numbers N = {1, 2, 3, ...} continues indefinitely and hence is infinite.

7. Two sets A and B are said to be equal if:

They have the same number of elements Every element of A is in B and every element of B is in A n(A) = n(B) only A and B are both finite

Explanation: Sets A and B are equal only when they contain exactly the same elements, i.e., A ⊆ B and B ⊆ A.

8. A set having exactly one element is called a:

Null set Power set Singleton set Universal set

Explanation: A set with precisely one element, such as {5}, is called a singleton set.

9. Two finite sets are called equivalent if:

They contain identical elements They have the same cardinal number (number of elements) One is a subset of the other Their union is the universal set

Explanation: Equivalent sets need not have identical elements; they simply must have equal cardinality, i.e., n(A) = n(B).

10. The set {x : x is a letter in the word 'SCHOOL'} in roster form is:

{S, C, H, O, O, L} {S, C, H, O, L} {S, C, H} {S, C, O, L}

Explanation: Roster form lists distinct elements only, so repeated letters like 'O' are written once: {S, C, H, O, L}.

SECTION II: Union, Intersection, Difference and Complement [Questions 11 to 20]

11. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A ∪ B is:

{3, 4} {1, 2, 3, 4, 5, 6} {1, 2} {5, 6}

Explanation: A ∪ B contains all elements present in A or B or both, taken without repetition: {1,2,3,4,5,6}.

12. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A ∩ B is:

{1, 2} {5, 6} {3, 4} {1, 2, 3, 4, 5, 6}

Explanation: A ∩ B contains elements common to both sets, i.e., {3, 4}.

13. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A − B is:

{1, 2} {5, 6} {3, 4} {1, 2, 3, 4}

Explanation: A − B consists of elements of A that are not in B, i.e., {1, 2}.

14. If U = {1,2,3,...,10} is the universal set and A = {2,4,6,8,10}, then A′ (complement of A) is:

{1,3,5,7,9} {2,4,6,8,10} {1,2,3,...,10} ∅

Explanation: The complement of A consists of elements of U not in A, i.e., the odd numbers {1,3,5,7,9}.

15. Two sets A and B are called disjoint if:

A ∪ B = ∅ A ∩ B = ∅ A − B = ∅ A = B

Explanation: Disjoint sets have no elements in common, so their intersection is the empty set.

16. For any set A, A ∪ A′ equals:

∅ A A′ U (the universal set)

Explanation: A set together with its complement covers the entire universal set: A ∪ A′ = U.

17. For any set A, A ∩ A′ equals:

U A ∅ A′

Explanation: A set and its complement share no common elements, so A ∩ A′ = ∅.

18. If A ⊂ B, then A ∪ B equals:

A B ∅ A ∩ B

Explanation: When A is a subset of B, every element of A is already in B, so their union is simply B.

19. If A ⊂ B, then A ∩ B equals:

A B ∅ U

Explanation: When A ⊆ B, the elements common to both sets are exactly the elements of A.

20. If n(A) = 20, n(B) = 15 and n(A ∩ B) = 5, then n(A ∪ B) is:

35 30 25 40

Explanation: Using n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 20 + 15 − 5 = 30.

SECTION III: Algebraic Properties of Sets and De Morgan's Laws [Questions 21 to 30]

21. Which law states that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)?

Associative Law Distributive Law Commutative Law Idempotent Law

Explanation: This is the distributive law of union over intersection.

22. Which law states that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)?

Distributive Law De Morgan's Law Commutative Law Absorption Law

Explanation: This is the distributive law of intersection over union.

23. According to De Morgan's Law, (A ∪ B)′ equals:

A′ ∪ B′ A′ ∩ B′ A ∩ B A ∪ B

Explanation: De Morgan's first law states that the complement of a union equals the intersection of the complements: (A ∪ B)′ = A′ ∩ B′.

24. According to De Morgan's Law, (A ∩ B)′ equals:

A′ ∩ B′ A′ ∪ B′ A ∪ B A ∩ B

Explanation: De Morgan's second law states that the complement of an intersection equals the union of the complements: (A ∩ B)′ = A′ ∪ B′.

25. The property A ∪ B = B ∪ A illustrates which law of set operations?

Associative Law Commutative Law Distributive Law Complement Law

Explanation: Since the order of the sets does not affect the union, this illustrates the Commutative Law.

26. The property (A ∪ B) ∪ C = A ∪ (B ∪ C) illustrates which law?

Commutative Law Distributive Law Associative Law Identity Law

Explanation: Grouping of sets does not change the result of repeated unions, illustrating the Associative Law.

27. For any set A, A ∪ ∅ equals:

∅ A U A′

Explanation: The Identity Law states that the union of a set with the empty set leaves the set unchanged: A ∪ ∅ = A.

28. For any set A, A ∩ U equals (where U is the universal set):

∅ U A A′

Explanation: The Identity Law for intersection states A ∩ U = A.

29. The law A ∪ A = A and A ∩ A = A is called the:

Idempotent Law Distributive Law Complement Law Associative Law

Explanation: Combining a set with itself under union or intersection leaves it unchanged, which is the Idempotent Law.

30. The law A ∪ (A ∩ B) = A is known as the:

Distributive Law Absorption Law De Morgan's Law Commutative Law

Explanation: This identity, where a set 'absorbs' the intersection term, is known as the Absorption Law.

SECTION IV: Power Set, Subsets and Cartesian Product [Questions 31 to 40]

31. The collection of all subsets of a set A is called the:

Universal set of A Power set of A Complement of A Subset family of A

Explanation: The power set of A, denoted P(A), is the set of all possible subsets of A, including ∅ and A itself.

32. If a set A has n elements, the number of elements in its power set P(A) is:

n n² 2^n 2n

Explanation: A set with n elements has 2^n subsets, so |P(A)| = 2^n.

33. If A = {1, 2}, then P(A) is:

{∅, {1}, {2}} {{1}, {2}, {1,2}} {∅, {1}, {2}, {1,2}} {1, 2}

Explanation: The power set of A = {1,2} includes the empty set, both singletons, and the set itself: {∅, {1}, {2}, {1,2}}.

34. The empty set ∅ is a subset of:

Only itself Every set Only finite sets Only the universal set

Explanation: By convention, the empty set is a subset of every set, including itself.

35. If A has 3 elements and B has 4 elements, the number of elements in A × B (Cartesian product) is:

7 12 1 81

Explanation: The number of elements in A × B equals n(A) × n(B) = 3 × 4 = 12.

36. If (x+1, y-2) = (3, 1) as an ordered pair, then the values of x and y are:

x=2, y=3 x=3, y=1 x=4, y=-1 x=2, y=1

Explanation: Equating corresponding components: x+1=3 gives x=2, and y-2=1 gives y=3.

37. If n(A) = 5, the number of proper subsets of A is:

32 31 16 25

Explanation: Total subsets = 2^5 = 32; excluding the set A itself (proper subsets), we get 32 − 1 = 31.

38. A × (B ∪ C) is equal to:

(A × B) ∪ (A × C) (A × B) ∩ (A × C) (A ∪ B) × C A × B × C

Explanation: The Cartesian product distributes over union: A × (B ∪ C) = (A × B) ∪ (A × C).

39. If A = ∅, then the power set P(A) is:

∅ {∅} {0} {A}

Explanation: Even the empty set has one subset — itself — so P(∅) = {∅}, which has 2^0 = 1 element.

40. If A ⊆ B and B ⊆ A, then which of the following must be true?

A ∩ B = ∅ A = B n(A) ≠ n(B) A and B are disjoint

Explanation: Mutual containment (A ⊆ B and B ⊆ A) is precisely the condition for two sets to be equal.

SECTION V: Relations - Definitions and Types [Questions 41 to 50]

41. A relation R from set A to set B is defined as:

Any subset of A × B Any subset of B × A A function from A to B The Cartesian product A × B

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

42. If n(A) = 3 and n(B) = 4, the total number of relations that can be defined from A to B is:

12 2^12 7 81

Explanation: Since there are n(A)×n(B) = 12 ordered pairs in A × B, the total number of relations (subsets) is 2^12.

43. A relation R on set A is called reflexive if:

(a, b) ∈ R implies (b, a) ∈ R for all a, b ∈ A (a, a) ∈ R for every a ∈ A (a, b), (b, c) ∈ R implies (a, c) ∈ R R = A × A

Explanation: Reflexivity requires every element to be related to itself: (a, a) ∈ R for all a ∈ A.

44. A relation R on set A is called symmetric if:

(a, a) ∈ R for all a (a, b) ∈ R implies (b, a) ∈ R (a, b), (b, c) ∈ R implies (a, c) ∈ R R has no elements

Explanation: Symmetry means whenever a is related to b, b must also be related to a.

45. A relation R on set A is called transitive if:

(a, b) ∈ R implies (b, a) ∈ R (a, a) ∈ R for all a (a, b) ∈ R and (b, c) ∈ R imply (a, c) ∈ R R is empty

Explanation: Transitivity means if a is related to b and b is related to c, then a must be related to c.

46. Let R = {(1,1), (2,2), (3,3)} be a relation on A = {1,2,3}. R is:

Only symmetric Only transitive Reflexive, symmetric and transitive Neither reflexive nor symmetric

Explanation: R contains (a,a) for every element (reflexive), trivially satisfies symmetry, and trivially satisfies transitivity.

47. The relation R = {(1,2), (2,1)} on A = {1,2,3} is:

Reflexive Symmetric but not reflexive Transitive but not symmetric An equivalence relation

Explanation: R is symmetric since (1,2) and (2,1) both appear, but it is not reflexive as (3,3) ∉ R (and (1,1), (2,2) are also missing).

48. A relation which is reflexive, symmetric, but not necessarily transitive is called:

A tolerance relation An equivalence relation A partial order A function

Explanation: A relation satisfying only reflexivity and symmetry (without requiring transitivity) is termed a tolerance relation.

49. The relation 'is less than' (<) defined on the set of real numbers is:

Reflexive and symmetric Transitive but not reflexive or symmetric An equivalence relation Reflexive only

Explanation: '<' is transitive (a<b, b<c ⟹ a<c) but not reflexive (a < a is false) and not symmetric (a<b does not imply b<a).

50. A relation R on a set A is called an antisymmetric relation if:

(a,b) ∈ R and (b,a) ∈ R imply a = b (a,b) ∈ R implies (b,a) ∈ R R is never symmetric (a,a) ∉ R for any a

Explanation: Antisymmetry requires that if both (a,b) and (b,a) belong to R, then a must equal b.

SECTION VI: Equivalence Relations [Questions 51 to 60]

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

A partial order relation An equivalence relation A universal relation An identity relation

Explanation: By definition, a relation satisfying all three properties—reflexivity, symmetry, and transitivity—is an equivalence relation.

52. The relation 'is congruent to' defined on the set of all triangles in a plane is:

Only reflexive Only symmetric An equivalence relation Neither reflexive nor transitive

Explanation: Congruence of triangles is reflexive (a triangle is congruent to itself), symmetric, and transitive, so it is an equivalence relation.

53. The relation 'is parallel to' defined on the set of all lines in a plane (assuming every line is parallel to itself) is:

An equivalence relation Only symmetric Only reflexive Not transitive

Explanation: Parallelism is reflexive (each line parallel to itself), symmetric, and transitive, making it an equivalence relation.

54. For an equivalence relation R on set A, the equivalence class of element a is defined as:

{x ∈ A : (a, x) ∈ R} {x ∈ A : x ∉ R} A × A {a}

Explanation: The equivalence class of a, denoted [a], consists of all elements related to a under R.

55. The equivalence classes formed by an equivalence relation on a set A always:

Overlap with each other Partition the set A into disjoint subsets whose union is A Contain exactly one element each Are empty

Explanation: A fundamental property of equivalence relations is that their equivalence classes partition the set into mutually disjoint, exhaustive subsets.

56. Let R be a relation on the set of integers Z defined by aRb if a − b is divisible by 5. R is:

Reflexive only Symmetric only An equivalence relation Not reflexive

Explanation: This 'congruence modulo 5' relation is reflexive, symmetric, and transitive, hence an equivalence relation.

57. For the equivalence relation 'congruence modulo n' on Z, the number of distinct equivalence classes is:

n n-1 Infinite 1

Explanation: Congruence modulo n partitions Z into exactly n equivalence classes: [0], [1], ..., [n-1].

58. The relation R = {(a,b) : a, b ∈ Z, a = b} on the set of integers is called the:

Universal relation Void relation Identity relation Inverse relation

Explanation: The relation where each element is related only to itself (a = b) is the identity relation.

59. The relation R = A × A on a set A is called the:

Void relation Universal relation Identity relation Empty relation

Explanation: When every element of A is related to every element of A, R equals A × A, called the universal relation.

60. If R and S are two equivalence relations on a set A, then R ∩ S is:

Always empty Always an equivalence relation on A Never reflexive Never symmetric

Explanation: The intersection of two equivalence relations retains reflexivity, symmetry, and transitivity, and is therefore also an equivalence relation.

SECTION VII: Functions - Basic Concepts, Domain and Range [Questions 61 to 70]

61. A function f from set A to set B is a relation in which:

Every element of A has at least two images in B Every element of A has a unique image in B Every element of B has a pre-image in A A and B must have the same number of elements

Explanation: By definition, a function assigns exactly one (unique) image in B to every element of A.

62. For a function f : A → B, the set A is called the:

Codomain Range Domain Image set

Explanation: The set A, from which inputs of the function are taken, is called the domain.

63. For a function f : A → B, the set B is called the:

Domain Codomain Range Pre-image set

Explanation: The set B, which contains all possible outputs, is called the codomain of the function.

64. The range of a function f : A → B is:

Always equal to B The set of all images of elements of A, a subset of B The set of all elements of A Always equal to A × B

Explanation: The range is the actual set of output values obtained, which is a subset of (or equal to) the codomain B.

65. The domain of the real function f(x) = 1/(x-3) is:

R R − {3} R − {0} (3, ∞)

Explanation: The function is undefined when x − 3 = 0, i.e., x = 3, so the domain excludes 3: R − {3}.

66. The domain of the real function f(x) = √(x − 5) is:

R [5, ∞) (−∞, 5) R − {5}

Explanation: For the square root to be defined (real-valued), we need x − 5 ≥ 0, i.e., x ≥ 5, giving domain [5, ∞).

67. If f(x) = x², the range of f when the domain is R (the set of all real numbers) is:

R [0, ∞) (−∞, 0] (0, ∞)

Explanation: Since x² is always non-negative for real x, the range consists of all non-negative real numbers, [0, ∞).

68. Which of the following is NOT a function from A = {1,2,3} to B = {4,5,6}?

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

Explanation: In option C, the element 1 has two images (4 and 5), which violates the uniqueness requirement of a function.

69. A function in which the domain and codomain are both subsets of real numbers is called a:

Real function Complex function Vector function Discrete function

Explanation: A function whose domain and codomain are subsets of R is called a real-valued function of a real variable, or simply a real function.

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

6 8 5 4

Explanation: Substituting x = 2: f(2) = 3(2) + 2 = 6 + 2 = 8.

SECTION VIII: One-One (Injective) Functions [Questions 71 to 80]

71. A function f : A → B is called one-one (injective) if:

Different elements of A can have the same image in B Distinct elements of A always have distinct images in B Every element of B has a pre-image f(A) = B

Explanation: Injectivity means no two distinct inputs map to the same output: f(a1) = f(a2) implies a1 = a2.

72. The function f : R → R defined by f(x) = 2x + 1 is:

Not one-one One-one Neither one-one nor onto Constant

Explanation: If f(x1) = f(x2), then 2x1+1 = 2x2+1, which gives x1 = x2, so f is one-one.

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

One-one Not one-one, since f(2) = f(-2) = 4 Onto only Both one-one and onto

Explanation: f(2) = 4 and f(-2) = 4 shows that two distinct inputs give the same output, so f is not one-one.

74. If f : A → B is one-one, then which of the following must be true?

n(A) ≤ n(B) (for finite sets) n(A) > n(B) n(A) = n(B) always f cannot be onto

Explanation: For a one-one function between finite sets, the domain cannot have more elements than the codomain: n(A) ≤ n(B).

75. The function f : N → N defined by f(x) = x + 5 is:

Not one-one One-one Neither one-one nor a function Onto

Explanation: If x1 + 5 = x2 + 5, then x1 = x2, so distinct inputs always give distinct outputs, meaning f is one-one.

76. A one-one function is also known as a/an:

Surjective function Injective function Bijective function Identity function

Explanation: The term 'injective' is synonymous with 'one-one'.

77. Graphically, a function f(x) is one-one if:

Its graph passes through the origin Every vertical line intersects the graph at most once Every horizontal line intersects the graph at most once The graph is symmetric about the y-axis

Explanation: The Horizontal Line Test states that a function is one-one if no horizontal line intersects its graph more than once.

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

6 8 9 3

Explanation: The number of one-one functions from a set of 2 elements to a set of 3 elements is P(3,2) = 3 × 2 = 6.

79. The function f : R → R defined by f(x) = |x| is:

One-one Not one-one, since f(1) = f(-1) = 1 Injective and surjective A strictly increasing function

Explanation: Since f(1) = 1 and f(-1) = 1, two distinct values produce the same output, so f is not one-one.

80. A strictly increasing (or strictly decreasing) function on its domain is always:

Not a function Many-one One-one Onto only

Explanation: A strictly monotonic function never repeats a value, so distinct inputs always yield distinct outputs, making it one-one.

SECTION IX: Onto and Into Functions, Bijections [Questions 81 to 90]

81. A function f : A → B is called onto (surjective) if:

Every element of B has at least one pre-image in A Every element of A has a unique image f(A) is a proper subset of B n(A) = n(B)

Explanation: Surjectivity means the range of f equals the entire codomain B, i.e., every element of B is 'hit' by some element of A.

82. A function f : A → B is called an into function if:

f(A) = B f(A) is a proper subset of B (range ≠ codomain) f is one-one A = B

Explanation: An into function does not cover the entire codomain; its range is a proper subset of B.

83. The function f : R → R defined by f(x) = x³ is:

Onto but not one-one One-one and onto (bijective) Neither one-one nor onto Only into

Explanation: f(x) = x³ is strictly increasing (hence one-one) and covers all real numbers as x ranges over R (hence onto), so it is bijective.

84. Consider f : R → R+ ∪ {0} defined by f(x) = x² (with restricted codomain). This function is:

Into but not onto Onto, since every non-negative real number has a pre-image One-one Neither one-one nor onto

Explanation: Restricting the codomain to non-negative reals, every value y ≥ 0 has a pre-image (x = √y), so f becomes onto (though still not one-one).

85. A function which is both one-one and onto is called a:

Constant function Bijective function Identity relation Into function

Explanation: A function satisfying both injectivity and surjectivity is called a bijective function or bijection.

86. If f : A → B is onto and A, B are finite sets, then:

n(A) < n(B) n(B) ≤ n(A) n(A) = n(B) always n(B) = 0

Explanation: For an onto function between finite sets, the codomain cannot have more elements than the domain: n(B) ≤ n(A).

87. The function f : R → R defined by f(x) = 5 (a constant function) is:

Onto Into, since only one value is attained in the range Bijective One-one

Explanation: A constant function maps every input to the single value 5, so it is neither one-one nor onto—it is an into function.

88. If A = {1,2,3} and B = {1,2}, the number of onto functions from A to B is:

6 8 2 4

Explanation: Total functions from A to B = 2³ = 8. Subtracting the 2 constant (non-onto) functions gives 8 − 2 = 6 onto functions.

89. The function f : Z → Z defined by f(x) = 2x is:

Onto One-one but not onto, since odd integers have no pre-image Bijective Neither one-one nor onto

Explanation: f is one-one (2x1=2x2 ⟹ x1=x2), but odd integers in the codomain Z are never obtained as 2x for integer x, so f is not onto.

90. If f : A → B is a bijection, then which of the following is true?

f has no inverse f⁻¹ : B → A exists and is also a bijection n(A) ≠ n(B) f is not one-one

Explanation: Only a bijective function has a well-defined inverse function, and that inverse is itself a bijection.

SECTION X: Composition of Functions and Inverse Functions [Questions 91 to 100]

91. If f : A → B and g : B → C are two functions, their composition gof : A → C is defined by:

gof(x) = g(x) + f(x) gof(x) = g(f(x)) gof(x) = f(g(x)) gof(x) = f(x) · g(x)

Explanation: The composite function gof is defined as gof(x) = g(f(x)), applying f first and then g.

92. If f(x) = x + 2 and g(x) = 3x, then (gof)(x) is:

3x + 2 3x + 6 3(x+2) x + 6

Explanation: (gof)(x) = g(f(x)) = g(x+2) = 3(x+2) = 3x + 6.

93. If f(x) = x + 2 and g(x) = 3x, then (fog)(x) is:

3x + 2 3x + 6 x + 6 3(x+2)

Explanation: (fog)(x) = f(g(x)) = f(3x) = 3x + 2.

94. In general, for two functions f and g, gof and fog are:

Always equal Not necessarily equal, i.e., composition is not commutative in general Always undefined Equal only when f and g are constants

Explanation: Function composition is generally not commutative: gof ≠ fog in most cases.

95. The composition of two one-one functions is always:

Onto One-one Neither one-one nor onto A constant function

Explanation: If f and g are both one-one, then gof is also one-one, since distinct inputs remain distinct through both mappings.

96. The composition of two onto functions is always:

One-one Onto Into Undefined

Explanation: If f and g are both onto, their composition gof is also onto, since every element of the final codomain has a pre-image.

97. A function f : A → A defined by f(x) = x for every x ∈ A is called the:

Constant function Identity function Inverse function Zero function

Explanation: The function that maps every element to itself is called the identity function, usually denoted I(x) = x.

98. If f : A → B is a bijection with inverse f⁻¹ : B → A, then f⁻¹of equals:

The zero function on A The identity function on A The identity function on B f itself

Explanation: By definition of inverse functions, f⁻¹of(x) = x for all x ∈ A, i.e., f⁻¹of = I_A, the identity function on A.

99. If f(x) = 2x + 3, then f⁻¹(x) is:

(x − 3)/2 (x + 3)/2 2x − 3 (x)/2 − 3

Explanation: Setting y = 2x+3 and solving for x gives x = (y−3)/2, so f⁻¹(x) = (x−3)/2.

100. For composition of functions to be defined for gof, it is necessary that:

Domain of g = Domain of f Range of f is a subset of the domain of g f and g must be one-one A = B = C

Explanation: gof is defined only when the outputs (range) of f can be validly taken as inputs of g, i.e., range of f ⊆ domain of g.

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