Relations & Functions – Chapter 2

1MCQ If (x + 1, y – 2) = (3, 1), then x and y are:

A. x = 2, y = 3

B. x = 3, y = 1

C. x = 4, y = 3

D. x = 2, y = 1

Correct Answer: A

x + 1 = 3 → x = 2

y − 2 = 1 → y = 3

Two ordered pairs are equal iff corresponding elements are equal.

2MCQ If n(A) = 3 and n(B) = 4, then n(A × B) is:

A. 7

B. 12

C. 81

D. 1

Correct Answer: B

n(A × B) = n(A) × n(B) = 3 × 4 = 12

The Cartesian product A × B has p × q elements when n(A) = p and n(B) = q.

3MCQ If A × B = {(p,q), (p,r), (m,q), (m,r)}, then A and B are:

A. A = {p,q,r}, B = {m}

B. A = {p,m}, B = {q,r}

C. A = {q,r}, B = {p,m}

D. A = {p,r}, B = {q,m}

Correct Answer: B

A = set of all first elements = {p, m}. B = set of all second elements = {q, r}. Check: {p,m} × {q,r} = {(p,q),(p,r),(m,q),(m,r)} ✓

4MCQ Which of the following statements is TRUE about A × B?

A. A × B = B × A always

B. If A and B are non-empty, A × B is a non-empty set of ordered pairs (x,y), x ∈ A, y ∈ B

C. A × φ ≠ φ

D. n(A × B) = n(A) + n(B)

Correct Answer: B

Option A: A × B ≠ B × A in general (order matters in ordered pairs). Option C: A × φ = φ. Option D: n(A×B) = n(A)·n(B). Only Option B is correct by definition of Cartesian product.

5MCQ The Cartesian product R × R represents:

A. Set of all real numbers

B. Coordinates of all points on a line

C. Coordinates of all points in 2D plane

D. Coordinates of all points in 3D space

Correct Answer: C

R × R = {(x,y) : x,y ∈ R} = all points in the 2D plane

R × R × R = {(x,y,z) : x,y,z ∈ R} = all points in 3D space

6MCQ A relation R from set A to set B is defined as:

A. A = B

B. A subset of A × B

C. A × B itself

D. A superset of A × B

Correct Answer: B

A relation R from A to B is a subset of the Cartesian product A × B. This subset is derived by describing a relationship between the first and second elements of the ordered pairs.

7MCQ For R = {(x,y) : y = x + 1, x ∈ {1,2,3,4,5,6}}, the domain is:

A. {2, 3, 4, 5, 6}

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

C. {1, 2, 3, 4, 5}

D. {2, 3, 4, 5, 6, 7}

Correct Answer: C

R = {(1,2),(2,3),(3,4),(4,5),(5,6)}. Note: 6 has no image in A = {1,...,6} since 6+1=7 ∉ A. Domain = {1,2,3,4,5}. Range = {2,3,4,5,6}. Codomain = {1,...,6}.

8MCQ Total number of relations that can be defined from A to B, where n(A)=2, n(B)=3 is:

A. 6

B. 2⁶ = 64

C. 2³ = 8

D. 2² = 4

Correct Answer: B

n(A × B) = 2 × 3 = 6

Total relations = 2^(n(A×B)) = 2⁶ = 64

Every subset of A × B is a relation, and a set with 6 elements has 2⁶ subsets.

9MCQ A function f from A to B requires:

A. Every element of B to have an image in A

B. Some elements of A to have images in B

C. Every element of A to have one and only one image in B

D. Elements of A to have more than one image

Correct Answer: C

A function f: A → B requires every element of A to have exactly one (unique) image in B. If any element of A has no image, or has more than one image, it is NOT a function.

10MCQ Which of the following is NOT a function?

A. R = {(2,1),(3,1),(4,2)}

B. R = {(2,2),(2,4),(3,3),(4,4)}

C. R = {(1,2),(2,3),(3,4),(4,5)}

D. R = {(1,1),(2,2),(3,3)}

Correct Answer: B

In Option B, element 2 maps to both 2 and 4 (two different images). This violates the definition of a function — each element must have only one image. Hence B is not a function.

11MCQ The identity function f: R → R is defined as:

A. f(x) = c (constant)

B. f(x) = x for all x ∈ R

C. f(x) = |x|

D. f(x) = 1/x

Correct Answer: B

The identity function is f(x) = x for all x ∈ R. Its domain and range are both R. Its graph is a straight line through the origin at 45° to both axes.

12MCQ The range of the signum function f: R → R is:

A. R

B. {0, 1}

C. {–1, 0, 1}

D. {–1, 1}

Correct Answer: C

f(x) = 1 if x > 0; f(x) = 0 if x = 0; f(x) = –1 if x < 0

The signum function has domain R but range = {–1, 0, 1} only. It essentially captures the "sign" of a number.

13MCQ The greatest integer function f(x) = [x] for x = 3.7 gives:

A. 4

B. 3.7

C. 3

D. –4

Correct Answer: C

The greatest integer function [x] returns the greatest integer less than or equal to x. For x = 3.7, the greatest integer ≤ 3.7 is 3. Note: [3.7] = 3, but [–3.7] = –4.

14MCQ For the modulus function f(x) = |x|, f(–5) equals:

A. –5

B. 0

C. 5

D. 25

Correct Answer: C

f(x) = |x| = x if x ≥ 0; = –x if x < 0

f(–5) = –(–5) = 5

The modulus function always returns non-negative values. Domain = R, Range = [0, ∞).

15MCQ If f(x) = 2x + 1, then (f + g)(x) where g(x) = x² is:

A. x² – 2x – 1

B. x² + 2x + 1

C. 2x³ + x²

D. (2x+1)/x²

Correct Answer: B

(f + g)(x) = f(x) + g(x) = (2x + 1) + x² = x² + 2x + 1

Algebra of functions: (f+g)(x) = f(x) + g(x). Also note: x² + 2x + 1 = (x+1)².

16MCQ The domain of f(x) = √(9 – x²) is:

A. [–3, 3]

B. (–3, 3)

C. R

D. [0, 3]

Correct Answer: A

For f(x) = √(9 – x²) to be defined: 9 – x² ≥ 0

x² ≤ 9 → –3 ≤ x ≤ 3 → Domain = [–3, 3]

17MCQ The range of f(x) = x² + 2 (where x is a real number) is:

A. R

B. [2, ∞)

C. (2, ∞)

D. [0, ∞)

Correct Answer: B

Since x² ≥ 0 for all real x, we have x² + 2 ≥ 2. Minimum value is 2 (at x = 0), and f(x) can be any value ≥ 2. So range = [2, ∞).

18MCQ The domain of the rational function f(x) = (x² + 3x + 5)/(x² – 5x + 4) is:

A. R

B. R – {1}

C. R – {1, 4}

D. R – {4}

Correct Answer: C

Denominator: x² – 5x + 4 = (x–1)(x–4) = 0 → x = 1 or x = 4

Domain = R – {1, 4}

A rational function is undefined where its denominator equals zero.

19MCQ A function f: N → N defined by f(x) = 2x. What is its range?

A. N (all natural numbers)

B. Set of odd natural numbers

C. Set of even natural numbers

D. {0, 2, 4, 6, ...}

Correct Answer: C

f(x) = 2x doubles every natural number: f(1)=2, f(2)=4, f(3)=6, …. So the range is the set of all even natural numbers = {2, 4, 6, 8, …}.

20MCQ If P = {1,2} and Q = {a, b}, then P × P × P contains how many elements?

A. 4

B. 6

C. 8

D. 9

Correct Answer: C

n(P × P × P) = n(P)³ = 2³ = 8

P × P × P = {(1,1,1),(1,1,2),(1,2,1),(1,2,2),(2,1,1),(2,1,2),(2,2,1),(2,2,2)}. Each element is an ordered triplet.

21MCQ The range of the constant function f(x) = 5 for all x ∈ R is:

A. R

B. {5}

C. [0, 5]

D. φ

Correct Answer: B

A constant function f(x) = c always outputs the same value c regardless of input. Domain = R, Range = {c} = {5}. Its graph is a horizontal line parallel to the x-axis.

22MCQ If f(x) = x² and g(x) = 2x + 1, then (fg)(x) is:

A. x² + 2x + 1

B. 2x³ + x²

C. x²/(2x+1)

D. x² – 2x – 1

Correct Answer: B

(fg)(x) = f(x) · g(x) = x² · (2x+1) = 2x³ + x²

Multiplication of two real functions: (fg)(x) = f(x)·g(x) for all x in the domain.

23MCQ For which relation is the codomain different from the range?

A. f: {1,2} → {3,4} defined by f(1)=3, f(2)=4

B. f: {1,2} → {3,4,5} defined by f(1)=3, f(2)=4

C. f(x) = x for x ∈ R

D. f: {a,b} → {1,2} where f(a)=1, f(b)=2

Correct Answer: B

In option B, codomain = {3,4,5} but range = {3,4} (only 3 and 4 are images; 5 has no preimage). Range ⊆ Codomain, and here they are unequal. Note that range ⊂ codomain always.

24MCQ The word "FUNCTION" was first used analytically by:

A. Cantor

B. Euler

C. Gottfried Wilhelm Leibnitz

D. Descartes

Correct Answer: C

The word FUNCTION first appeared in Gottfried Wilhelm Leibnitz's Latin manuscript (1673). Johan Bernoulli later assigned it a specialized analytical use in 1698 in a letter to Leibnitz.

25MCQ Let A = {1,2} and B = {3,4}. The number of subsets of A × B is:

A. 4

B. 8

C. 16

D. 6

Correct Answer: C

A × B = {(1,3),(1,4),(2,3),(2,4)} → n(A×B) = 4

Number of subsets = 2⁴ = 16

These 16 subsets include φ, 4 singleton sets, 6 pairs, 4 triples, and A×B itself.

P1Practice If A = {1,2,3}, B = {3,4} and C = {4,5,6}, find A × (B ∩ C) and verify it equals (A×B) ∩ (A×C).

Solution

B ∩ C = {4} → A × (B ∩ C) = {(1,4),(2,4),(3,4)}

A × B = {(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)}

A × C = {(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)}

(A×B) ∩ (A×C) = {(1,4),(2,4),(3,4)} ✓

P2Practice Let A = {1,2,3,...,14}. Define R from A to A by R = {(x,y): 3x – y = 0}. Write the domain, codomain and range.

Solution

y = 3x, so pairs: (1,3),(2,6),(3,9),(4,12) [since 5×3=15>14]

R = {(1,3),(2,6),(3,9),(4,12)}

Domain = {1,2,3,4} | Range = {3,6,9,12} | Codomain = {1,...,14}

P3Practice Let A = {1,2,3,4,6}. Let R be the relation: {(a,b): a,b ∈ A, b is exactly divisible by a}. Write R in roster form, domain and range.

Solution

R = {(1,1),(1,2),(1,3),(1,4),(1,6),(2,2),(2,4),(2,6),(3,3),(3,6),(4,4),(6,6)}

Domain = {1,2,3,4,6} = A

Range = {1,2,3,4,6} = A

P4Practice A function f is defined by f(x) = 2x – 5. Find f(0), f(7), f(–3).

Solution

f(0) = 2(0) – 5 = –5

f(7) = 2(7) – 5 = 14 – 5 = 9

f(–3) = 2(–3) – 5 = –6 – 5 = –11

P5Practice The temperature function is t(C) = 9C/5 + 32. Find t(0), t(28), t(–10) and the value of C when t(C) = 212.

Solution

t(0) = 9(0)/5 + 32 = 32°F

t(28) = 9(28)/5 + 32 = 50.4 + 32 = 82.4°F

t(–10) = 9(–10)/5 + 32 = –18 + 32 = 14°F

t(C) = 212: 9C/5 = 180 → C = 180×5/9 = 100°C

P6Practice Find the domain and range of f(x) = –|x|.

Solution

f(x) = –|x| is defined for all x ∈ R → Domain = R

Since |x| ≥ 0, we have –|x| ≤ 0 → Range = (–∞, 0] = {y : y ≤ 0}

The function returns 0 at x = 0 and negative values everywhere else.

P7Practice Let f(x) = x² and g(x) = 2x + 1. Find (f+g)(x), (f–g)(x), (fg)(x), (f/g)(x).

Solution

(f+g)(x) = x² + 2x + 1 = (x+1)²

(f–g)(x) = x² – 2x – 1

(fg)(x) = x²(2x+1) = 2x³ + x²

(f/g)(x) = x²/(2x+1), where x ≠ –1/2

P8Practice The Cartesian product A × A has 9 elements, and (–1, 0) and (0, 1) are among them. Find set A and the remaining elements of A × A.

Solution

n(A × A) = 9 → n(A)² = 9 → n(A) = 3

From (–1,0): –1 ∈ A and 0 ∈ A. From (0,1): 1 ∈ A.

∴ A = {–1, 0, 1}

Remaining elements: (–1,–1),(–1,1),(0,–1),(0,0),(1,–1),(1,0),(1,1)

P9Practice Write the relation R = {(x, x³): x is a prime number less than 10} in roster form.

Solution

Primes less than 10: {2, 3, 5, 7}

R = {(2, 8), (3, 27), (5, 125), (7, 343)}

Each ordered pair (x, x³) where x is a prime < 10. Domain = {2,3,5,7}, Range = {8,27,125,343}.

P10Practice Let f = {(1,1),(2,3),(0,–1),(–1,–3)} be a linear function from Z to Z. Find f(x).

Solution

Let f(x) = mx + c (linear function)

From (0, –1): f(0) = c = –1 → c = –1

From (1, 1): f(1) = m + c = m – 1 = 1 → m = 2

∴ f(x) = 2x – 1

Verify: f(2) = 3 ✓, f(–1) = –3 ✓

P11Practice Find the range of each: (i) f(x) = 2 – 3x, x > 0 (ii) f(x) = x² + 2 (iii) f(x) = x (all real).

Solution

(i) x > 0 → 3x > 0 → 2 – 3x < 2 → Range = (–∞, 2)

(ii) x² ≥ 0 → x²+2 ≥ 2 → Range = [2, ∞)

(iii) f(x) = x, x ∈ R → Range = R

P12Practice Let A = {x,y,z} and B = {1,2}. Find the number of relations from A to B. List any two of them.

Solution

n(A) = 3, n(B) = 2 → n(A × B) = 6

Number of relations = 2⁶ = 64

Example 1: R₁ = φ (empty relation)

Example 2: R₂ = {(x,1),(y,2)} (a function from A to B using 2 pairs)

P13Practice Define R on N by R = {(x,y): y = x+5, x < 4, x,y ∈ N}. Write R in roster form, domain and range.

Solution

x ∈ N, x < 4 → x = 1, 2, 3

R = {(1,6),(2,7),(3,8)}

Domain = {1,2,3} | Range = {6,7,8}

P14Practice Determine the domain and range of R = {(x, x+5): x ∈ {0,1,2,3,4,5}}.

Solution

R = {(0,5),(1,6),(2,7),(3,8),(4,9),(5,10)}

Domain = {0,1,2,3,4,5}

Range = {5,6,7,8,9,10}

P15Practice Let f, g: R → R be defined by f(x) = x+1 and g(x) = 2x–3. Find f+g, f–g, and f/g.

Solution

(f+g)(x) = (x+1) + (2x–3) = 3x – 2

(f–g)(x) = (x+1) – (2x–3) = x + 1 – 2x + 3 = –x + 4

(f/g)(x) = (x+1)/(2x–3), x ≠ 3/2

📐 Relations & Functions — Ch. 2

Section A — Multiple Choice Questions

Section B — Practice Questions

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