Class 9 Maths Relations and Functions Notes

Class 9 Maths Relations and Functions Notes

Chapter 3 — Relations and Functions

1. Ordered Pairs

An ordered pair is written:

(a, b)

The order matters.

Generally:

(a,b) ≠ (b,a)

Two ordered pairs are equal exactly when their corresponding components are equal:

(a,b) = (c,d) ⇔ a = c and b = d

2. Cartesian Product

For sets A and B:

A × B = {(a,b) : a ∈ A, b ∈ B}

It contains every possible ordered pair formed by taking the first element from A and the second from B.

If:

n(A) = m and n(B) = n

then:

n(A × B) = mn

3. Relations

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

A relation can connect one element with:

  • no element,
  • one element,
  • or several elements,

depending on the relation.

The domain consists of first coordinates.

The range consists of second coordinates that actually occur.

4. Functions

A function is a special type of relation.

For a function:

Every element of the domain must have exactly one image.

So one input cannot have two different output values.

However, different inputs may have the same output.

This distinction is crucial.

5. Domain and Range

For a function:

f : A → B

  • A = domain
  • outputs actually obtained = range

The codomain and range need not always be the same.

6. Graph of a Function

A function can be represented graphically by plotting its ordered pairs.

The vertical line test helps determine whether a graph represents a function:

If any vertical line intersects the graph at more than one point, the graph does not represent a function of x.

7. Transformations of Graphs

If the original graph is:

y = f(x)

then:

y = f(x) + c

moves the graph vertically:

  • c > 0: upward
  • c < 0: downward

For:

y = f(x + k)

the graph shifts horizontally:

  • k > 0: left
  • k < 0: right
  • k < 0: rightward movement by |k|

Exam Focus

The most common conceptual trap is confusing:

  • relation with function
  • domain with range
  • element with image
  • horizontal and vertical graph shifts

Chapter 3 — Relations and Functions: Complete Question Bank

A. Multiple Choice Questions (MCQs)

1. Which of the following is an ordered pair?

A. {2,3}
B. (2,3)
C. [2,3]
D. 2,3

2. Two ordered pairs (a,b) and (c,d) are equal when:

A. a = c only
B. b = d only
C. a = c and b = d
D. a = d and b = c

3. If A = {1,2} and B = {a,b,c}, how many elements are in A × B?

A. 2
B. 3
C. 5
D. 6

4. Which of the following belongs to A × B when a ∈ A and b ∈ B?

A. (b,a)
B. {a,b}
C. (a,b)
D. a+b

5. A relation from A to B is a subset of:

A. A ∪ B
B. A ∩ B
C. A × B
D. B × A only

6. In an ordered pair (x,y), x is called the:

A. Image
B. Second component
C. First component
D. Range

7. The set of first components of the ordered pairs in a relation is called its:

A. Range
B. Domain
C. Codomain
D. Image set

8. A function is a relation in which every element of the domain has:

A. At least two images
B. Exactly one image
C. No image
D. Different images from every other element

9. Which statement is true for a function?

A. One input may have two outputs
B. Every input has exactly one output
C. Every output must have exactly one input
D. Domain and range must be equal

10. If f(x) = 2x + 1, then f(3) is:

A. 5
B. 6
C. 7
D. 8

11. Which test is commonly used to determine whether a graph represents a function of x?

A. Horizontal line test
B. Vertical line test
C. Diagonal line test
D. Parallel line test

12. The graph of y = f(x) + 3 is obtained by shifting the graph of y = f(x):

A. 3 units left
B. 3 units right
C. 3 units upward
D. 3 units downward

B. Fill in the Blanks

13. An ordered pair is written in the form __________.

14. In (a,b), the first coordinate is called the __________.

15. The Cartesian product of A and B is denoted by __________.

16. If n(A)=m and n(B)=n, then n(A×B)=__________.

17. A relation from A to B is a __________ of A × B.

18. The first components of a relation form its __________.

19. The second components that actually occur form its __________.

20. A function assigns exactly __________ output to each input.

21. The set from which the inputs of a function are taken is called the __________.

22. In f : A → B, B is called the __________.

23. The vertical line test is used to identify whether a graph represents a __________.

24. y=f(x)+k shifts the graph vertically by __________ units when k is positive.

C. True or False

25. (2,3) and (3,2) are always the same ordered pair.

26. The Cartesian product depends on the order of the two sets.

27. If A has 3 elements and B has 4 elements, then A×B has 12 elements.

28. Every relation is necessarily a function.

29. A function can assign the same output to two different inputs.

30. A function can assign two different outputs to the same input.

31. The domain consists of the first coordinates of the ordered pairs in a relation.

32. The range must always be equal to the codomain.

33. A vertical line crossing a graph twice means that graph is not a function of x.

34. y=f(x)-4 shifts the graph four units upward.

D. Match the Following

Column AColumn B
35. (a,b)a. Set of actual outputs
36. A × Bb. Input set
37. Domainc. Ordered pair
38. Ranged. Cartesian product
39. Functione. Exactly one output for each input

E. Cartesian Product Questions

40. Let A={1,2} and B={x,y}. Write A×B.

41. For the same sets, write B×A.

42. Explain why A×B and B×A need not be equal.

43. If n(A)=4 and n(B)=5, find n(A×B).

44. If A={a,b,c} and B={1,2}, how many ordered pairs are contained in A×B?

45. Determine whether (2,5) belongs to A×B when 2∈A and 5∈B.

F. Relations

46. Let A={1,2,3} and B={4,5}. Give any three ordered pairs belonging to A×B.

47. Consider the relation:

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

Find its domain and range.

48. Consider:

R={(a,1),(b,2),(c,2)}.

Is R a function from {a,b,c} to {1,2}? Give a reason.

49. Consider:

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

Is this a function? Explain.

50. Construct a relation from {1,2,3} to {a,b} containing exactly four ordered pairs.

G. Functions

51. Determine whether the relation

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

is a function.

52. Determine whether

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

is a function.

53. If f(x)=3x−2, find f(0), f(2) and f(5).

54. If g(x)=x²+1, find g(−2).

55. If f(x)=5x+3 and f(x)=18, find x.

56. A function maps 1→4, 2→4, and 3→7. What is its range?

57. Can two different elements of the domain have the same image? Explain with an example.

58. Can one element of the domain have two different images in a function? Explain.

H. Domain and Range

59. Find the domain and range of:

R={(2,5),(3,7),(4,9)}.

60. Find the domain and range of:

R={(a,b),(c,d),(e,b)}.

61. A function maps {1,2,3,4} to {5,6,7} as:

1→5, 2→6, 3→6, 4→7.

Find its domain and range.

62. Explain the difference between range and codomain.

I. Graphs and Transformations

63. What does the vertical line test tell us about a graph?

64. Describe the transformation from y=f(x) to y=f(x)+5.

65. Describe the transformation from y=f(x) to y=f(x)−2.

66. Describe the transformation from y=f(x) to y=f(x+4).

67. Describe the transformation from y=f(x) to y=f(x−3).

68. If the point (2,5) lies on y=f(x), what point lies on y=f(x)+4?

69. If (3,7) lies on y=f(x), what corresponding point appears after shifting the graph 2 units downward?

J. Assertion–Reason Questions

Choose:

A. Both Assertion and Reason are true, and Reason correctly explains Assertion.
B. Both are true, but Reason does not correctly explain Assertion.
C. Assertion is true, Reason is false.
D. Assertion is false, Reason is true.

70. Assertion: A×B and B×A are generally different.
Reason: The order of components in an ordered pair matters.

71. Assertion: Every function is a relation.
Reason: A function is a special type of relation.

72. Assertion: A function may map two different inputs to the same output.
Reason: A function requires each input to have exactly one output, not necessarily a unique output.

73. Assertion: Every relation is a function.
Reason: A relation may assign more than one output to a single input.

74. Assertion: y=f(x)+3 shifts a graph upward.
Reason: 3 is added to every output value.

K. Higher-Order Thinking Questions

75. A relation contains five ordered pairs. Its domain contains only four distinct elements. Can the relation still be a function? Explain.

76. A student claims that a function cannot have two different inputs with the same output. Is this claim correct? Justify.

77. If A contains 3 elements and B contains 2 elements, how many different relations from A to B are possible?

78. If a relation contains (2,5) and (2,8), can it be a function? Explain without using a graph.

79. A graph passes the vertical line test but has several points with the same y-coordinate. Can it still represent a function? Explain.

80. If the graph of y=f(x) contains (4,−2), determine the corresponding point after the transformation y=f(x)+6.

Answer Key

MCQs

  1. B
  2. C
  3. D
  4. C
  5. C
  6. C
  7. B
  8. B
  9. B
  10. C
  11. B
  12. C

Fill in the Blanks

  1. (a,b)
  2. first component / abscissa
  3. A×B
  4. mn
  5. subset
  6. domain
  7. range
  8. one
  9. domain
  10. codomain
  11. function
  12. k

True/False

  1. False
  2. True
  3. True
  4. False
  5. True
  6. False
  7. True
  8. False
  9. True
  10. False

Match

35–c
36–d
37–b
38–a
39–e

Numerical / Direct Answers

40. {(1,x),(1,y),(2,x),(2,y)}

41. {(x,1),(x,2),(y,1),(y,2)}

43. 20

44. 6

47. Domain = {1,2,3}; Range = {2,4}

48. Yes, it is a function.

49. No. Input 1 has two different outputs.

51. Yes.

52. No.

53.

  • f(0) = −2
  • f(2) = 4
  • f(5) = 13

54. 5

55. x=3

56. {4,7}

59. Domain = {2,3,4}; Range = {5,7,9}

61. Domain = {1,2,3,4}; Range = {5,6,7}

68. (2,9)

69. (3,5)

77. 2⁶ = 64 possible relations.

78. No.

80. (4,4).

Key Revision Rule

For this chapter, remember:

Cartesian product → Relation → Function → Domain/Range → Graph