Class 11 Mathematics Notes Chapter 2 (Chapter 2) – Examplar Problems (English) Book

Alright class, let's get started with Chapter 2: Relations and Functions from your NCERT Exemplar. This is a foundational chapter, crucial not just for Class 11 and 12, but also for various competitive government exams where quantitative aptitude or mathematics is tested. Pay close attention to the definitions and especially to how we find the domain and range of functions.

Chapter 2: Relations and Functions - Detailed Notes for Exam Preparation

1. Cartesian Product of Sets

• Definition: Given two non-empty sets A and B, the Cartesian product A × B is the set of all possible ordered pairs (a, b) such that 'a' belongs to A and 'b' belongs to B.
  • Symbolically: A × B = {(a, b) | a ∈ A and b ∈ B}
• Ordered Pair: An ordered pair consists of two elements in a specific order. (a, b) is different from (b, a) unless a = b.
• Equality of Ordered Pairs: Two ordered pairs (a, b) and (c, d) are equal if and only if a = c and b = d.
• Key Properties:
  • If either A or B is the null set (∅), then A × B = ∅.
  • In general, A × B ≠ B × A. (Order matters!)
  • If n(A) = p and n(B) = q (where n(X) denotes the number of elements in set X), then n(A × B) = p * q.
  • If A and B are non-empty sets and either A or B is an infinite set, then A × B is an infinite set.
  • A × A × A = {(a, b, c) | a, b, c ∈ A}. Here, (a, b, c) is called an ordered triplet.

Example: If A = {1, 2} and B = {x, y}, then
A × B = {(1, x), (1, y), (2, x), (2, y)}
B × A = {(x, 1), (x, 2), (y, 1), (y, 2)}
n(A × B) = 2 * 2 = 4

2. Relations

• Definition: A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
  • Symbolically: R ⊆ A × B.
  • If (a, b) ∈ R, we say 'a' is related to 'b' under the relation R, written as a R b.
• Representation:
  • Roster Form: List all the ordered pairs belonging to the relation. R = {(a, b) | a R b}.
  • Set-builder Form: Define the relation using a rule. R = {(a, b) | a ∈ A, b ∈ B, and 'rule relating a and b'}.
  • Arrow Diagram: Draw two ovals representing sets A and B. Draw arrows from elements 'a' in A to elements 'b' in B if (a, b) ∈ R.
• Domain: The set of all first elements of the ordered pairs in the relation R. Domain(R) = {a | (a, b) ∈ R}. Note: Domain(R) ⊆ A.
• Range: The set of all second elements of the ordered pairs in the relation R. Range(R) = {b | (a, b) ∈ R}. Note: Range(R) ⊆ B.
• Codomain: The entire set B is called the codomain of the relation R from A to B.
• Total Number of Relations: If n(A) = p and n(B) = q, then n(A × B) = pq. The total number of possible subsets (relations) of A × B is 2^(pq).

Example: Let A = {1, 2, 3}, B = {1, 4, 9}. Let R be the relation "is the square root of" from A to B.
A × B = {(1,1), (1,4), (1,9), (2,1), (2,4), (2,9), (3,1), (3,4), (3,9)}
R = {(1, 1), (2, 4), (3, 9)} (Roster Form)
R = {(a, b) | a ∈ A, b ∈ B, and a = √b} or {(a, b) | a ∈ A, b ∈ B, and a² = b} (Set-builder Form)
Domain(R) = {1, 2, 3}
Range(R) = {1, 4, 9}
Codomain = B = {1, 4, 9}

3. Functions

• Definition: A relation 'f' from a set A to a set B is said to be a function if every element of set A has one and only one image in set B.
  • In other words:
    • Every element in A must be related to some element in B.
    • No element in A can be related to more than one element in B.
• Notation: If f is a function from A to B, we write f: A → B. If (a, b) ∈ f, we write f(a) = b, where 'b' is called the image of 'a' under f, and 'a' is called the pre-image of 'b' under f.
• Key Distinction: All functions are relations, but not all relations are functions.
• Domain, Codomain, Range of a Function:
  • Domain: The entire set A.
  • Codomain: The entire set B.
  • Range: The set of all images of elements of A under the function f. Range(f) = {f(a) | a ∈ A}. Range(f) ⊆ Codomain(B).
• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

4. Real Valued Functions

• Definition: A function which has either ℝ (the set of real numbers) or one of its subsets as its range is called a real valued function. Further, if its domain is also either ℝ or a subset of ℝ, it is called a real function. (In Class 11, we primarily deal with real functions).

5. Finding Domain and Range of Real Functions (Crucial for Exams!)

• Domain: The set of all real numbers 'x' for which the function f(x) is defined (i.e., yields a real number).
  • Polynomial Functions: Domain is always ℝ. (e.g., f(x) = 2x³ - x + 5)
  • Rational Functions (f(x) = P(x)/Q(x)): Domain is ℝ - {x | Q(x) = 0}. (Exclude values that make the denominator zero).
  • Square Root Functions (f(x) = √g(x)): Domain requires g(x) ≥ 0. (Expression under the square root must be non-negative).
  • Combined Functions: Consider restrictions from all parts of the function. For example, for f(x) = √(x-1) / (x-2), we need x-1 ≥ 0 (i.e., x ≥ 1) AND x-2 ≠ 0 (i.e., x ≠ 2). So, Domain = [1, ∞) - {2} or [1, 2) U (2, ∞).
• Range: The set of all possible output values (y or f(x)) that the function can produce.
  • Method 1: Algebraic Manipulation: Set y = f(x) and try to express x in terms of y (x = g(y)). The values 'y' can take (for which 'x' is real and in the domain) form the range.
  • Method 2: Using Properties:
    • For f(x) = ax + b (linear), Range = ℝ.
    • For f(x) = ax² + bx + c (quadratic), find the vertex (-b/2a, f(-b/2a)). Range is [f(-b/2a), ∞) if a > 0, or (-∞, f(-b/2a)] if a < 0. (Completing the square helps).
    • x² ≥ 0, |x| ≥ 0, √x ≥ 0 (for x ≥ 0).
    • For trigonometric functions (covered later, but useful): -1 ≤ sin x ≤ 1, -1 ≤ cos x ≤ 1.
  • Method 3: Graphical: The projection of the graph onto the y-axis gives the range.
  • Method 4: Standard Function Ranges: Know the ranges of common functions (see below).

6. Some Specific Types of Functions (Graphs, Domain, Range)

• Identity Function: f(x) = x
  • Domain = ℝ
  • Range = ℝ
  • Graph: Straight line passing through the origin, making a 45° angle with the positive x-axis (y = x).
• Constant Function: f(x) = c (where c is a constant)
  • Domain = ℝ
  • Range = {c} (A single value)
  • Graph: Horizontal line parallel to the x-axis at y = c.
• Polynomial Function: f(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ (n is a non-negative integer)
  • Domain = ℝ
  • Range: Depends on the degree and coefficients. (ℝ for odd degree, restricted interval for even degree like quadratics).
• Rational Function: f(x) = P(x) / Q(x), where P(x), Q(x) are polynomials and Q(x) ≠ 0.
  • Domain = ℝ - {x | Q(x) = 0}
  • Range: Varies, often found using algebraic manipulation (y = f(x) -> x = g(y)).
• Modulus Function: f(x) = |x| = { x, if x ≥ 0; -x, if x < 0 }
  • Domain = ℝ
  • Range = [0, ∞) (Non-negative real numbers)
  • Graph: V-shaped, symmetric about the y-axis.
• Signum Function: f(x) = { 1, if x > 0; 0, if x = 0; -1, if x < 0 }. Also f(x) = |x|/x for x ≠ 0, and f(0)=0.
  • Domain = ℝ
  • Range = {-1, 0, 1} (Only three values)
  • Graph: Discontinuous at x=0, looks like steps.
• Greatest Integer Function (Floor Function): f(x) = [x] or ⌊x⌋ = The greatest integer less than or equal to x.
  • Example: [2.7] = 2, [3] = 3, [-1.5] = -2, [0.8] = 0.
  • Domain = ℝ
  • Range = ℤ (Set of all integers)
  • Graph: Step-wise function, discontinuous at every integer.

7. Algebra of Real Functions

Let f: X → ℝ and g: X → ℝ be two real functions where X ⊆ ℝ.

• Addition: (f + g): X → ℝ defined by (f + g)(x) = f(x) + g(x) for all x ∈ X.
• Subtraction: (f - g): X → ℝ defined by (f - g)(x) = f(x) - g(x) for all x ∈ X.
• Multiplication by a Scalar: (αf): X → ℝ defined by (αf)(x) = α * f(x) for all x ∈ X, where α is a scalar (real number).
• Multiplication: (fg): X → ℝ defined by (fg)(x) = f(x) * g(x) for all x ∈ X.
• Quotient: (f/g): X' → ℝ defined by (f/g)(x) = f(x) / g(x), provided g(x) ≠ 0.
  • The domain X' for f/g is {x | x ∈ X and g(x) ≠ 0}.

Important Note on Domains of Combined Functions:
If f has domain D₁ and g has domain D₂, then:

• Domain(f ± g) = D₁ ∩ D₂
• Domain(fg) = D₁ ∩ D₂
• Domain(f/g) = {x | x ∈ D₁ ∩ D₂ and g(x) ≠ 0}

Multiple Choice Questions (MCQs)

1. If A = {a, b} and B = {1, 2, 3}, then the number of relations from A to B is:
   a) 6
   b) 8
   c) 32
   d) 64

2. Let R = {(x, y) | x, y ∈ ℕ and x + 2y = 10}. The range of the relation R is:
   a) {1, 2, 3, 4}
   b) {2, 4, 6, 8}
   c) {1, 3, 5, 7}
   d) {1, 2, 3, 4, 5}

3. The domain of the function f(x) = √(4 - x²) is:
   a) (-∞, -2] U [2, ∞)
   b) [-2, 2]
   c) (-2, 2)
   d) ℝ

4. The range of the function f(x) = |x - 3| is:
   a) ℝ
   b) (0, ∞)
   c) [0, ∞)
   d) (3, ∞)

5. Which of the following relations is a function?
   a) {(1, a), (2, b), (1, c)}
   b) {(x, y) | x² + y² = 1, x, y ∈ ℝ}
   c) {(1, 1), (2, 4), (3, 9), (4, 16)}
   d) {(a, p), (b, q), (c, r), (a, s)}

6. The domain of the function f(x) = (x² + 1) / (x² - 3x + 2) is:
   a) ℝ - {1, 2}
   b) ℝ - {-1, -2}
   c) ℝ
   d) ℝ - {0}

7. If f(x) = x² and g(x) = √x, then the domain of (f + g)(x) is:
   a) ℝ
   b) (0, ∞)
   c) [0, ∞)
   d) (-∞, 0]

8. The value of [π] + [-π] where [.] denotes the greatest integer function is:
   (Use π ≈ 3.14)
   a) 0
   b) -1
   c) 1
   d) 6

9. Let f(x) = 1 / (1 - x). The range of f(x) is:
   a) ℝ
   b) ℝ - {0}
   c) ℝ - {1}
   d) ℝ - {0, 1}

10. If f: R → R is defined by f(x) = x² + 1, then the pre-images of 17 are:
    a) {4}
    b) {-4}
    c) {4, -4}
    d) {16}

Answer Key for MCQs:

1. d (n(A)=2, n(B)=3, n(A×B)=6. No. of relations = 2⁶ = 64)
2. a (Pairs are (8,1), (6,2), (4,3), (2,4). Range = {1, 2, 3, 4})
3. b (Need 4 - x² ≥ 0 => x² ≤ 4 => -2 ≤ x ≤ 2)
4. c (Modulus function output is always non-negative)
5. c (Each first element has a unique second element)
6. a (Need x² - 3x + 2 ≠ 0 => (x-1)(x-2) ≠ 0 => x ≠ 1 and x ≠ 2)
7. c (Domain(f)=ℝ, Domain(g)=[0, ∞). Intersection is [0, ∞))
8. b ([3.14] = 3. [-3.14] = -4. Sum = 3 + (-4) = -1)
9. b (Let y = 1/(1-x) => 1-x = 1/y => x = 1 - 1/y. For x to be real, y ≠ 0. Also x cannot be 1, which corresponds to y being undefined, but y can approach infinity. So Range = ℝ - {0})
10. c (f(x) = 17 => x² + 1 = 17 => x² = 16 => x = ±4)

Study these concepts thoroughly, especially focusing on calculating domain and range, as they are very common in exams. Practice problems from the Exemplar book itself. Let me know if any part needs further clarification.