Relations and Functions – Class 11 Maths Chapter 2 Notes & NCERT PDF

Class 11MathematicsChapter 2NCERT book: Mathematics - Class 11

Notes and a simple summary of Chapter 2 of the NCERT Class 11 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 2: Relations and Functions

Introduction

In everyday life we constantly pair things: a student with a roll number, a person with a father, a number with its square. This chapter formalises pairing using ordered pairs and the Cartesian product, then defines a relation as a subset of such a product and a function as a special relation in which every input has exactly one output. Functions are the central idea of calculus in later classes.

Key Concepts

2.2 Cartesian Products of Sets

An ordered pair (a, b) is a pair of elements in a definite order; (a, b) ≠ (b, a) unless a = b.

Two ordered pairs are equal if and only if their corresponding entries are equal: (a, b) = (x, y) ⇔ a = x and b = y.

The Cartesian product A × B = {(a, b) : a ∈ A, b ∈ B}.

If either A or B is empty, A × B = φ.

If n(A) = p and n(B) = q, then n(A × B) = pq.

In general A × B ≠ B × A.

R × R = {(x, y) : x, y ∈ R} is the coordinate plane; R × R × R represents three-dimensional space.

A × B × C = {(a, b, c) : a ∈ A, b ∈ B, c ∈ C} is a set of ordered triplets.

2.3 Relations

A relation R from A to B is a subset of A × B, chosen by some rule linking the first element x and the second element y.

If (x, y) ∈ R, y is the image of x and x is the pre-image of y.

Domain — the set of all first elements of the ordered pairs in R.

Range — the set of all second elements; the whole set B is the codomain. Range ⊆ codomain.

A relation can be shown in roster form, set-builder form or as an arrow diagram.

Since every subset of A × B is a relation, the total number of relations from A to B is 2^(pq) when n(A) = p and n(B) = q.

2.4 Functions

A function f from A to B is a relation in which every element of A has one and only one image in B. We write f : A → B, y = f(x).

A is the domain, B the codomain; the set of images is the range.

A real function has its domain and range as subsets of R. If only a formula is given, the domain is the largest set of real numbers for which the formula makes sense.

2.4.1 Some functions and their graphs

Identity function f(x) = x: domain and range R; graph is the line y = x.

Constant function f(x) = c: range {c}; graph is a horizontal line.

Polynomial function f(x) = a₀ + a₁x + … + aₙxⁿ with real coefficients and n a non-negative integer (e.g. x³ – x² + 2 is one; x^(2/3) + 2x is not).

Rational function f(x)/g(x) where f, g are polynomials and g(x) ≠ 0.

Modulus function f(x) = |x| = x if x ≥ 0 and –x if x < 0; range: non-negative reals.

Signum function f(x) = 1 if x > 0, 0 if x = 0, –1 if x < 0; range {–1, 0, 1}.

Greatest integer function f(x) = [x], the greatest integer less than or equal to x; e.g. [2.5] = 2 and [–2.5] = –3. Its graph looks like steps.

2.4.2 Algebra of real functions — for f, g : X → R:

(f + g)(x) = f(x) + g(x)

(f – g)(x) = f(x) – g(x)

(αf)(x) = α f(x) for a real α

(fg)(x) = f(x) g(x)

(f/g)(x) = f(x)/g(x), provided g(x) ≠ 0

Worked Examples

Example 1 (equality of ordered pairs). If (x/3 + 1, y – 2/3) = (5/3, 1/3), then x/3 + 1 = 5/3 gives x = 2, and y – 2/3 = 1/3 gives y = 1.

Example 2 (Cartesian product). G = {7, 8}, H = {5, 4, 2}.

G × H = {(7,5), (7,4), (7,2), (8,5), (8,4), (8,2)} — 2 × 3 = 6 pairs.

H × G has the same number of pairs but different ones, e.g. (5, 7).

Example 3 (counting relations). A = {1, 2}, B = {3, 4}: n(A × B) = 4, so there are 2⁴ = 16 relations from A to B. For A = {x, y, z}, B = {1, 2}: 2⁶ = 64 relations.

Example 4 (algebra of functions). f(x) = x + 1, g(x) = 2x – 3:

(f + g)(x) = 3x – 2

(f – g)(x) = –x + 4

(fg)(x) = (x + 1)(2x – 3) = 2x² – x – 3

(f/g)(x) = (x + 1)/(2x – 3), x ≠ 3/2

Example 5 (domain of a rational function). f(x) = (x² + 2x + 1)/(x² – 8x + 12). The denominator factorises as (x – 2)(x – 6), so the domain is R – {2, 6}.

Example 6 (domain and range). f(x) = √(x – 1) needs x ≥ 1: domain [1, ∞), range [0, ∞). f(x) = |x – 1|: domain R, range [0, ∞).

Formula Sheet

(a, b) = (x, y) ⇔ a = x, b = y

n(A × B) = n(A) · n(B)

Number of relations from A to B = 2^(n(A)·n(B))

|x| = x (x ≥ 0), –x (x < 0)

[x] = greatest integer ≤ x

(f ± g)(x) = f(x) ± g(x); (fg)(x) = f(x)g(x); (f/g)(x) = f(x)/g(x), g(x) ≠ 0

Exam Focus

"Is this relation a function?" Check two things: every element of the domain appears, and no first element repeats with a different second element.

Domain of a formula: remove values that make a denominator zero and values that make an expression under a square root negative.

Arrow diagrams: draw the relation, then read off domain, range and codomain.

Key Terms

Ordered pair (क्रमित युग्म), Cartesian product (कार्तीय गुणन)

Relation (संबंध), Image (प्रतिबिंब), Pre-image (पूर्व प्रतिबिंब)

Domain (प्रांत), Codomain (सहप्रांत), Range (परिसर)

Function (फलन), Real function (वास्तविक फलन)

Identity function (तत्समक फलन), Constant function (अचर फलन)

Polynomial function (बहुपद फलन), Rational function (परिमेय फलन)

Modulus function (मापांक फलन), Signum function (चिह्न फलन), Greatest integer function (महत्तम पूर्णांक फलन)

Common Mistakes

Treating (a, b) and (b, a) as the same pair.

Confusing range with codomain — the range can be smaller.

Calling a relation a function when some element of the domain has no image or two images.

Evaluating [–2.5] as –2 instead of –3.

Forgetting to exclude zeros of the denominator when stating the domain of f/g.

💡 Key Learning Points

✓Use ordered pairs and form the Cartesian product A × B, knowing n(A × B) = pq
✓Define a relation as a subset of A × B and find its domain, range and codomain
✓Recognise a function as a relation where each input has exactly one image
✓Know the graphs and ranges of standard real functions: identity, constant, polynomial, rational, modulus, signum, greatest integer
✓Add, subtract, multiply and divide real functions and state the resulting domain

👨‍🏫 Teaching Tips

→Begin with real pairings (student–roll number, city–state) before introducing notation
→Use arrow diagrams to show why some relations are functions and others are not
→Let students list A × B and B × A for small sets to see why they differ
→Plot each standard function on graph paper and discuss its domain and range
→Show the step graph of the greatest integer function with open and closed dots
→Give practice on finding domains by asking 'where does the formula break?'

📋 Assessment Questions

Can the student solve for unknowns using equality of ordered pairs?
Can the student list A × B and count the number of relations from A to B?
Can the student decide whether a given relation is a function?
Can the student state the domain and range of |x|, [x], signum and simple rational functions?
Can the student find f + g, f – g, fg and f/g for given f and g?

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