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CBSE Class 11 Mathematics★ 6-8 marks · CBSE Class 11 Mathematics Chapter 2

Relations and Functions

How are things CONNECTED? Relations and functions formalise the idea of association — from the Cartesian product of sets to the special case where every input has EXACTLY ONE output. This chapter covers domain, range, and types of functions. CBSE Class 11 Mathematics Chapter 2.

⏱ 24 min readMedium difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Compute Cartesian products of two sets and determine the number of ordered pairs in A×B
2Define a relation as a subset of A×B and identify its domain, range, and codomain
3Distinguish functions from general relations using the one-input-one-output criterion
4Classify functions as one-one, many-one, onto, into, or bijective with algebraic justification
5Find the domain and range of polynomial, rational, modulus, and square-root real functions

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Why this chapter matters

Functions are the central language of all mathematics — every subsequent chapter in Class 11 and 12 uses functional notation and thinking. This chapter carries very high JEE weightage as domain, range, and type-of-function questions appear in almost every paper.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Sets

Relations are defined as subsets of Cartesian products; set notation (∈, ⊆, ∪, ∩) is used throughout

Relations and Functions

"A function is a rule that takes an input and gives EXACTLY ONE output. Everything else is just a relation."

1. Chapter Overview

Sets tell us WHAT elements exist. RELATIONS tell us how elements of different sets are ASSOCIATED. FUNCTIONS are a special kind of relation where each input has EXACTLY ONE output. This chapter covers: Cartesian product, relations (definition, domain, range), and functions (definition, types — one-one, onto, into).

2. Cartesian Product of Sets

Definition

The Cartesian product of two sets A and B: A × B = {(a, b) : a ∈ A and b ∈ B}
It's the set of ALL POSSIBLE ORDERED PAIRS (a, b) where a comes from A and b from B
A × B ≠ B × A (generally — order matters!)
n(A × B) = n(A) × n(B) (number of elements)

Examples

If A = {1, 2} and B = {x, y}: A × B = {(1,x), (1,y), (2,x), (2,y)}
R × R = the COORDINATE PLANE (all ordered pairs of real numbers)

3. Relations

Definition

A relation R from set A to set B is a SUBSET of A × B
(a, b) ∈ R means 'a is related to b under relation R'
Domain of R: set of all FIRST elements
Range of R: set of all SECOND elements
Codomain of R: set B (the 'target' set)

Number of Relations

Total possible relations from A to B = 2^(n(A) × n(B))

4. Functions

Definition

A function (mapping) f from set A to set B (f : A → B) is a relation where EVERY element of A is related to EXACTLY ONE element of B
A = DOMAIN. B = CODOMAIN.
f(x) = the IMAGE of x under f
Range = {f(x) : x ∈ A} — the ACTUAL outputs. Range ⊆ Codomain.

Key Distinction: Relation vs Function

A function IS a relation (every function is a relation)
NOT every relation is a function
A relation is a function if: (a) every element of the domain has an image (is mapped somewhere), AND (b) no element of the domain has MORE THAN ONE image

5. Types of Functions

Based on Mapping Pattern

Type	Definition	Visual
One-One (Injective)	Different inputs → DIFFERENT outputs. f(x₁) = f(x₂) ⇒ x₁ = x₂.	Horizontal line test: any horizontal line cuts the graph at MOST ONCE.
Many-One	Two or MORE different inputs map to the SAME output.	Horizontal line cuts graph MORE THAN ONCE.
Onto (Surjective)	Range = Codomain. Every element of B is the image of SOME element of A.	All of B is 'covered'.
Into	Range ⊂ Codomain (strictly). Some elements of B are NOT images of any element of A.	Codomain is not fully covered.
One-One and Onto (Bijective)	BOTH one-one AND onto. The 'perfect' function.	Invertible.

Some Standard Functions

Identity function f(x) = x. Domain = R. Range = R. One-one and onto.
Constant function f(x) = c. Many-one (all inputs → same output).
Polynomial function f(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ
Rational function f(x) = P(x)/Q(x), Q(x) ≠ 0
Modulus function f(x) = |x|. Many-one.

6. Algebra of Real Functions

For functions f and g defined on a common domain:

(f + g)(x) = f(x) + g(x)
(f — g)(x) = f(x) — g(x)
(f · g)(x) = f(x) · g(x)
(f / g)(x) = f(x) / g(x), where g(x) ≠ 0

7. Exam Focus

Cartesian product — notation, counting elements
Relation — definition as subset of Cartesian product. Domain, range.
Function — definition (EVERY input → EXACTLY ONE output)
One-one, many-one, onto, into — definitions with examples
One-one onto (bijective) — significance
Domain and range of real functions (especially rational, modulus, square root)

8. Key Concepts

A relation from A to B is a subset of A × B
A function f : A → B assigns exactly ONE element of B to each element of A
One-one: f(x₁) = f(x₂) ⇒ x₁ = x₂
Onto: Range = Codomain (for every y ∈ B, ∃ x ∈ A such that f(x) = y)

9. Conclusion

Functions are the MOST IMPORTANT CONCEPT in mathematics — they appear in every subsequent chapter:

RELATION: A subset of A × B. Any association.
FUNCTION: A special relation. Every input → ONE output. Deterministic.
TYPES: One-one (injective). Onto (surjective). Bijective = both.
The idea: Mathematics is about MAPS between sets. A function IS a map. Understanding functions is understanding the central metaphor of all of mathematics.

'Functions are the verbs of mathematics. They are what things DO.'

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Cartesian Product

A×B = {(a,b) : a∈A and b∈B}, n(A×B) = n(A)×n(B)

A×B ≠ B×A in general; they are equal only when A=B or one is empty

Number of Relations from A to B

Total relations = 2^(n(A)×n(B))

Every subset of A×B is a valid relation

One-One (Injective) Condition

f(x₁) = f(x₂) ⟹ x₁ = x₂

Equivalently, x₁ ≠ x₂ implies f(x₁) ≠ f(x₂); test with horizontal line on graph

Onto (Surjective) Condition

Range = Codomain: for every y∈B, there exists x∈A such that f(x) = y

Every element of B must have at least one pre-image in A

Algebra of Functions

(f+g)(x) = f(x)+g(x); (fg)(x) = f(x)·g(x); (f/g)(x) = f(x)/g(x) where g(x)≠0

Operations defined on the intersection of the domains of f and g

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Treating Range as equal to Codomain for all functions

✓ Range ⊆ Codomain. Range equals Codomain ONLY for onto (surjective) functions. For into functions, the range is a proper subset of the codomain.

WATCH OUT

✗ Confusing many-one with into — thinking they describe the same property

✓ Many-one vs one-one describes the INPUT side (do two inputs share one output?). Into vs onto describes the OUTPUT side (is every codomain element used?). These are independent properties.

WATCH OUT

✗ Writing A×B = B×A

✓ (1,2) ∈ A×B but (2,1) ∈ B×A; they are different ordered pairs. A×B = B×A only when A = B.

WATCH OUT

✗ Accepting a relation as a function when one domain element maps to two range elements

✓ A function requires EXACTLY ONE image per domain element. If any x maps to two different y-values, it is a relation but NOT a function.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 2.1

Exercise 2.1

Cartesian products — computing A×B, listing ordered pairs, finding number of elements

Questions

Ex 2.2

Exercise 2.2

Relations — domain, range, codomain, arrow diagrams

Questions

Ex 2.3

Exercise 2.3

Functions — identifying functions from relations, algebra of functions, types of functions

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Cartesian Product

If A = {1, 2} and B = {a, b, c}, find A×B and state n(A×B).

▶ Show solution

A×B = {(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)}. n(A×B) = n(A)×n(B) = 2×3 = 6.

Q2MEDIUM· Domain and Range

Find the domain and range of f(x) = √(9 − x²).

▶ Show solution

Step 1: Require 9−x² ≥ 0, so x² ≤ 9, giving −3 ≤ x ≤ 3. Domain = [−3, 3]. Step 2: At x=0, f(0)=3 (maximum). At x=±3, f=0 (minimum). f is non-negative throughout. Range = [0, 3].

Q3HARD· Types of Functions

Let f: R→R be defined by f(x) = 2x+3. Prove that f is bijective.

▶ Show solution

One-one: Suppose f(x₁)=f(x₂). Then 2x₁+3 = 2x₂+3, so 2x₁=2x₂, giving x₁=x₂. Hence f is one-one. Onto: Let y∈R be arbitrary. Set x=(y−3)/2. Then x∈R and f(x) = 2·(y−3)/2+3 = y−3+3 = y. So every y has a pre-image. Hence f is onto. Since f is both one-one and onto, f is bijective.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Cartesian product A×B consists of ordered pairs (a,b); order matters, so (1,2) ≠ (2,1)
•A function f:A→B maps every element of A to EXACTLY ONE element of B
•Domain = set A (all inputs); Codomain = set B (target); Range = actual outputs ⊆ Codomain
•One-one (injective): f(x₁)=f(x₂) ⟹ x₁=x₂ — no two distinct inputs share the same output
•Onto (surjective): Range = Codomain — every element of B has at least one pre-image in A
•Bijective = one-one AND onto — such functions have inverse functions
•Modulus function f(x)=|x| is many-one (f(2)=f(−2)=2) and into for f:R→R (negative reals have no image)
•Identity function f(x)=x is both one-one and onto on R — it is bijective

CBSE marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 6-8 marks

Question type	Marks each	Typical count	What it tests
Short Answer	2	1-2	Cartesian product computation, identifying function type from arrow diagram
Long Answer	4-6	1	Proving a function is one-one/onto, finding domain and range

Prep strategy

Practice identifying function types using the horizontal line test on graphs — quick and visual
For domain problems, always check three restrictions: square roots (radicand ≥ 0), denominators (≠ 0), logarithms (argument > 0)
When proving one-one, always start with 'assume f(x₁)=f(x₂)' and derive x₁=x₂ algebraically — never just claim it

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Machine Learning Models

Every ML model is mathematically a function mapping input features to output predictions — bijective functions are used in normalisation and invertible transforms.

Cryptographic Hash Functions

Hash functions in cybersecurity are many-one functions — multiple inputs can produce the same hash — and understanding injectivity is key to analysing collision resistance.

Coordinate Transformations in GPS

GPS systems use bijective functions (rotations and translations) to map between different coordinate systems, ensuring every point has a unique representation.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

For 'is this a function?' questions, check the arrow diagram — if any element of A has two arrows leaving it, it is not a function
When finding range, identify the behaviour of the function (increasing/decreasing/bounded) and test boundary/critical values
For one-one proofs, the algebraic method is required — the graphical horizontal-line argument is intuition, not a proof
For JEE, be prepared for domain/range of composite functions — work inside-out applying each restriction in turn

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Composition of functions f∘g and its injectivity/surjectivity — f∘g one-one implies g is one-one; f∘g onto implies f is onto
Inverse functions: f⁻¹ exists if and only if f is bijective — found by solving y=f(x) for x, then writing x as a function of y

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 11 BoardHigh

JEE MainVery High

JEE AdvancedHigh

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

No. A relation is a function only when every element of the domain has exactly one image. If any element of A maps to two different elements of B, it is a relation but not a function.

Yes. Such functions are called 'into'. For example f:R→R defined by f(x)=x² has range [0,∞) which is a proper subset of R.

The image of a specific element x is the single value f(x). The range is the set of ALL images: {f(x) : x ∈ Domain}. The image is an element; the range is a set.

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