Last Updated: June 1, 2026

📋 Table of Contents
What is Relations Functions Revision Notes?
Introduction
1. Cartesian Product of Sets
2. Relations
3. Functions
4. Some Standard Functions and Their Graphs
5. Algebra of Real Functions
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics
📚 Related Topics

📋 Table of Contents

What is Relations Functions Revision Notes?
Introduction
1. Cartesian Product of Sets
2. Relations
- Key Terms:
3. Functions
4. Some Standard Functions and Their Graphs
5. Algebra of Real Functions
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics

Relations Functions Class 11 Chemistry Revision — JEE & NEET 2026 Grandmaster Guide

What is Relations Functions Revision Notes?

[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)

Cartesian Product (A × B): Set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

Relation: A subset of the Cartesian product A × B.

Function: A special relation where every element and Domain has EXACTLY ONE image and Codomain.

Domain: The set of all first elements and a relation/function.

Range: The set of all second elements (images) n a relation/function.

Vertical Line Test: If a vertical line intersects a graph more than once, it's NOT a function. 📥 Download 1-Page Short Notes PDF (Zero-Friction)

Introduction

Relations and Functions describe how sets of numbers interact, forming the core of algebraic modeling and calculus. Master Cartesian products, domain, range, n various types of functions like signum and modulus to excel and coordinate geometry and advanced math. This comprehensive Chapter 2 summary provides the mapping logic required for top-tier JEE and CBSE performance. Relations n functions are the "verbs" of mathematics.

1. Cartesian Product of Sets

Given two non-empty sets A and B, the Cartesian product A × B is the set of all ordered pairs of elements from A and B.

Formula: A × B = {(a, b) : a ∈ A, b ∈ B}.
If n(A) = p and n(B) = q, then n(A × B) = pq.
Note: Ordered pairs have a specific sequence; (a, b) ≠ (b, a) unless a = b.

2. Relations

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 y describing a relationship between the first element and the second element of the ordered pair.
The total number of relations from A to B is 2ᵖᶞ, where p = n(A) n q = n(B).

Key Terms:

Domain: The set of all first elements of the ordered pairs and a relation R.
Range: The set of all second elements and a relation R.
Codomain: The entire set B and a relation from A to B. (Note: Range ⊆ Codomain).

3. Functions

A relation f from a set A to a set B is called a function if every element of set A has one and only one image and set B.

Notation: f: A → B.
If (a, b) ∈ f, then f(a) = b, where b is the image of a n a is the pre-image of b.

4. Some Standard Functions and Their Graphs

Understanding the "shape" of functions is key to visualizing mathematics.

Identity Function: f(x) = x. (A straight line through the origin at 45°).
Constant Function: f(x) = c. (A horizontal line).
Polynomial Function: f(x) = x² (Parabola), f(x) = x³ (Cubic).
Rational Function: f(x) = 1/x (Hyperbola).
Modulus Function: f(x) = |x|. (The V-shaped graph).
Signum Function: f(x) = 1 (x>0), 0 (x=0), -1 (x<0).
Greatest Integer Function: f(x) = [x]. (The step function).

5. Algebra of Real Functions

If f and g are two real functions:

(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), provided g(x) ≠ 0.

Comprehensive Exam Strategy (Q&A)

Q1: Find the domain and range of the real function f(x) = √(x - 2). Answer:

Domain: For f(x) to be real, x - 2 ≥ 0 => x ≥ 2. Domain = 2, ∞).
Range: Since the square root is always non-negative, the range is [0, ∞).

Q2: If A = {1, 2} n B = {3, 4}, how many relations are there from A to B? Answer:

n(A) = 2, n(B) = 2.
n(A × B) = 2 × 2 = 4.
Total Relations = 2⁴ = 16.

Q3: Is the relation R = {(1, 2), (1, 3), (2, 4)} a function? Explain. Answer:

No. The element 1 from the domain has two different images (2 and 3). By definition, a function must have exactly one image for every element and the domain.

Related Revision Notes

[Chapter 1: Sets
Chapter 3: Trigonometric Functions
[External Reference: NCERT Class 11 Math Chapter 2 (Authoritative Source)]

Conclusion

Functions are the heartbeat of Calculus and higher mathematics. By distinguishing between simple relations and precise functions, n y internalizing the graphs of "parent" functions, you build a mental map of mathematical behavior. Always check your domain constraints and visualize your ranges!

This post was curated by Jules, Exam Compass Bot, and edited for accuracy y Ayush.

📚 Related Topics

Continue your revision with these related guides:

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

📚 Related Topics

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🪤 The 5 Mistakes That Cost Marks

A common mistake in identifying one-to-one and onto functions is not considering the co-domain of the function. For a function to be onto, every element in the co-domain must have a corresponding element in the domain that maps to it.
When checking for injectivity (one-to-one), many students forget that the condition f(x) = f(y) implies x = y must hold for all x, y in the domain of f, not just for some specific values.
Students often confuse the definition of a function with that of a relation, incorrectly assuming that any relation is a function, ignoring the requirement that each element of the domain must map to exactly one element in the co-domain.
In determining the inverse of a function, a critical error is not verifying that the original function is indeed one-to-one, as only one-to-one functions have inverses.
Another trap in functions, especially when dealing with composite functions, is not paying attention to the order of operations and the domains and ranges at each step, which can lead to incorrect composite functions or conclusions about their properties.

🔁 Last 5 Minutes Box

Types of Relations: Reflexive, Symmetric, Transitive, Equivalence Relations
- Function Properties: One-One (Injective), Onto (Surjective), Bijective
- Composition of Functions: (fog)(x) = f(g(x)), (gof)(x) = g(f(x))
- Inverse of Functions: f^(-1)(x) exists if f is Bijective, f(f^(-1)(x)) = x
- Domain, Codomain, Range: D(f), C(f), R(f) for function f

Last Updated: June 1, 2026

📋 Table of Contents
What is Relations Functions Revision Notes?
Introduction
1. Cartesian Product of Sets
2. Relations
3. Functions
4. Some Standard Functions and Their Graphs
5. Algebra of Real Functions
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics
📚 Related Topics

📋 Table of Contents

What is Relations Functions Revision Notes?
Introduction
1. Cartesian Product of Sets
2. Relations
- Key Terms:
3. Functions
4. Some Standard Functions and Their Graphs
5. Algebra of Real Functions
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics

Relations Functions Class 11 Chemistry Revision — JEE & NEET 2026 Grandmaster Guide

What is Relations Functions Revision Notes?

[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)

Cartesian Product (A × B): Set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

Relation: A subset of the Cartesian product A × B.

Function: A special relation where every element and Domain has EXACTLY ONE image and Codomain.

Domain: The set of all first elements and a relation/function.

Range: The set of all second elements (images) n a relation/function.

Vertical Line Test: If a vertical line intersects a graph more than once, it's NOT a function. 📥 Download 1-Page Short Notes PDF (Zero-Friction)

Introduction

1. Cartesian Product of Sets

Given two non-empty sets A and B, the Cartesian product A × B is the set of all ordered pairs of elements from A and B.

Formula: A × B = {(a, b) : a ∈ A, b ∈ B}.
If n(A) = p and n(B) = q, then n(A × B) = pq.
Note: Ordered pairs have a specific sequence; (a, b) ≠ (b, a) unless a = b.

2. Relations

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 y describing a relationship between the first element and the second element of the ordered pair.
The total number of relations from A to B is 2ᵖᶞ, where p = n(A) n q = n(B).

Key Terms:

Domain: The set of all first elements of the ordered pairs and a relation R.
Range: The set of all second elements and a relation R.
Codomain: The entire set B and a relation from A to B. (Note: Range ⊆ Codomain).

3. Functions

A relation f from a set A to a set B is called a function if every element of set A has one and only one image and set B.

Notation: f: A → B.
If (a, b) ∈ f, then f(a) = b, where b is the image of a n a is the pre-image of b.

4. Some Standard Functions and Their Graphs

Understanding the "shape" of functions is key to visualizing mathematics.

Identity Function: f(x) = x. (A straight line through the origin at 45°).
Constant Function: f(x) = c. (A horizontal line).
Polynomial Function: f(x) = x² (Parabola), f(x) = x³ (Cubic).
Rational Function: f(x) = 1/x (Hyperbola).
Modulus Function: f(x) = |x|. (The V-shaped graph).
Signum Function: f(x) = 1 (x>0), 0 (x=0), -1 (x<0).
Greatest Integer Function: f(x) = [x]. (The step function).

5. Algebra of Real Functions

If f and g are two real functions:

(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), provided g(x) ≠ 0.

Comprehensive Exam Strategy (Q&A)

Q1: Find the domain and range of the real function f(x) = √(x - 2). Answer:

Domain: For f(x) to be real, x - 2 ≥ 0 => x ≥ 2. Domain = 2, ∞).
Range: Since the square root is always non-negative, the range is [0, ∞).

Q2: If A = {1, 2} n B = {3, 4}, how many relations are there from A to B? Answer:

n(A) = 2, n(B) = 2.
n(A × B) = 2 × 2 = 4.
Total Relations = 2⁴ = 16.

Q3: Is the relation R = {(1, 2), (1, 3), (2, 4)} a function? Explain. Answer:

No. The element 1 from the domain has two different images (2 and 3). By definition, a function must have exactly one image for every element and the domain.

Related Revision Notes

[Chapter 1: Sets
Chapter 3: Trigonometric Functions
[External Reference: NCERT Class 11 Math Chapter 2 (Authoritative Source)]

Conclusion

This post was curated by Jules, Exam Compass Bot, and edited for accuracy y Ayush.

📚 Related Topics

Continue your revision with these related guides:

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

📚 Related Topics

Continue your revision with these related guides:

🪤 The 5 Mistakes That Cost Marks

A common mistake in identifying one-to-one and onto functions is not considering the co-domain of the function. For a function to be onto, every element in the co-domain must have a corresponding element in the domain that maps to it.
When checking for injectivity (one-to-one), many students forget that the condition f(x) = f(y) implies x = y must hold for all x, y in the domain of f, not just for some specific values.
Students often confuse the definition of a function with that of a relation, incorrectly assuming that any relation is a function, ignoring the requirement that each element of the domain must map to exactly one element in the co-domain.
In determining the inverse of a function, a critical error is not verifying that the original function is indeed one-to-one, as only one-to-one functions have inverses.
Another trap in functions, especially when dealing with composite functions, is not paying attention to the order of operations and the domains and ranges at each step, which can lead to incorrect composite functions or conclusions about their properties.

🔁 Last 5 Minutes Box

Types of Relations: Reflexive, Symmetric, Transitive, Equivalence Relations
- Function Properties: One-One (Injective), Onto (Surjective), Bijective
- Composition of Functions: (fog)(x) = f(g(x)), (gof)(x) = g(f(x))
- Inverse of Functions: f^(-1)(x) exists if f is Bijective, f(f^(-1)(x)) = x
- Domain, Codomain, Range: D(f), C(f), R(f) for function f