Relations and Functions Class 11 Math Quick Recall / Short Notes (2026-27)

Mapping the Mathematical Universe: Relations and Functions

[!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 in Domain has EXACTLY ONE image in Codomain.

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

Range: The set of all second elements (images) in 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, and various types of functions like signum and modulus to excel in coordinate geometry and advanced math. This comprehensive Chapter 2 summary provides the mapping logic required for top-tier JEE and CBSE performance. Relations and 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 by 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) and q = n(B).

Key Terms:

Domain: The set of all first elements of the ordered pairs in a relation R.
Range: The set of all second elements in a relation R.
Codomain: The entire set B in 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 in set B.

Notation: f: A → B.
If (a, b) ∈ f, then f(a) = b, where b is the image of a and 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} and 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 in 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, and by internalizing the graphs of "parent" functions, you build a mental map of mathematical behavior. Always check your domain constraints and visualize your ranges!