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Classes XI & XII Mathematics

Chapter 2: Relations and Functions

Standard NCERT & CBSE aligned study curriculum. Master concepts, track accuracy, revise weak areas, and challenge yourself with 9 customized practice modes.

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Chapter Overview

Welcome to Class XI Mathematics: Relations and Functions. This chapter forms a core structural component of the math syllabus, designed to build analytical rigor and key formula models.

Use the detailed subtopic guide below to review standard definitions, key mathematical rules, and study guidelines.

Prerequisite Concepts

Sets

Detailed Subtopics Study Guide

Review detailed conceptual explanations, mathematical equations, and guidelines for each subtopic in this chapter:

1Ordered pairs and Cartesian product

Concept Explanation

An ordered pair (a, b) consists of two elements in a specific order. The Cartesian product A x B is the set of all ordered pairs (a, b) such that a belongs to A and b belongs to B.

Mathematical Representation
A \times B = \{(a, b) \mid a \in A \land b \in B\}, \quad n(A \times B) = n(A) \cdot n(B)
Study Guideline: The order matters: (a, b) is not equal to (b, a) unless a = b. If A or B is empty, A x B is empty.

2Domain, Co-domain, and Range of relation

Concept Explanation

A relation R from set A to set B is a subset of A x B. The Domain is the set of all first elements in R. The Range is the set of all second elements in R. The Co-domain is the entire set B.

Mathematical Representation
\text{Domain} = \{x \mid (x,y) \in R\}, \quad \text{Range} = \{y \mid (x,y) \in R\}, \quad \text{Range} \subseteq \text{Co-domain}
Study Guideline: The Range is always a subset of the Co-domain. The Co-domain is the set of all *possible* outputs, while the Range is the set of *actual* outputs.

3Function as special relation

Concept Explanation

A function f from set A to B is a relation where every element of A has exactly one image in B. No two distinct ordered pairs in f can share the same first element.

Mathematical Representation
f: A \to B \quad \text{where } \forall x \in A, \, \exists ! y \in B \text{ such that } f(x) = y
Study Guideline: Vertical Line Test: if any vertical line intersects the graph of a relation more than once, the relation is not a function.

4Types of real functions (modulus, signum, greatest integer)

Concept Explanation

Real functions map real inputs to real outputs. Special functions include: Modulus |x| (returns absolute value), Signum sgn(x) (returns -1, 0, or 1 based on sign), and Greatest Integer [x] (returns the largest integer less than or equal to x).

Mathematical Representation
|x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}, \quad \text{sgn}(x) = \begin{cases} 1 & x > 0 \\ 0 & x = 0 \\ -1 & x < 0 \end{cases}, \quad [x] = n \iff n \le x < n+1
Study Guideline: Greatest Integer Function (floor function) always rounds down. For example, [2.7] = 2, and [-1.3] = -2.

5Sum and quotient of functions

Concept Explanation

Operations on real functions. For two functions f and g, their sum, difference, product, and quotient are defined on the intersection of their individual domains. For quotient, the divisor function cannot be zero.

Mathematical Representation
(f+g)(x) = f(x) + g(x), \quad \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \quad \text{where } D_{f/g} = D_f \cap D_g \setminus \{x \mid g(x)=0\}
Study Guideline: Always find the intersection of the domains of f and g first, and then exclude points where the denominator function g(x) equals zero.