Login / Register
Follow Us
0
0
Register Free!
Register Free!
Online Mock Tests & Exam Preparation | Target Exams
Select Medium / माध्यम चुनें:
Eng (English) Hindi (हिन्दी)
CBSE • Class XI • Mathematics • Ch 2
Estimated Time: 45 Mins
Study Progress: In Progress

Relations and Functions

In Class 11 Mathematics, "Relations and Functions" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

Take Chapter CBT Test View All Chapters
Overview Core Concepts Summary Self Check Take MCQ Test

🔄 Have You Ever Wondered?

What is the exact mathematical definition of a 'black box' machine that takes an input number, processes it, and spits out a unique output without fai...

What is the exact mathematical definition of a 'black box' machine that takes an input number, processes it, and spits out a unique output without fail? Functions are the operational gears of higher calculus.

Why This Chapter Matters

In Class 11 Mathematics, "Relations and Functions" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

Before You Begin (Prerequisites)

  • Ordered pairs and Cartesian plane.
  • Set theory from Chapter 1.
  • Algebraic equations.

What You Will Learn (Core Objectives)

  • Define Cartesian Product of Sets ($A \times B$) and compute its cardinality ($|A \times B| = pq$).
  • Define a Relation as a subset of $A \times B$; identify Domain, Codomain, and Range.
  • Define a Function as a special relation where every element of domain has a unique image.
  • Analyze standard functions and graphs: Identity, Constant, Polynomial, Rational, Modulus ($|x|$), Signum, Greatest Integer ($[x]$).
  • Perform Algebra of Real Functions: $(f + g)(x)$, $(f - g)(x)$, $(fg)(x)$, and $(f/g)(x)$.

Chapter Roadmap & Progression

1 1. Cartesian Product & Relations
2 2. Definition of a Function
3 3. Special Real Functions

Complete Concept Guide (100% Curriculum Coverage)

1. Cartesian Product & Relations

Given non-empty sets $A$ and $B$, the Cartesian Product is: $$\mathbf{A \times B = \{(a, b) : a \in A, b \in B\}} \quad (|A \times B| = |A| \cdot |B|)$$ A Relation $R$ from $A$ to $B$ is any subset of $A \times B$. The total number of possible relations is $\mathbf{2^{pq}}$ where $|A|=p, |B|=q$.

2. Definition of a Function

A relation $f$ from $A$ to $B$ is a Function if and only if:
• Every element of $A$ has an image in $B$, AND
• No element of $A$ has more than one image in $B$ (Vertical Line Test!).

3. Special Real Functions

  • Modulus Function: $f(x) = |x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}$, Domain $=\mathbb{R}$, Range $=[0, \infty)$.
  • Signum Function: $f(x) = \begin{cases} 1, & x > 0 \\ 0, & x = 0 \\ -1, & x < 0 \end{cases}$, Domain $=\mathbb{R}$, Range $=\{-1, 0, 1\}$.
  • Greatest Integer Function: $f(x) = [x]$ (step function rounding down to largest integer $\le x$).

Visual Learning & Conceptual Map

Relations and Functions Master Matrix

Conceptual framework, core mechanisms, and analytical relationships
Academic Architecture

1. Cartesian Product & Relations • 2. Definition of a Function

Chapter Summary & 10 Key Takeaways

Takeaway 1
Cartesian Product: Ordered pairs $(a, b)$ forming total relational coordinate space.
Takeaway 2
Function: Mapping where every domain element is assigned a strictly unique range element.
Takeaway 3
Vertical Line Test: Geometric check where any vertical line intersects a function graph at most once.
Takeaway 4
Modulus: V-shaped graph outputting non-negative magnitude.
Takeaway 5
Greatest Integer Function: Step-stair graph rounding down to floor integer.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
If $A = \{1, 2\}$ and $B = \{3, 4\}$, write $A \times B$ and state the total number of relations from $A$ to $B$.
Reveal Answer & Explanation
Answer: $A \times B = \{(1, 3), (1, 4), (2, 3), (2, 4)\}$. Number of elements $= 2 \times 2 = 4$. Total relations $= 2^4 = 16$.
A × B has 4 pairs; 16 relations.
2
Find the domain and range of the real function $f(x) = \sqrt{9 - x^2}$.
Reveal Answer & Explanation
Answer: For $f(x)$ to be real, $9 - x^2 \ge 0 \implies x^2 \le 9 \implies -3 \le x \le 3$. Domain $=[-3, 3]$. Since $0 \le \sqrt{9-x^2} \le 3$, Range $=[0, 3]$.
Domain = [-3, 3]; Range = [0, 3].
3
What is the Signum Function? State its domain and range.
Reveal Answer & Explanation
Answer: The signum function is defined as $f(x) = \frac{|x|}{x}$ for $x \ne 0$ and $f(0) = 0$. Domain is $\mathbb{R}$; Range is $\{-1, 0, 1\}$.
Domain = R; Range = {-1, 0, 1}.
4
Evaluate $[3.7]$, $[-2.3]$, and $[5]$ for the greatest integer function $[x]$.
Reveal Answer & Explanation
Answer: $[3.7] = 3$; $[-2.3] = -3$ (largest integer $\le -2.3$); $[5] = 5$.
[3.7]=3, [-2.3]=-3, [5]=5.
5
Why is every function a relation, but not every relation a function?
Reveal Answer & Explanation
Answer: Because a function is a specialized relation with strict constraints: every element of the first set must relate to exactly one unique element in the second set; general relations permit one-to-many pairings.
Functions require unique single outputs for each input.
Finished Studying This Chapter?
READY TO PRACTICE?

Timed CBT Practice Tests (Exam Simulator)

Put your concepts to the test with official curriculum-aligned Foundation and Advanced practice tests. Get instant accuracy scores, time metrics, and step-by-step verified explanations.

Foundation Test
15 MCQs • Core Concepts
Advanced Test
15 MCQs • HOTS & Traps

All Class 11 Mathematics Chapters

AI Study Friend

Instant Concept Tutor

Have a doubt in Relations and Functions? Ask our AI study tutor for rapid explanations or customized quizzes.