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JAC • 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.

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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.
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Timed CBT Practice Tests (Exam Simulator)

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