Study Material

Relations and Functions Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Q: What are Relations and Functions in Class 11 Maths?

Relations and Functions is the second chapter in Class 11 Maths that establishes the crucial connection between elements of sets. A relation is a subset of the Cartesian product of two sets, while a function is a special type of relation where every element of the domain has exactly one image in the codomain. This chapter is essential for advanced calculus and coordinate geometry in JEE.

Q: What are the different types of relations?

The main types of relations are: Reflexive (aRa for all a), Symmetric (aRb implies bRa), Transitive (aRb and bRc implies aRc), Antisymmetric (aRb and bRa implies a=b), and Equivalence (reflexive + symmetric + transitive). Equivalence relations partition a set into disjoint equivalence classes.

Q: What are the different types of functions?

Functions are classified as: One-one (injective) — distinct inputs have distinct outputs; Onto (surjective) — range equals codomain; Bijective (one-one and onto); Into — range is a proper subset of codomain. For counting: all functions from A to B = |B|^|A|; injective = P(|B|, |A|); bijective = |A|! when |A|=|B|.

Q: What is the difference between domain, codomain and range?

The domain is the set of all possible input values (first elements) of a relation or function. The codomain is the set of all possible output values (second elements). The range is the set of actual output values that occur in the relation or function. The range is always a subset of the codomain.

Q: Can I download the Relations and Functions formula sheet PDF for free?

Yes. You can download the complete Relations and Functions formula sheet PDF for free using the download button on this page. It covers Cartesian product, types of relations, domain, codomain, range, types of functions, composite functions, inverse functions, even and odd functions, and graphical transformations in one comprehensive place for quick revision before JEE and board exams.

By Rohit Gupta Aug 22, 2026 17 min read

Relations and Functions Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Relations and Functions — Competishun

Relations and Functions Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Cartesian Product · Types of Relations · Domain · Range · Types of Functions · Composite · Inverse

Download Formula Sheet PDF

Relations and Functions is the second chapter in Class 11 Maths and one of the most foundational topics for JEE and board exams. It establishes the crucial connection between elements of sets and forms the basis for advanced calculus, coordinate geometry, and many other areas of mathematics.

This chapter covers Cartesian product, relations (definition, domain, range, types), functions (definition, domain, codomain, range, types, counting), composite functions, inverse functions, even and odd functions, periodic functions, and graphical transformations. Understanding these concepts is essential for solving complex problems in JEE and beyond.

This page gives you the complete guide to Relations and Functions with all concepts explained in depth. You will find clear definitions, formulas, visual diagrams, and practice questions. Download the free PDF below and keep it handy for quick revision before your JEE Main, board exams, or any other test.

Cartesian ProductA × B

Types of RelationsReflexive · Symmetric · Transitive

Types of FunctionsOne-one · Onto · Bijective

Free PDFDownload

Download the Relations and Functions Complete Guide PDF

Get all Relations and Functions concepts, formulas, types, and practice questions in one clean PDF, free. Perfect for JEE and board revision.

Download Free PDF

Opens in Google Drive · No sign-up needed

What are Relations and Functions?

Definition: A relation from a set A to a set B is a subset of the Cartesian product A × B. A function is a special type of relation where every element of the domain has exactly one image in the codomain.

Before understanding relations and functions, we must first understand the Cartesian product of two sets.

Cartesian Product (A × B)

The set of all ordered pairs (a, b) such that a ∈ A and b ∈ B.

Relation vs Function

A relation is any subset of A × B. A function is a relation where each element of A has a unique image in B.

Key Insight: Relations and Functions establish the crucial connection between elements of sets. They are essential for advanced calculus and coordinate geometry. Every function is a relation, but not every relation is a function.

Glossary of Terms — Complete A to Z

Before diving deep into each topic, let's understand the key terminology used in this chapter:

Term	Definition
Ordered Pair	A pair (a, b) where the order matters. (a, b) = (x, y) iff a = x and b = y.
Cartesian Product	A × B = {(a, b) : a ∈ A, b ∈ B}.
Relation	A subset of A × B. A relation from A to B.
Domain	The set of all first elements (a) of ordered pairs in a relation or function.
Codomain	The set B in a relation or function from A to B.
Range	The set of all second elements (b) that actually occur.
Reflexive Relation	(a, a) ∈ R for all a ∈ A.
Symmetric Relation	(a, b) ∈ R ⇒ (b, a) ∈ R.
Transitive Relation	(a, b), (b, c) ∈ R ⇒ (a, c) ∈ R.
Equivalence Relation	Reflexive + Symmetric + Transitive.
Function	A relation where each element of A has exactly one image in B.
One-one (Injective)	f(a₁) = f(a₂) ⇒ a₁ = a₂.
Onto (Surjective)	Range = Codomain.
Bijective	One-one + Onto.
Composite Function	(f ∘ g)(x) = f(g(x)).
Inverse Function	Exists iff f is bijective.
Even Function	f(−x) = f(x).
Odd Function	f(−x) = −f(x).
Periodic Function	f(x + T) = f(x) for some T > 0.
Mastering these terms is essential for understanding Relations and Functions.

Ordered Pairs and Cartesian Product

Definition: An ordered pair (a, b) has two elements where the order matters: (a, b) = (x, y) if and only if a = x and b = y. The Cartesian product of two sets A and B is defined as A × B = {(a, b) : a ∈ A, b ∈ B}.

Cartesian Product A × B

The Cartesian product A × B consists of all ordered pairs (a, b) where a ∈ A and b ∈ B.

Key Properties

n(A × B) = n(A) × n(B): The number of elements in the Cartesian product is the product of the cardinalities of the two sets.
A × B ≠ B × A (unless A = B or one is empty).
A × (B ∩ C) = (A × B) ∩ (A × C) (Distributive property).
A × (B ∪ C) = (A × B) ∪ (A × C).

Important: The Cartesian product is the foundation for relations and functions. A relation is simply a subset of the Cartesian product.

Relations — Definition and Types

Definition: A relation R from a set A to a set B is a subset of A × B. If R ⊆ A × B, then R is a relation from A to B. For a relation R, Domain(R) = {a : (a, b) ∈ R}, Range(R) = {b : (a, b) ∈ R}, and Codomain = B.

Number of Relations

Formulas:
• Number of relations from A to B = 2^{|A| × |B|}
• Number of relations on a set A = 2^|A|²

Types of Relations on a Set

Reflexive

(a, a) ∈ R for all a ∈ A

Example: "is equal to"

Symmetric

(a, b) ∈ R ⇒ (b, a) ∈ R

Example: "is sibling of"

Transitive

(a, b), (b, c) ∈ R ⇒ (a, c) ∈ R

Example: "is ancestor of"

Antisymmetric

(a, b), (b, a) ∈ R ⇒ a = b

Example: "is less than or equal to"

Equivalence Relation

Reflexive + Symmetric + Transitive

Example: "is congruent to" (mod n)

Counting Relations:
• Reflexive relations on a set with n elements = 2^{n² − n}
• Symmetric relations on a set with n elements = 2^n(n+1)/2

Equivalence Classes

An equivalence relation partitions a set into disjoint equivalence classes. Any two classes are either equal or disjoint; their union is the whole set. The number of equivalence relations on a set of n elements is given by the Bell number.

Key Insight: Equivalence relations are one of the most important types of relations. They allow us to partition a set into equivalence classes, which is fundamental in many areas of mathematics.

Functions — Definition and Types

Definition: A function f from A to B (denoted f: A → B) is a relation from A to B such that every element of A has exactly one image in B. In other words, f(x) is defined for all x ∈ A, and each x has a unique image.

Domain, Codomain, and Range

Domain: The set A of all possible inputs.
Codomain: The set B of all possible outputs.
Range: The set of all actual outputs: {f(x) : x ∈ A} ⊆ B.

Function f: A → B

A function maps every element of A to exactly one element of B. The range is a subset of the codomain.

Number of Functions

Formulas:
• Total functions from A to B = |B|^|A|
• Injective (one-one) functions (when |A| ≤ |B|) = P(|B|, |A|) = |B|! / (|B| − |A|)!
• Surjective (onto) functions (when |A| ≥ |B|) = Σ_i=0^|B| (−1)ⁱ C(|B|, i) (|B| − i)^|A|
• Bijective functions (when |A| = |B|) = |A|!

Types of Functions

One-one (Injective)

f(a₁) = f(a₂) ⇒ a₁ = a₂

Horizontal line test: at most one intersection

Onto (Surjective)

Range = Codomain

Every element of B has a pre-image

Bijective

One-one + Onto

Perfect pairing between A and B

Into

Range ⊂ Codomain

Not every element of B is used

Key Insight: One-one functions preserve distinctness (no two inputs give the same output). Onto functions cover the entire codomain. Bijective functions are invertible.

Standard Functions and Their Properties

Function	Rule	Domain	Range
Identity	f(x) = x	R	R
Constant	f(x) = c	R	{c}
Polynomial	f(x) = a₀ + a₁x + ... + aₙxⁿ	R	Varies
Rational	f(x) = g(x)/h(x), h(x) ≠ 0	R − {roots of h}	Varies
Modulus	f(x) = \|x\|	R	[0, ∞)
Signum	f(x) = sgn(x) = x/\|x\|	R	{-1, 0, 1}
Greatest Integer	f(x) = ⌊x⌋	R	Z
Fractional Part	f(x) = {x} = x − ⌊x⌋	R	[0, 1)
The greatest integer function is constant on [n, n+1) and jumps at every integer.

Modulus and Signum Functions

|x|: Modulus function. |x| = x for x ≥ 0, and |x| = −x for x < 0.
sgn(x): Signum function. sgn(x) = 1 for x > 0, 0 for x = 0, and −1 for x < 0.
|x| = x · sgn(x).

Important: The greatest integer function and fractional part function are frequently tested in JEE. Remember that ⌊x⌋ ≤ x < ⌊x⌋ + 1 and {x} ∈ [0, 1).

Composite Functions — f ∘ g

Definition: The composite function (f ∘ g)(x) is defined as f(g(x)). The domain of f ∘ g is the set of all x in the domain of g such that g(x) is in the domain of f.

Composite Function f ∘ g

First apply g, then apply f to the result. Composition is not commutative: f ∘ g ≠ g ∘ f in general.

Properties of Composite Functions

Associative: (f ∘ g) ∘ h = f ∘ (g ∘ h).
Not commutative: f ∘ g ≠ g ∘ f (in general).
If f and g are injective, then f ∘ g is injective.
If f and g are surjective, then f ∘ g is surjective.

Key Insight: Composition of functions is associative but not commutative. The order matters: f ∘ g means "apply g first, then f".

Inverse Functions — The Reverse Mapping

Definition: A function f: A → B is invertible if there exists a function f⁻¹: B → A such that f⁻¹(f(x)) = x for all x ∈ A and f(f⁻¹(y)) = y for all y ∈ B. A function has an inverse if and only if it is bijective.

Key Properties:
• Domain of f⁻¹ = Range of f; Range of f⁻¹ = Domain of f.
• The graph of f⁻¹ is the reflection of the graph of f in the line y = x.
• If f is increasing, then f⁻¹ is increasing.

How to find the inverse: Solve y = f(x) for x in terms of y, then interchange x and y.

Even and Odd Functions — Symmetry

Definition: A function f is even if f(−x) = f(x) for all x in its domain. A function is odd if f(−x) = −f(x) for all x in its domain.

Even functions are symmetric about the y-axis.
Odd functions are symmetric about the origin.
Every function can be split uniquely into an even and an odd part: f(x) = fₑ(x) + fₒ(x), where fₑ(x) = (f(x) + f(−x))/2 and fₒ(x) = (f(x) − f(−x))/2.

Operation	Result
Even + Even	Even
Odd + Odd	Odd
Even × Even	Even
Odd × Odd	Even
Even × Odd	Odd
These properties are useful for quickly determining the symmetry of combined functions.

Key Insight: Even functions are symmetric about the y-axis; odd functions are symmetric about the origin. These symmetries can simplify integration and graphing.

Periodic Functions — Repeating Behaviour

Definition: A function f is periodic if there exists a positive constant T such that f(x + T) = f(x) for all x in its domain. The smallest such T is called the fundamental period.

sin x: Period = 2π
cos x: Period = 2π
tan x: Period = π
|sin x|: Period = π
|cos x|: Period = π
sin(kx): Period = 2π/|k|
tan(kx): Period = π/|k|

Important: If f(x) has period T, then f(ax) has period T/|a|. The period of the sum of two periodic functions exists if the ratio of their periods is rational.

Graphical Transformations — Shifting and Scaling

Definition: Graphical transformations are operations that change the position, size, or orientation of the graph of a function.

y = f(x) + a

Shift up by a (if a > 0) / down by |a|

y = f(x + a)

Shift left by a (if a > 0) / right by |a|

y = a · f(x)

Vertical stretch (|a| > 1) / compress (0 < |a| < 1)

y = f(ax)

Horizontal compress (|a| > 1) / stretch (0 < |a| < 1)

y = −f(x)

Reflect in x-axis

y = f(−x)

Reflect in y-axis

y = |f(x)|

Flip the part below x-axis upward

y = f(|x|)

Mirror the right part to the left

Key Insight: Graphical transformations are a powerful tool for quickly sketching graphs. Understanding them saves time in JEE and board exams.

Practice Questions — From JEE and Boards

Question	Answer
Q1: If A = {1, 2, 3} and B = {4, 5}, how many functions are possible from A to B?	\|B\|^\|A\| = 2³ = 8
Q2: Find the number of relations from A = {1, 2} to B = {3, 4}.	2^\|A\|×\|B\| = 2^2×2 = 2⁴ = 16
Q3: Let R = {(1, 2), (2, 3), (1, 3)}. Is R transitive?	Yes, because (1,2) and (2,3) ⇒ (1,3) ∈ R.
Q4: What is the domain of f(x) = 1/(x − 2)?	All real numbers except 2: R − {2}
Q5: Is f(x) = x² an even or odd function?	Even, because f(−x) = (−x)² = x² = f(x).
Q6: What is the period of f(x) = sin(3x)?	2π/3
Q7: If f(x) = 2x + 1, find f⁻¹(x).	y = 2x + 1 → x = (y − 1)/2 → f⁻¹(x) = (x − 1)/2
Q8: How many bijective functions exist from A = {1, 2, 3} to B = {4, 5, 6}?	3! = 6
Practise these types of questions to become comfortable with applying Relations and Functions concepts in exam scenarios.

All Relations and Functions Formulas at a Glance

Formula	What It Means
A × B = {(a, b) : a ∈ A, b ∈ B}	Cartesian product
n(A × B) = n(A) × n(B)	Cardinality of Cartesian product
Domain(R) = {a : (a, b) ∈ R}	Domain of a relation
Range(R) = {b : (a, b) ∈ R}	Range of a relation
No. of relations = 2^\|A\|×\|B\|	Total relations from A to B
No. of functions = \|B\|^\|A\|	Total functions from A to B
Injective = P(\|B\|, \|A\|)	One-one functions
Bijective = \|A\|! (when \|A\|=\|B\|)	One-one onto functions
(f ∘ g)(x) = f(g(x))	Composite function
f⁻¹(f(x)) = x, f(f⁻¹(y)) = y	Inverse function property
f(−x) = f(x)	Even function
f(−x) = −f(x)	Odd function
f(x + T) = f(x)	Periodic function
Memorise these formulas for Relations and Functions. They are the key to scoring full marks in this chapter.

Common Mistakes in Relations and Functions

Confusing domain, codomain, and range: Domain is the set of inputs, codomain is the set of possible outputs, and range is the set of actual outputs.
Forgetting that every function is a relation but not every relation is a function: A function requires each input to have exactly one output.
Misapplying the vertical line test: A graph is a function if every vertical line meets it at most once.
Confusing one-one and onto: One-one means distinct inputs have distinct outputs. Onto means every element of the codomain is used.
Forgetting the order in composition: (f ∘ g)(x) = f(g(x)) — apply g first, then f.
Not checking whether a function is bijective before finding its inverse: Only bijective functions have inverses.

Golden Rule: In Relations and Functions, always clearly define the domain and codomain. Use the vertical line test for functions and the horizontal line test for one-one functions.

Why Relations and Functions Matters for JEE and Boards

Foundation for higher mathematics: Relations and Functions is the building block for calculus, coordinate geometry, and many other topics.
Moderate weightage: This chapter appears in 1-2 questions in JEE Main and is a guaranteed source of marks in board exams.
Conceptual clarity: This chapter rewards students who understand the definitions and properties rather than just memorizing formulas.
Practical relevance: Functions are used everywhere, from physics to economics to computer science.

Why this guide helps: A comprehensive Relations and Functions guide with all concepts, definitions, formulas, and practice questions saves you time during revision and helps you quickly recall everything during the exam. You won't need to look anywhere else.

Get the Complete Relations and Functions PDF for Free

Download the full Relations and Functions guide with all concepts, definitions, formulas, and practice questions. Perfect for last-minute revision before JEE and board exams.

Download Free PDF

Opens in Google Drive · No sign-up needed

Master Class 11 Maths with Competishun

Competishun teaches complete JEE mathematics with structured lectures, DPPs, PYQs and an All India Test Series.

All taught by faculty with 18 to 25 years of experience.

Class 11 · JEE 2028

Pratham

Two-year foundation program that builds strong Class 11 maths, physics and chemistry concepts.

Explore Pratham

Class 12 · JEE 2027

Prakhar

Class 12 program balancing boards, JEE Main and Advanced preparation together.

Explore Prakhar

Droppers · JEE 2027

Praveen & Fast Lane

Dropper program for a focused repeat attempt, including the accelerated Fast Lane batch.

Explore Praveen

Live Batch

Pragyaan

Live interactive batch with real-time classes and mentorship for JEE and NEET.

Explore Pragyaan

Distance Learning

DLP Study Material

Complete printed material with DPPs and PYQs for Class 11, Class 12 and droppers, delivered home.

Explore DLP

All Stages

All India Test Series

Regular JEE and NEET level tests with detailed analysis to track and improve your rank.

Explore Test Series

Free Maths Lectures on YouTube

Want to see Relations and Functions taught step by step with clear explanations? Competishun's full concept lectures are free on its YouTube channel.

More than 2.2 million+ students follow the channel for free concept lectures, PYQ solutions and exam strategy.

Learn the concepts, then use this Relations and Functions guide to revise.

Build Your Maths Foundation

Class 11, Class 12 or dropper, explore every Competishun program.

Or start free with full concept lectures on YouTube.

Explore All Courses Watch Free Lectures

About Competishun

Competishun is an online JEE and NEET coaching platform with programs for Class 11, Class 12 and droppers.

Complete DLP study material and an All India Test Series covering JEE Main, Advanced and NEET.

Taught by faculty with 18 to 25 years of experience.

More than 2.2 million+ students follow the Competishun YouTube channel.

Explore Courses YouTube Channel

Free JEE and Boards Concept Lectures

Follow Competishun for free Physics, Chemistry and Maths lectures, PYQ solutions and strategy.

Read These Next

Chemistry Isomerism: Complete Guide and Formula Sheet

Structural, geometrical, optical, and conformational isomerism with R/S configuration explained in depth.

Chemistry Atomic Structure: Complete Guide and Formula Sheet

Subatomic particles, Bohr's model, quantum numbers, de Broglie wavelength and Heisenberg uncertainty explained in depth.

Maths Sets Class 11 Formulas: Complete Formula List, Symbols & Types with Free PDF Download

All set formulas, symbols, types, and Venn diagrams for JEE and boards.

Frequently Asked Questions — Relations and Functions

What are Relations and Functions in Class 11 Maths?

Relations and Functions is the second chapter in Class 11 Maths that establishes the crucial connection between elements of sets. A relation is a subset of the Cartesian product of two sets, while a function is a special type of relation where every element of the domain has exactly one image in the codomain. This chapter is essential for advanced calculus and coordinate geometry in JEE.

What are the different types of relations?

The main types of relations are: Reflexive (aRa for all a), Symmetric (aRb implies bRa), Transitive (aRb and bRc implies aRc), Antisymmetric (aRb and bRa implies a=b), and Equivalence (reflexive + symmetric + transitive). Equivalence relations partition a set into disjoint equivalence classes.

What are the different types of functions?

Functions are classified as: One-one (injective) — distinct inputs have distinct outputs; Onto (surjective) — range equals codomain; Bijective (one-one and onto); Into — range is a proper subset of codomain. For counting: all functions from A to B = |B|^|A|; injective = P(|B|, |A|); bijective = |A|! when |A|=|B|.

What is the difference between domain, codomain and range?

The domain is the set of all possible input values (first elements) of a relation or function. The codomain is the set of all possible output values (second elements). The range is the set of actual output values that occur in the relation or function. The range is always a subset of the codomain.

Can I download the Relations and Functions formula sheet PDF for free?

Yes. You can download the complete Relations and Functions formula sheet PDF for free using the download button on this page. It covers Cartesian product, types of relations, domain, codomain, range, types of functions, composite functions, inverse functions, even and odd functions, and graphical transformations in one comprehensive place for quick revision before JEE and board exams.

Relations and Functions Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Download the Relations and Functions Complete Guide PDF

What are Relations and Functions?

Cartesian Product (A × B)

Relation vs Function

Glossary of Terms — Complete A to Z

Ordered Pairs and Cartesian Product

Key Properties

Relations — Definition and Types

Number of Relations

Types of Relations on a Set

Reflexive

Symmetric

Transitive

Antisymmetric

Equivalence Relation

Equivalence Classes

Functions — Definition and Types

Domain, Codomain, and Range

Number of Functions

Types of Functions

One-one (Injective)

Onto (Surjective)

Bijective

Into

Standard Functions and Their Properties

Modulus and Signum Functions

Composite Functions — f ∘ g

Properties of Composite Functions

Inverse Functions — The Reverse Mapping

Even and Odd Functions — Symmetry

Periodic Functions — Repeating Behaviour

Graphical Transformations — Shifting and Scaling

y = f(x) + a

y = f(x + a)

y = a · f(x)

y = f(ax)

y = −f(x)

y = f(−x)

y = |f(x)|

y = f(|x|)

Practice Questions — From JEE and Boards

All Relations and Functions Formulas at a Glance

Common Mistakes in Relations and Functions

Why Relations and Functions Matters for JEE and Boards

Get the Complete Relations and Functions PDF for Free