Class 12 Maths Chapters List:- JEE & NEET Revision Notes

Chapter 1 Relations and Functions

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JEE Maths

Class 12 Chapter 1

Class 12 Maths Chapter 1: Relations and Functions

From assigning students to roll numbers and mapping cities to their pin codes to defining mathematical models used in engineering and computer science, Relations and Functions help establish meaningful connections between different quantities. This chapter extends the concepts learned in Class 11 by introducing types of relations, equivalence relations, composite functions, inverse functions, and binary operations. These concepts serve as the building blocks for advanced topics such as Calculus, Matrices, Vector Algebra, and Linear Transformations, making this chapter highly important for JEE Main, JEE Advanced, and CBSE Board examinations.

At ALLEN, our expert faculty have prepared these Class 12 Maths Chapter 1: Relations and Functions smart study material to help students revise the chapter efficiently with complete conceptual clarity. Class 12 Maths Chapter 1 notes include concise theory, important definitions, standard formulas, shortcut techniques, solved illustrations, common exam traps, and PYQ-based insights. Whether you're preparing for CBSE Board exams or competitive exams like JEE Main and JEE Advanced, these revision notes will help you strengthen your fundamentals and solve questions with confidence.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate to High
Revision Time	20–25 Minutes
Key Concepts Covered	Types of relations and functions, algebra of functions, binary operations

Important topics/subtopics covered:-

Topic	Overview
Types of Relations	Reflexive, Symmetric, Transitive and Equivalence Relations
Types of Functions	One-One, Onto, Bijective Functions
Composite Functions	Composition of two or more functions
Inverse Functions	Finding and verifying inverse functions
Binary Operations	Definition and properties
Identity Function	Concept and applications
Algebra of Functions	Addition, subtraction, multiplication and division of functions
Applications	Foundation for Calculus and Higher Mathematics

2.0Chapter 1 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Class 12 Maths Chapter 1 NCERT Solutions	Available Soon
Relations and Functions Important Questions	Available Soon
Formula Sheet PDF	Available Soon
JEE Main & Advanced Previous Year Questions	Available Soon
Chapter-wise Mock Test	Available Soon
Practice Worksheets	Available Soon
Quick Revision Notes PDF	Available Soon
Mind Maps	Available Soon

Relations and Functions

Relation:

The relation is a linear operation that establishes a relationship between the elements of two sets according to some definite rule of relationship.

R: {(a, b)| (a, b) ∈ A × B and a R b}

Function:

Let A and B be two sets, and let a rule or manner or correspondence ‘f’ exist that associates each element of A with a unique element in B. Then, f is called a mapping or function from A to B.

f: A → B or $A f B$

Which reads f maps A to B or f is a function from A to B.

Related Video:

1.0Definition of Relation

A Relation R from set A to set B is established as a subset of the Cartesian product A × B. In other words, relation R consists of ordered pairs where the first element belongs to set A and the second element belongs to set B. Therefore, we can say that R is a relation from A to B, represented as $R \subseteq A \times B$ .

Total number of relations if n(A) = m and n(B) = n is 2^mn.

2.0Domain and Range of a Relation

Let R be a relation from a set A to set B. The domain of R comprises all the first elements of the ordered pairs in R.

The Range of R is the set of all second elements of the ordered pairs.

Thus, Domain = {a: (a, b) ∈ R}

and Range = {b: (a, b) ∈ R}

It is clear from the above definition that the domain of a relation from set A to B is a subset of A, and its Range is a subset of B.

1.0Types of Relations

Different types of relations are:

Universal Relations
Void Relations
Identity Relations
Reflexive Relations
Symmetric Relations
Transitive Relations
Equivalence Relations

Let us discuss all types of relations one by one.

Universal Relations

A relation R on set A is termed a Universal Relation if every element of A is related to every other element within set A.

i.e., R = A × A.

Void Relations

A relation R on set A is called Void/ Empty Relation if no element of A is related to any element of A.

i.e., $R \subseteq A \times B .$

Identity Relations

A relation defined on a set A is called an Identity Relation if each element of A is related to only and only itself.

i.e., I = {(a, a): ∀ a ∈ A}.

Reflexive Relations

A relation defined on a set A is called Reflexive Relation if each element of A is related to itself.

I.e., (a, a) ∈ R ∀ a ∈ A.

Symmetric Relations

A relation defined on a set A is said to be Symmetric Relation if

(a, b) ∈ R ⇒ (b, a) ∈ R for all a, b ∈ A.

i.e., a R b ⇒ b R a ∀ a, b, ∈ A.

Transitive Relations

Let A be any set. A relation R on set A is said to be a Transitive Relation if

(a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R for all a, b, c ∈ A.

i.e., a R b, b R c ⇒ a R c ∀ a, b, c ∈ A.

Equivalence Relations

A relation R on a set A is said to be an Equivalence Relation if:

It is Reflexive Relation
It is Symmetric Relation
It is Transitive Relation

2.0Function

A function f from set A to set B is defined as a relation where each element in A maps to exactly one element in B. This means that for every input from A, B has a unique output. We can represent this function as f: A → B. If (a, b) ∈ f, then f(a) = b, where b represents the image of a under f, while a is referred to as the preimage of b under f.

3.0Types of Function

Identity Function
Constant Function
Polynomial Function
Modulus Function
Signum Function
Greatest Integer Function

Let us discuss these functions in detail

Identity Function

Consider the set R of real numbers. Define the real-valued function f: R → R as y = f(x) = x for all x ∈ R. This type of function is known as the identity function, denoted by f(x) = x. Both domain and range of f are R. Its graph is a straight line that passes through the origin.

Constant Function

The function f: R → R can be expressed as y = f(x) = c for all x ∈ R, where c is a constant. In this function, the domain of f comprises all real numbers, denoted as R, and its range is the singleton set {c}.

Polynomial Function

A polynomial function f: R → R is defined such that for each x in R, y = f(x) = a₀ + a₁x + a₂x²+ …. + a_nxⁿ, where n represents a non-negative integer, while a₀, a₁, a₂ , … , a_n are real numbers.

Modulus Function

The modulus function f: R → R, defined as f(x) = |x| ∀ x ∈ R, is called the absolute value function. When x is non-negative, f(x) equals x, but when x is negative, f(x) becomes the negative of x, i.e., f(x)= ${x, x \geq 0 - x, x < 0$

Signum Function

The function f: R → R is defined by

$f (x) = ⎩ ⎨ ⎧ 1, if x > 0 0, if x = 0 - 1, if x < 0$

Is called the signum function.

Greatest Integer Function

The function f: R → R defined as f(x) = [x], where x ∈ R, represents the greatest integer function. Here, the notation [x] denotes the greatest integer less than or equal to x.

$[x] = ⎩ ⎨ ⎧ - 1 for 0 for 1 for 2 for - 1 \leq x < 0 0 \leq x < 1 1 \leq x < 2 2 \leq x < 3 \dots\dots .$

and so on.

4.0Difference Between Relation and Function

Differentiating /Parameter	Relation	Function
Definition	A relation is a connection between sets of values, often represented as a subset of the Cartesian product.	A function is a relation in which each input is associated with exactly one output.
Denotation	The symbol "R" is used to denote a relation.	A function is represented by the symbols "F" or "f".
Example	R = {(p, 1), (p, 2), (q, 1)}	F = {(p, 1), (q, 2)}
Note	Not every relation is a function.	Every function is a type of relation.

5.0Relation and Function Formulas (R: A → B)

Relation (R): R ⊆ A × B
Domain (D): D = { x ∣ (x, y) ∈ R}
Codomain (C): C = B
Range (R): R = { y ∣ ∃ x ∈ D, (x, y) ∈ R}
Cartesian Product (A × B): A × B = {(a, b) ∣ a ∈ A and b ∈ B}
Vertical Line Test: The vertical line test confirms that a relation is a function if each vertical line crosses the graph at most once.
One-to-One Function: f(x₁) = f(x₂) ⇒ x₁ = x₂ only
Onto Function (Surjective): Every element in the codomain has at least one preimage.
Bijective: Both one-to-one and onto.
Composition of Functions (f o g): (fog)(x) = f( g(x))

6.0Relations and Functions Examples

Question 1. In the set N × N consider the relation R defined as (a, b) R (c, d) ⇔ ad = bc; a, b, c, d ∈ N

Show that this is an equivalence relation.

Solution:

For proving the relation R on N × N to be an equivalence relation, we will prove that the relation R is reflexive, symmetric and transitive.

Reflexive:

Let x, y ∈ N, then

⇒ (x, y) ∈ N × N ⇒ xy = yx, by commutativity in N

⇒ (x, y) R (x, y) ∀ (x, y) ∈ N × N

∴ R is reflexive.

Symmetric:

Let (x, y) R (p, q) where x, y, p, q ∈ N, then

(x, y) R (p, q) ⇒ xq = yp ⇒ yp = xq

⇒ py = qx [by commutativity in N]

⇒ (p, q) R (x, y)

∴R is symmetric.

Transitive:

Let (x₁, y₁) R (x₂, y₂) and (x₂, y₂) R (x₃, y₃) where

⇒ (x₁, y₁) (x₂, y₂), (x₃, y₃) ∈ N × N, then

⇒ (x₁, y₁) R (x₂, y₂) ⇒ x₁ y₂ = y₁ x₂ ...(1)

∴ (x₂, y₂) R (x₃, y₃) ⇒ x₂ y₃ = y₂ x₃ ...(2)

putting the value of x₂ from (2) into (1)

⇒ x₁ y₃ = y₁ x₃ ...(3)

⇒ (x₁, y₁) R (x₃, y₃)

Therefore, R is transitive.

Hence the relation defined on N × N is an equivalence relation.

Question 2. Find the domain of function y=\sqrt{5-2 x} .

Solution:

5 – 2x > 0 ⇒ x \leq \frac{5}{2}

∴ Domain is (–∞, 5/2]

Question 3. Find the range of function f(x) = 3 – cos x.

Solution:

f(x) = 3 – cos x

⇒ –1 ≤ cos x ≤ 1

⇒ 2 ≤ 3 – cos x ≤ 4

∴ Range of f(x) = [2, 4]

Question 4. Find the domain $f (x) = \frac{1}{∣ [ ∣ x ∣ - 5 ] ∣ - 11}$ where [.] denotes greatest integer function.

Solution:

[|x|-5]|>11

So, [|x|-5]>11 or [|x|-5]<-11

$\Rightarrow [∣ x ∣] > 16$ [|x|]<-6

$\Rightarrow ∣ x ∣ \geq 17$ |x|<-6 (Not Possible)

⇒ x ≤ -17 or x ≥ 17

So, x ∈ (–∞, –17] ∪ [17, ∞)

Also Read

Asymptotes	Sets, Relation and Function Previous Year Questions with Solutions	Implicit Functions
Composite Functions	One to One Function	Exponential Functions
Bijective Function	Algebraic Function	Logarithmic Functions

1.0Learning Outcomes

After completing these revision notes, you will be able to:

Differentiate between various types of relations.
Identify reflexive, symmetric, transitive, and equivalence relations.
Classify functions as one-one, onto, and bijective.
Evaluate composite functions and determine inverse functions.
Solve problems involving binary operations.
Understand the algebra of functions and identity functions.
Apply concepts of relations and functions to JEE Main, JEE Advanced, and CBSE Board-level questions.
Build a strong foundation for higher Mathematics and Calculus.

2.05-Minute Quick Revision

Must-Revise Concepts

Relation
Function
Reflexive Relation
Symmetric Relation
Transitive Relation
Equivalence Relation
One-One Function
Onto Function
Bijective Function
Composite Function
Inverse Function
Binary Operation

Important Formula Recall

Composite Function: $(f \circ g) (x) = f (g (x))$
Identity Function: $I_{A} (x) = x$
Inverse Function $f^{- 1} (f (x)) = x f (f^{- 1} (x)) = x$
Condition for Invertibility: A function must be both one-one and onto (bijective) to possess an inverse.

Important Properties

Every equivalence relation is reflexive, symmetric, and transitive.
Every inverse function exists only for a bijective function.
Composition of functions is generally not commutative.
Identity function acts as the neutral element under composition.
Binary operations must satisfy the closure property on the given set.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Types of Relations	High	Easy–Moderate
Equivalence Relations	High	Moderate
One-One & Onto Functions	Very High	Moderate
Composite Functions	Very High	Moderate
Inverse Functions	Very High	Moderate
Binary Operations	High	Moderate

4.0Formula Handbook

The following formulas and standard results are essential for solving questions from Relations and Functions in JEE Main, JEE Advanced, and CBSE examinations.

5.0Common Mistakes & JEE Tips

Relations and Functions is a concept-driven chapter. Most mistakes occur due to confusion between different types of relations and functions or incorrect verification of conditions.

Common Mistake	Correct Approach
Confusing one-one and onto functions	Check injectivity and surjectivity separately before concluding.
Assuming every function has an inverse	Verify that the function is bijective first.
Ignoring the order while composing functions	Compute (g(x)) first, then apply (f).
Assuming (f\circ g=g\circ f)	Remember that function composition is generally not commutative.
Incorrectly identifying equivalence relations	Verify reflexive, symmetric, and transitive properties individually.
Forgetting to check closure in binary operations	Ensure the result always belongs to the given set.
Confusing domain, codomain, and range	Write each set clearly before solving questions.
Skipping verification after finding the inverse	Confirm using (f(f^{-1}(x))=x) and (f^{-1}(f(x))=x).

6.0ALLEN Faculty Tips

Master the definitions before attempting numerical problems.
Draw mapping diagrams to visualize different types of functions.
Learn standard examples of one-one, onto, bijective, and many-one functions.
Verify all three conditions while checking equivalence relations.
Practice inverse function questions from both algebraic and graphical perspectives.
Revise important properties of binary operations regularly.
Solve Previous Years' Questions (PYQs) to identify common question patterns.

7.0PYQ Trend Analysis

Relations and Functions is a foundational chapter for JEE Main and JEE Advanced. Questions are frequently asked on inverse functions, composite functions, one-one and onto mappings, and equivalence relations.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Types of Relations	1–2	Easy
Equivalence Relations	1–2	Moderate
One-One & Onto Functions	2–3	Moderate
Composite Functions	2–3	Moderate
Inverse Functions	2–3	Moderate
Binary Operations	1–2	Moderate

8.0Most Common Question Types

Identifying reflexive, symmetric, transitive, and equivalence relations.
Classifying functions as one-one, onto, bijective, or many-one.
Finding the composition of two or more functions.
Determining the inverse of a given function.
Verifying whether an inverse exists.
Solving problems involving binary operations.
Finding the domain and range of composite functions.
Mixed conceptual questions based on properties of relations and functions.

9.0Smart Revision Strategy

Follow this revision sequence for effective preparation:

Revise the definitions of relations and different types of functions.
Learn the properties of reflexive, symmetric, transitive, and equivalence relations.
Practice identifying one-one, onto, and bijective functions.
Solve questions on composite and inverse functions.
Revise binary operations and their properties.
Attempt mixed conceptual problems involving multiple function types.
Practice Previous Years' Questions (PYQs).
Finish with a timed chapter test to improve speed and accuracy.

Class 12 Maths Chapters List:- JEE & NEET Revision Notes

Chapter 1 Relations and Functions

Chapter 2 Inverse Trigonometric Functions

Chapter 3 Matrices

Chapter 4 Determinants

Chapter 5 Continuity and Differentiability

Chapter 6 Application of Derivatives

Chapter 7 Integrals

Chapter 8 Application of Integrals

Chapter 9 Differential Equations

Chapter 10 Vector Algebra

Chapter 11 Three Dimensional Geometry

Chapter 12 Linear Programming

Chapter 13 Probability

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I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Class 12 Maths Chapter 1: Relations and Functions

From assigning students to roll numbers and mapping cities to their pin codes to defining mathematical models used in engineering and computer science, Relations and Functions help establish meaningful connections between different quantities. This chapter extends the concepts learned in Class 11 by introducing types of relations, equivalence relations, composite functions, inverse functions, and binary operations. These concepts serve as the building blocks for advanced topics such as Calculus, Matrices, Vector Algebra, and Linear Transformations, making this chapter highly important for JEE Main, JEE Advanced, and CBSE Board examinations.

At ALLEN, our expert faculty have prepared these Class 12 Maths Chapter 1: Relations and Functions smart study material to help students revise the chapter efficiently with complete conceptual clarity. Class 12 Maths Chapter 1 notes include concise theory, important definitions, standard formulas, shortcut techniques, solved illustrations, common exam traps, and PYQ-based insights. Whether you're preparing for CBSE Board exams or competitive exams like JEE Main and JEE Advanced, these revision notes will help you strengthen your fundamentals and solve questions with confidence.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate to High
Revision Time	20–25 Minutes
Key Concepts Covered	Types of relations and functions, algebra of functions, binary operations

Important topics/subtopics covered:-

Topic	Overview
Types of Relations	Reflexive, Symmetric, Transitive and Equivalence Relations
Types of Functions	One-One, Onto, Bijective Functions
Composite Functions	Composition of two or more functions
Inverse Functions	Finding and verifying inverse functions
Binary Operations	Definition and properties
Identity Function	Concept and applications
Algebra of Functions	Addition, subtraction, multiplication and division of functions
Applications	Foundation for Calculus and Higher Mathematics

2.0Chapter 1 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Class 12 Maths Chapter 1 NCERT Solutions	Available Soon
Relations and Functions Important Questions	Available Soon
Formula Sheet PDF	Available Soon
JEE Main & Advanced Previous Year Questions	Available Soon
Chapter-wise Mock Test	Available Soon
Practice Worksheets	Available Soon
Quick Revision Notes PDF	Available Soon
Mind Maps	Available Soon

Relations and Functions

Relation:

The relation is a linear operation that establishes a relationship between the elements of two sets according to some definite rule of relationship.

R: {(a, b)| (a, b) ∈ A × B and a R b}

Function:

Let A and B be two sets, and let a rule or manner or correspondence ‘f’ exist that associates each element of A with a unique element in B. Then, f is called a mapping or function from A to B.

f: A → B or $A f B$

Which reads f maps A to B or f is a function from A to B.

Related Video:

1.0Definition of Relation

A Relation R from set A to set B is established as a subset of the Cartesian product A × B. In other words, relation R consists of ordered pairs where the first element belongs to set A and the second element belongs to set B. Therefore, we can say that R is a relation from A to B, represented as $R \subseteq A \times B$ .

Total number of relations if n(A) = m and n(B) = n is 2^mn.

2.0Domain and Range of a Relation

Let R be a relation from a set A to set B. The domain of R comprises all the first elements of the ordered pairs in R.

The Range of R is the set of all second elements of the ordered pairs.

Thus, Domain = {a: (a, b) ∈ R}

and Range = {b: (a, b) ∈ R}

It is clear from the above definition that the domain of a relation from set A to B is a subset of A, and its Range is a subset of B.

1.0Types of Relations

Different types of relations are:

Universal Relations
Void Relations
Identity Relations
Reflexive Relations
Symmetric Relations
Transitive Relations
Equivalence Relations

Let us discuss all types of relations one by one.

Universal Relations

A relation R on set A is termed a Universal Relation if every element of A is related to every other element within set A.

i.e., R = A × A.

Void Relations

A relation R on set A is called Void/ Empty Relation if no element of A is related to any element of A.

i.e., $R \subseteq A \times B .$

Identity Relations

A relation defined on a set A is called an Identity Relation if each element of A is related to only and only itself.

i.e., I = {(a, a): ∀ a ∈ A}.

Reflexive Relations

A relation defined on a set A is called Reflexive Relation if each element of A is related to itself.

I.e., (a, a) ∈ R ∀ a ∈ A.

Symmetric Relations

A relation defined on a set A is said to be Symmetric Relation if

(a, b) ∈ R ⇒ (b, a) ∈ R for all a, b ∈ A.

i.e., a R b ⇒ b R a ∀ a, b, ∈ A.

Transitive Relations

Let A be any set. A relation R on set A is said to be a Transitive Relation if

(a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R for all a, b, c ∈ A.

i.e., a R b, b R c ⇒ a R c ∀ a, b, c ∈ A.

Equivalence Relations

A relation R on a set A is said to be an Equivalence Relation if:

It is Reflexive Relation
It is Symmetric Relation
It is Transitive Relation

2.0Function

A function f from set A to set B is defined as a relation where each element in A maps to exactly one element in B. This means that for every input from A, B has a unique output. We can represent this function as f: A → B. If (a, b) ∈ f, then f(a) = b, where b represents the image of a under f, while a is referred to as the preimage of b under f.

3.0Types of Function

Identity Function
Constant Function
Polynomial Function
Modulus Function
Signum Function
Greatest Integer Function

Let us discuss these functions in detail

Identity Function

Consider the set R of real numbers. Define the real-valued function f: R → R as y = f(x) = x for all x ∈ R. This type of function is known as the identity function, denoted by f(x) = x. Both domain and range of f are R. Its graph is a straight line that passes through the origin.

Constant Function

The function f: R → R can be expressed as y = f(x) = c for all x ∈ R, where c is a constant. In this function, the domain of f comprises all real numbers, denoted as R, and its range is the singleton set {c}.

Polynomial Function

A polynomial function f: R → R is defined such that for each x in R, y = f(x) = a₀ + a₁x + a₂x²+ …. + a_nxⁿ, where n represents a non-negative integer, while a₀, a₁, a₂ , … , a_n are real numbers.

Modulus Function

The modulus function f: R → R, defined as f(x) = |x| ∀ x ∈ R, is called the absolute value function. When x is non-negative, f(x) equals x, but when x is negative, f(x) becomes the negative of x, i.e., f(x)= ${x, x \geq 0 - x, x < 0$

Signum Function

The function f: R → R is defined by

$f (x) = ⎩ ⎨ ⎧ 1, if x > 0 0, if x = 0 - 1, if x < 0$

Is called the signum function.

Greatest Integer Function

The function f: R → R defined as f(x) = [x], where x ∈ R, represents the greatest integer function. Here, the notation [x] denotes the greatest integer less than or equal to x.

$[x] = ⎩ ⎨ ⎧ - 1 for 0 for 1 for 2 for - 1 \leq x < 0 0 \leq x < 1 1 \leq x < 2 2 \leq x < 3 \dots\dots .$

and so on.

4.0Difference Between Relation and Function

Differentiating /Parameter	Relation	Function
Definition	A relation is a connection between sets of values, often represented as a subset of the Cartesian product.	A function is a relation in which each input is associated with exactly one output.
Denotation	The symbol "R" is used to denote a relation.	A function is represented by the symbols "F" or "f".
Example	R = {(p, 1), (p, 2), (q, 1)}	F = {(p, 1), (q, 2)}
Note	Not every relation is a function.	Every function is a type of relation.

5.0Relation and Function Formulas (R: A → B)

Relation (R): R ⊆ A × B
Domain (D): D = { x ∣ (x, y) ∈ R}
Codomain (C): C = B
Range (R): R = { y ∣ ∃ x ∈ D, (x, y) ∈ R}
Cartesian Product (A × B): A × B = {(a, b) ∣ a ∈ A and b ∈ B}
Vertical Line Test: The vertical line test confirms that a relation is a function if each vertical line crosses the graph at most once.
One-to-One Function: f(x₁) = f(x₂) ⇒ x₁ = x₂ only
Onto Function (Surjective): Every element in the codomain has at least one preimage.
Bijective: Both one-to-one and onto.
Composition of Functions (f o g): (fog)(x) = f( g(x))

6.0Relations and Functions Examples

Question 1. In the set N × N consider the relation R defined as (a, b) R (c, d) ⇔ ad = bc; a, b, c, d ∈ N

Show that this is an equivalence relation.

Solution:

For proving the relation R on N × N to be an equivalence relation, we will prove that the relation R is reflexive, symmetric and transitive.

Reflexive:

Let x, y ∈ N, then

⇒ (x, y) ∈ N × N ⇒ xy = yx, by commutativity in N

⇒ (x, y) R (x, y) ∀ (x, y) ∈ N × N

∴ R is reflexive.

Symmetric:

Let (x, y) R (p, q) where x, y, p, q ∈ N, then

(x, y) R (p, q) ⇒ xq = yp ⇒ yp = xq

⇒ py = qx [by commutativity in N]

⇒ (p, q) R (x, y)

∴R is symmetric.

Transitive:

Let (x₁, y₁) R (x₂, y₂) and (x₂, y₂) R (x₃, y₃) where

⇒ (x₁, y₁) (x₂, y₂), (x₃, y₃) ∈ N × N, then

⇒ (x₁, y₁) R (x₂, y₂) ⇒ x₁ y₂ = y₁ x₂ ...(1)

∴ (x₂, y₂) R (x₃, y₃) ⇒ x₂ y₃ = y₂ x₃ ...(2)

putting the value of x₂ from (2) into (1)

⇒ x₁ y₃ = y₁ x₃ ...(3)

⇒ (x₁, y₁) R (x₃, y₃)

Therefore, R is transitive.

Hence the relation defined on N × N is an equivalence relation.

Question 2. Find the domain of function y=\sqrt{5-2 x} .

Solution:

5 – 2x > 0 ⇒ x \leq \frac{5}{2}

∴ Domain is (–∞, 5/2]

Question 3. Find the range of function f(x) = 3 – cos x.

Solution:

f(x) = 3 – cos x

⇒ –1 ≤ cos x ≤ 1

⇒ 2 ≤ 3 – cos x ≤ 4

∴ Range of f(x) = [2, 4]

Question 4. Find the domain $f (x) = \frac{1}{∣ [ ∣ x ∣ - 5 ] ∣ - 11}$ where [.] denotes greatest integer function.

Solution:

[|x|-5]|>11

So, [|x|-5]>11 or [|x|-5]<-11

$\Rightarrow [∣ x ∣] > 16$ [|x|]<-6

$\Rightarrow ∣ x ∣ \geq 17$ |x|<-6 (Not Possible)

⇒ x ≤ -17 or x ≥ 17

So, x ∈ (–∞, –17] ∪ [17, ∞)

Also Read

Asymptotes	Sets, Relation and Function Previous Year Questions with Solutions	Implicit Functions
Composite Functions	One to One Function	Exponential Functions
Bijective Function	Algebraic Function	Logarithmic Functions

1.0Learning Outcomes

After completing these revision notes, you will be able to:

Differentiate between various types of relations.
Identify reflexive, symmetric, transitive, and equivalence relations.
Classify functions as one-one, onto, and bijective.
Evaluate composite functions and determine inverse functions.
Solve problems involving binary operations.
Understand the algebra of functions and identity functions.
Apply concepts of relations and functions to JEE Main, JEE Advanced, and CBSE Board-level questions.
Build a strong foundation for higher Mathematics and Calculus.

2.05-Minute Quick Revision

Must-Revise Concepts

Relation
Function
Reflexive Relation
Symmetric Relation
Transitive Relation
Equivalence Relation
One-One Function
Onto Function
Bijective Function
Composite Function
Inverse Function
Binary Operation

Important Formula Recall

Composite Function: $(f \circ g) (x) = f (g (x))$
Identity Function: $I_{A} (x) = x$
Inverse Function $f^{- 1} (f (x)) = x f (f^{- 1} (x)) = x$
Condition for Invertibility: A function must be both one-one and onto (bijective) to possess an inverse.

Important Properties

Every equivalence relation is reflexive, symmetric, and transitive.
Every inverse function exists only for a bijective function.
Composition of functions is generally not commutative.
Identity function acts as the neutral element under composition.
Binary operations must satisfy the closure property on the given set.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Types of Relations	High	Easy–Moderate
Equivalence Relations	High	Moderate
One-One & Onto Functions	Very High	Moderate
Composite Functions	Very High	Moderate
Inverse Functions	Very High	Moderate
Binary Operations	High	Moderate

4.0Formula Handbook

The following formulas and standard results are essential for solving questions from Relations and Functions in JEE Main, JEE Advanced, and CBSE examinations.

5.0Common Mistakes & JEE Tips

Relations and Functions is a concept-driven chapter. Most mistakes occur due to confusion between different types of relations and functions or incorrect verification of conditions.

Common Mistake	Correct Approach
Confusing one-one and onto functions	Check injectivity and surjectivity separately before concluding.
Assuming every function has an inverse	Verify that the function is bijective first.
Ignoring the order while composing functions	Compute (g(x)) first, then apply (f).
Assuming (f\circ g=g\circ f)	Remember that function composition is generally not commutative.
Incorrectly identifying equivalence relations	Verify reflexive, symmetric, and transitive properties individually.
Forgetting to check closure in binary operations	Ensure the result always belongs to the given set.
Confusing domain, codomain, and range	Write each set clearly before solving questions.
Skipping verification after finding the inverse	Confirm using (f(f^{-1}(x))=x) and (f^{-1}(f(x))=x).

6.0ALLEN Faculty Tips

Master the definitions before attempting numerical problems.
Draw mapping diagrams to visualize different types of functions.
Learn standard examples of one-one, onto, bijective, and many-one functions.
Verify all three conditions while checking equivalence relations.
Practice inverse function questions from both algebraic and graphical perspectives.
Revise important properties of binary operations regularly.
Solve Previous Years' Questions (PYQs) to identify common question patterns.

7.0PYQ Trend Analysis

Relations and Functions is a foundational chapter for JEE Main and JEE Advanced. Questions are frequently asked on inverse functions, composite functions, one-one and onto mappings, and equivalence relations.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Types of Relations	1–2	Easy
Equivalence Relations	1–2	Moderate
One-One & Onto Functions	2–3	Moderate
Composite Functions	2–3	Moderate
Inverse Functions	2–3	Moderate
Binary Operations	1–2	Moderate

8.0Most Common Question Types

Identifying reflexive, symmetric, transitive, and equivalence relations.
Classifying functions as one-one, onto, bijective, or many-one.
Finding the composition of two or more functions.
Determining the inverse of a given function.
Verifying whether an inverse exists.
Solving problems involving binary operations.
Finding the domain and range of composite functions.
Mixed conceptual questions based on properties of relations and functions.

9.0Smart Revision Strategy

Follow this revision sequence for effective preparation:

Revise the definitions of relations and different types of functions.
Learn the properties of reflexive, symmetric, transitive, and equivalence relations.
Practice identifying one-one, onto, and bijective functions.
Solve questions on composite and inverse functions.
Revise binary operations and their properties.
Attempt mixed conceptual problems involving multiple function types.
Practice Previous Years' Questions (PYQs).
Finish with a timed chapter test to improve speed and accuracy.

Frequently Asked Questions

Why is Relations and Functions important for JEE preparation?

Which topic from this chapter is asked most frequently in JEE Main?

What is the best strategy to revise Relations and Functions before the exam?

How can I quickly identify whether a function is one-one or onto?

Which chapters should I study after completing Relations and Functions?

Join ALLEN!

Class 12 Maths Chapter 1: Relations and Functions

1.0Chapter Snapshot

2.0Chapter 1 Smart Study Material by ALLEN Experts: Download PDFs

On this page

Relations and Functions

1.0Definition of Relation

2.0Domain and Range of a Relation

On this page

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

On this page

Frequently Asked Questions

Why is Relations and Functions important for JEE preparation?

Which topic from this chapter is asked most frequently in JEE Main?

What is the best strategy to revise Relations and Functions before the exam?

How can I quickly identify whether a function is one-one or onto?

Which chapters should I study after completing Relations and Functions?

Join ALLEN!

Class 12 Maths Chapter 1: Relations and Functions

1.0Chapter Snapshot

2.0Chapter 1 Smart Study Material by ALLEN Experts: Download PDFs

On this page

Relations and Functions

1.0Definition of Relation

2.0Domain and Range of a Relation

On this page

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

On this page