CBSE · Class 12 · Mathematics · Chapter 1

Relations and
Functions

Complete chapter resources for CBSE Class 12 Maths — topic breakdown, key concepts, sample questions, previous year board questions, and instant AI question paper generation.

4Topics
6–8Board marks
8Sample questions
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Key Concepts — Chapter 1
  • Equivalence relation: Reflexive + Symmetric + Transitive
  • One-one (injective): f(x₁) = f(x₂) ⟹ x₁ = x₂
  • Onto (surjective): Range = Codomain
  • Bijective: One-one AND onto
  • Composition: (gof)(x) = g(f(x))
  • Inverse: f⁻¹ exists iff f is bijective
Chapter Overview

What this chapter covers

Chapter 1 of CBSE Class 12 Mathematics extends the concept of relations and functions from Class 11. A relation R on a set A is a subset of A × A, and the chapter classifies relations as reflexive, symmetric, transitive, and the important combined category — equivalence relations. Every equivalence relation partitions its set into disjoint equivalence classes, a concept that underpins abstract algebra and is frequently tested in board exams.

The chapter then deepens the study of functions by introducing the formal classifications: injective (one-one), surjective (onto), and bijective functions. Proving these properties algebraically — by assuming f(x₁) = f(x₂) for injectivity, or constructing a pre-image for surjectivity — is the core skill tested. The chapter also introduces composition of functions (fog and gof) and establishes that the composition of two bijective functions is itself bijective.

The final topic is the invertible function: a function has an inverse if and only if it is bijective. Board questions on this chapter range from 1-mark MCQs identifying function types to 5-mark proofs requiring students to establish all three properties of an equivalence relation or show a function is bijective and find its inverse. This chapter provides the logical foundation for calculus, matrix theory, and abstract algebra studied later in Class 12.

Topics

What's inside Chapter 1

As per NCERT Class 12 Mathematics (CBSE syllabus)

Topic 1
Types of Relations
Empty, universal, and identity relations. Reflexive, symmetric, and transitive relations. Equivalence relations and equivalence classes with examples on sets of integers and geometric figures.
Topic 2
Types of Functions
Injective (one-one) functions: distinct inputs map to distinct outputs. Surjective (onto) functions: range equals codomain. Bijective functions: both one-one and onto. Methods for algebraic proof.
Topic 3
Composition of Functions
Definition of fog and gof, showing gof ≠ fog in general. Properties: associativity of composition, composition of two injective/surjective/bijective functions. Practical computation questions.
Topic 4
Invertible Functions
Conditions for invertibility: f is invertible iff f is bijective. Finding f⁻¹ explicitly by solving y = f(x) for x. Verifying fof⁻¹ = f⁻¹of = Identity. Binary operations — commutativity and associativity.
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Class 11 · Relations and Functions
Domain, codomain, range; graphs of standard functions
Class 11 · Sets
Cartesian products, subsets — basis of relation definition
Chapter 1 Relations &
Functions
Leads to
Ch 2 · Inverse Trigonometric Functions
Inverses exist only when trig functions are restricted to bijective domains
Ch 3 · Matrices & Ch 4 · Determinants
Matrix operations as functions; linear maps and bijections
Board Blueprint

Marks & question-type breakdown

Typical pattern based on CBSE Class 12 board papers from the last five years.

Question type Marks Typical count What's usually tested
MCQ / Assertion-Reason 1 1–2 Identifying function type, equivalence relation property, or composition value
Very Short Answer 2 1 Check if a given relation is reflexive/symmetric/transitive; compute fog or gof
Short Answer 3 1 Prove a function is one-one or onto; find inverse of a bijective function
Long Answer / Proof 5 1 Show a relation is an equivalence relation; prove bijection and find f⁻¹
Total (approximate) 6–8 4–5 Weightage varies across paper sets and years
Sample Questions

8 sample questions — generated by MarksZen AI

Aligned to CBSE Class 12 Maths Chapter 1. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
The relation R = {(a, b) : a ≤ b} defined on the set of real numbers is: (a) Reflexive and transitive but not symmetric (b) Symmetric and transitive but not reflexive (c) An equivalence relation (d) Neither reflexive nor symmetric
Q2 Easy 1 mark MCQ
Let f : R → R be defined by f(x) = 3x + 5. Then f is: (a) One-one but not onto (b) Onto but not one-one (c) Neither one-one nor onto (d) Bijective
Q3 Easy 2 marks Short Answer
If f(x) = 2x + 3 and g(x) = x² − 1, find (fog)(2) and (gof)(2).
Q4 Medium 2 marks Short Answer
Check whether the relation R defined on the set A = {1, 2, 3} by R = {(1,1), (2,2), (3,3), (1,2), (2,1)} is an equivalence relation.
Q5 Medium 3 marks Short Answer
Show that the function f : R → R defined by f(x) = (2x − 1) / 3 is bijective. Also find f⁻¹.
Q6 Medium 3 marks Short Answer
Prove that the function f : N → N defined by f(n) = n² is injective but not surjective. Justify your answer with appropriate reasoning.
Q7 Hard 5 marks Long Answer
Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that the relation R = {(a, b) : |a − b| is divisible by 4} is an equivalence relation. Also write the equivalence class of 1.
Q8 Hard 5 marks Long Answer
Consider the function f : R \ {−4/3} → R defined by f(x) = (4x + 3) / (3x + 4). (i) Show that f is one-one. (ii) Show that f is onto. (iii) Find f⁻¹ and verify that fof⁻¹ = Identity.
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Previous Year Questions

From CBSE board examinations

Actual questions from past Class 12 Maths board papers — Relations and Functions chapter.

Board 20235 marks
Let A = R − {3} and B = R − {1}. Consider the function f : A → B defined by f(x) = (x − 2)/(x − 3). Show that f is one-one and onto, and hence find f⁻¹. (All India 2023)
Board 20223 marks
Check whether the relation R in the set Z of integers defined as R = {(a, b) : a + b is even} is reflexive, symmetric and transitive. Is R an equivalence relation? (Delhi 2022)
Board 20202 marks
If f : R → R is defined by f(x) = x² − 3x + 2, find f(f(x)). (CBSE 2020)
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FAQ

Questions teachers ask

How many marks does Relations and Functions carry in the CBSE Class 12 board exam? +
Relations and Functions typically carries 6–8 marks in the CBSE Class 12 Mathematics board exam. Questions appear as 1-mark MCQs on identifying function types, 2-mark short answers on proving injectivity or surjectivity, and 3–5 mark questions on composition of functions or finding inverses. This chapter forms part of the Algebra unit which carries 13 marks overall.
What is the difference between injective, surjective, and bijective functions? +
An injective (one-one) function maps distinct elements of the domain to distinct elements of the codomain — no two inputs share the same output. A surjective (onto) function has every element of the codomain as the image of at least one domain element. A bijective function is both injective and surjective simultaneously; only bijective functions have a proper inverse function.
How do I prove a function is one-one or onto in a board exam? +
To prove one-one: assume f(x₁) = f(x₂) and show algebraically that x₁ = x₂. To prove onto: take an arbitrary y in the codomain, solve for x in terms of y, show x exists in the domain, and verify f(x) = y. Board marks are awarded for each logical step, so show all working explicitly rather than just stating conclusions.
What are equivalence relations and how often do they appear in board papers? +
An equivalence relation on a set A must be reflexive (aRa for all a), symmetric (aRb implies bRa), and transitive (aRb and bRc implies aRc). Board papers almost always include a 3–5 mark question asking students to verify all three properties for a given relation. Checking each property step-by-step with examples is the expected approach.
How do I generate a custom question paper for Relations and Functions using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 12 Mathematics, select Chapter 1 (Relations and Functions), set your preferred question-type mix (MCQ, short answer, long answer) and total marks — the AI generates a complete board-aligned paper with answer key in under 2 minutes, ready for PDF export.