Class 12 Mathematics Chapter 1: Relations and Functions

CBSE 2026–27 | NCERT-aligned free resource

1. What is a Relation?

A relation from set A to set B is a subset of A × B. It connects elements of one set with elements of another. Relations can be represented using ordered pairs, arrow diagrams and matrices.

A: {1,2,3} → B: {a,b}
1 → a 2 → a,b 3 → b
Relation = collection of selected ordered pairs

2. Types of Relations

Reflexive: every a∈A satisfies (a,a)∈R. Symmetric: (a,b)∈R implies (b,a)∈R. Transitive: (a,b) and (b,c) imply (a,c). A relation that is reflexive, symmetric and transitive is an equivalence relation.

3. Functions

A function f:A→B assigns every element of A exactly one image in B. The domain is A, the codomain is B, and the range is the set of actual images.

4. One-One and Onto

A function is one-one if distinct inputs have distinct outputs. It is onto if every element of the codomain is an image of at least one domain element. A function that is both is bijective.

One-one: no two arrows end at the same output
Onto: every codomain element receives ≥1 arrow
Bijective: both conditions

Worked Example

For f(x)=2x+3 on R→R, if f(a)=f(b), then 2a+3=2b+3, so a=b. Thus it is one-one. For any y∈R, x=(y−3)/2 is real, so it is onto. Hence f is bijective.

Common Mistakes

Range and codomain are not always the same.
A function must assign exactly one output to every input.
Check all three conditions before calling a relation an equivalence relation.

Practice

Test a relation for reflexivity, symmetry and transitivity.
Determine whether a given mapping is a function.
Classify a function as one-one, onto or bijective.
Construct an equivalence relation on a finite set.

Deep-Dive Revision & Worked Examples

Composition of Functions — Worked Example

Let f(x)=2x+1 and g(x)=x². Then (f∘g)(x)=f(x²)=2x²+1, whereas (g∘f)(x)=(2x+1)². Therefore composition is not generally commutative.

One-One Test

To prove f is one-one, assume f(a)=f(b) and show a=b. For a function defined on a real interval, a strictly increasing or strictly decreasing function is one-one.

Inverse Function

A function has an inverse function only when it is one-one on its domain and the inverse is defined on its range. A reliable method is to write y=f(x), interchange x and y, then solve for y.

Visual Concept Map

Function f: A → B
↓
one-one → distinct inputs have distinct outputs
onto → every element of B has a pre-image
bijective → one-one + onto
↓
bijective function ↔ inverse function

Exam-Level Practice

Determine whether a given relation is a function and justify your answer.
Test whether a function is one-one using f(a)=f(b).
Find the composition of two functions in both orders.
Find an inverse and verify both compositions.
Construct a real-life example of an onto but not one-one mapping.

CBSE Class 12 Mathematics – Chapter 1: Relations and Functions