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Short Intro

In this post, students can find complete step-by-step solutions for Class 12 Maths Chapter 1 Exercise 1.2 – Relations and Functions based on the latest NCERT syllabus. This exercise covers one-one functions, onto functions, bijective functions, injective and surjective mappings, modulus function, signum function, and greatest integer function in an easy and exam-oriented way.

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Quick Information Box

Item	Details
Board	NCERT / CBSE
Class	12
Subject	Mathematics
Chapter	Relations and Functions
Exercise	1.2
Main Topics	One-One & Onto Functions

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Concepts Used (Topics Covered)

• One-One (Injective) Function
• Onto (Surjective) Function
• Bijective Function
• Many-One Function
• Greatest Integer Function
• Modulus Function
• Signum Function
• Domain, Co-domain & Range

The chapter explains different types of functions and mappings between sets.

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Important Formulas

One-One Function

f(x₁) = f(x₂) ⇒ x₁ = x₂

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Onto Function

For every y ∈ Y, there exists x ∈ X such that f(x) = y

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Bijective Function

A function which is both one-one and onto

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Modulus Function

$f(x)=|x|$

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Greatest Integer Function

$f(x)=[x]$

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Questions & Step-by-step Solutions

Question 1

Show that the function

$f(x)=\frac{1}{x}$

is one-one and onto on R*.

Solution

Let:

f(x₁) = f(x₂)

Then:

1/x₁ = 1/x₂

Therefore:

x₁ = x₂

Hence, the function is one-one.

Now for any:

y ∈ R*

choose:

x = 1/y

Then:

f(x) = y

Therefore, the function is onto.

Hence:

f is bijective.

The result is not true if domain is N because negative real numbers cannot be images.

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Question 2

Check injectivity and surjectivity.

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(i)

$f(x)=x^2$

for:

f : N → N

Solution

If:

x₁² = x₂²

then:

x₁ = x₂

Hence one-one.

But:

2

is not image of any natural number.

Therefore:

One-one but not onto.

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(ii)

$f(x)=x^2$

for:

f : Z → Z

Solution

Since:

f(2) = f(−2)

the function is not one-one.

Negative integers are not images.

Hence:

Neither one-one nor onto.

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(iii)

$f(x)=x^2$

for:

f : R → R

Solution

Since:

f(1) = f(−1)

not one-one.

Negative real numbers are not images.

Therefore:

Neither one-one nor onto.

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(iv)

$f(x)=x^3$

for:

f : N → N

Solution

Cube function is strictly increasing.

Hence one-one.

But numbers like:

2

are not perfect cubes.

Hence:

Not onto.

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(v)

$f(x)=x^3$

for:

f : Z → Z

Solution

Cube function is one-one.

But integers like:

2

are not cubes.

Therefore:

One-one but not onto.

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Question 3

Greatest Integer Function

$f(x)=[x]$

Solution

Since:

[1.2] = [1.8] = 1

the function is not one-one.

Also numbers like:

1.5

are not images.

Hence:

Neither one-one nor onto.

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Question 4

Modulus Function

$f(x)=|x|$

Solution

Since:

|1| = |−1|

not one-one.

Negative numbers are not images.

Hence:

Neither one-one nor onto.

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Question 5

Signum Function

$f(x)=\begin{cases}1,&x>0\\0,&x=0\\-1,&x<0\end{cases}$​

Solution

Many numbers have same image.

Therefore not one-one.

Range is:

{−1,0,1}

Hence not onto on R.

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Question 6

Function from A = {1,2,3} to B = {4,5,6,7}

Solution

Given:

f = {(1,4),(2,5),(3,6)}

Different elements have different images.

Therefore:

f is one-one.

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Question 7

State whether functions are one-one or onto.

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(i)

$f(x)=3-4x$

Solution

Linear function with non-zero slope.

Hence:

One-one and onto.

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(ii)

$f(x)=1+x^2$

Solution

Since:

f(1)=f(−1)

not one-one.

Also values less than 1 are not images.

Hence:

Neither one-one nor onto.

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Question 8

Show that

$f(a,b)=(b,a)$

is bijective.

Solution

Different ordered pairs produce different images.

Hence one-one.

Every pair in:

B × A

has pre-image.

Hence onto.

Therefore:

f is bijective.

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Question 9

Function on natural numbers

Solution

The function maps odd and even numbers uniquely.

Every natural number has pre-image.

Therefore:

The function is bijective.

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Question 10

Check whether function is one-one and onto.

$f(x)=\frac{x-2}{x-3}$

Solution

Let:

f(x₁)=f(x₂)

After simplification:

x₁=x₂

Hence one-one.

Also every element in:

B = R − {1}

has pre-image.

Therefore:

f is onto.

Hence bijective.

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Question 11

Choose correct answer for

$f(x)=x^4$

Solution

Since:

f(1)=f(−1)

not one-one.

Negative values are not images.

Correct option:

(D) Neither one-one nor onto

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Question 12

Choose correct answer for

$f(x)=3x$

Solution

Linear function with non-zero slope.

Correct option:

(A) One-one and onto

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Common Mistakes

• Confusing onto with one-one
• Forgetting co-domain
• Ignoring repeated outputs
• Wrongly assuming every linear function is onto for all domains

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Exam Tips

• Always check injective and surjective separately.
• Use counterexamples carefully.
• Remember range while checking onto.
• Practice graph interpretation.

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Practice MCQs

MCQ 1

A function which is both one-one and onto is called:

A. Constant
B. Bijective
C. Many-one
D. Identity

Answer:

B. Bijective

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MCQ 2

Function:

$f(x)=x^2$

on R is:

A. One-one
B. Onto
C. Bijective
D. Neither one-one nor onto

Answer:

D. Neither one-one nor onto

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MCQ 3

The modulus function is:

A. Onto
B. One-one
C. Bijective
D. Neither one-one nor onto

Answer:

D. Neither one-one nor onto

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FAQ Section

What is an injective function?

A function in which different inputs have different outputs.

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What is a surjective function?

A function where every element of co-domain has a pre-image.

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What is a bijective function?

A function which is both injective and surjective.

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Why is modulus function not one-one?

Because:

|1| = |−1|

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