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RD Sharma books are easily the most preferred books in the country. They are widely used by CBSE schools and a majority of teachers to set up question papers. RD Sharma's books are comprehensive, content-rich, and informative. They are a one-stop shop for CBSE . There are many solved examples for every chapter, followed by Level 1 and Level 2 questions

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**Functions Excercise: 2.4**

functions exercise 2.4 question 1 (I)

No, is many one.

A function has an inverse if it is one-one and onto.

Let us check one-one

Since all elements have an image, they don't have a unique image.

is not one-one

Since, is not one-one, it doesn't have an inverse

Functions excercise 2.4 question 1 (II)

No, is many one.

A function has an inverse if it is one-one and onto.

Let us check one-one

Since and have same image, is not one-one.

Since is not one-one, it doesn't have an inverse.

functions exercise 2.4 question 1 (iii)

Yes, is a bijection.

A function has an inverse if it is one-one and onto.

Let us check one-one

Since all element has unique image is one-one.

Check onto,

Since each and every image there is a corresponding element.

is onto.

Since the function is both one-one and onto, it will have an inverse.

Functions exercise 2.4 question 2 (I)

Answer:

Given:

Hint:

A function has an inverse if it is one-one and onto.

Solution:

Let us check one-one.

Here the different elements of the domain have different images in the co-domain.

Thus, this is one-one.

Range of

Therefore is a bijection and clearlyexists.

Hence .

Functions exercise 2.4 question 2 (ii)

doesn't exist.

Let us check one-one

The different elements of the domain have different images in the codomain.

Thus, is one-one.

This is not onto because the element in the codomain has no preimage in the domain .

Therefore, doesn't exist.

Functions exercise 2.4 question 3

, clearly and are bijections.

Here, , and are invertible.

Now,

So,

So,

Clearly, is a bijection.

So, is invertible.

we get,

Functions exercise 2.4 question 4 (0)

Clearly and are bijections.

So,

Now,

Also

So, have the same domains.

Then,

So,

From

Function Exercise 2.4 Question 5

, defined by

Injectivity of :

Let and be two elements of the domain, such that,

So, is one-one.

Subjectivity of

Let be n the codomain, .

is onto.

So, is bijection.

Let …(i)

[From (i)]

Function Exercise 2.4 Question 6

.

Bijection function should be fulfil the injectivity, surjectivity.

Let be two elements of domain, such that

So, is one-one.

Let be in the co-domain, such that .

is onto.

So, is a bijection.

… (i)

Function Exercise 2.4 Question 7

.

Bijection function should be fulfil the injectivity and surjectivity.

Let be two elements of the domain .

So is one-one.

Let be in the co-domain such that

is onto, so is bijection.

… (i)

Function Exercise 2.4 Question 8

is invertible.

If the function is invertible, .

for all

Hence, the given function is invertible and inverse of is itself

Function Exercise 2.4 Question 9

is invertible with the function.

given by

Bijection function should fulfil the injection and surjection condition.

Let be two elements of domain, such that.

is one-one.

Subjectivity of :

Let is in the co-domain such that

[Adding on both sides]

is onto.

So, is a bijection and hence, it is invertible.

… (i)

[Adding on both sides]

[From (i)]

Function Exercise 2.4 Question 10

If be defined by .

Bijection function should be fulfil the injectivity, surjectivity condition.

Let be two elements in domain, such that .

So, is one-one.

Surjectivity of .

Let be in the co-domainsuch that

is onto.

So, is a bijection and hence it’s convertible.

…(i)

From (i)

Function Exercise 2.4 Question 11

Bijection, .

is defined as .

Bijection function should be fulfil the injection and surjection condition.

Let be two elements of domain.

is one-one.

Let be in the co-domain, such that

is onto.

So, is a bijection, hence is invertible.

…(i)

[from (i

Function Exercise 2.4 Question 12

defined by .

Bijective function should be fulfil the injective and surjective function.

be two elements of domain, such

is one-one.

Let be in the co-domain, such that

.

is onto.

So, is a bijection and hence, it’s invertible

Let

Injectivity of

Let be two elements of domain.

So, is one-one.

Subjectivity of :

Let be in the co-domain, such that .

(domain)

is onto.

So, is a bijection, hence it’s invertible.

Let

…(ii)

So, [from (ii)]

.

…(iii)

Let …(vi)

.

Function Exercise 2.4 Question 13

defined by .

One-one means every domain has a distinct range. One means every image has some preimage in the domain function.

We have,.

The function defined by .

Let such that then

is one-one.

Let , then .

The function is onto if there exists such that .

Thus, for any , there exists such that

is onto.

So, is one-one and onto function.

Function Exercise 2.4 Question 14

given by .

We need to prove that

Let

Dividing by,

Let

Now,

Identity function,

Also,

So, is invertible, an identity function.

Function Exercise 2.4 Question 15

is a function defined as .

The function should be an identity function.

Let

(Let )

Now,

=y+6-6

Identity function.

Identity function

Since andare identity functions.

Thus is invertible.

So,

Now,

Function Exercise 2.4 Question 16

is given by

Injectivity,

Let be such that

is a one-one function.

Let be an arbitrary element of .

Then

As

Also, because,

Which is not possible.

Thus,

So every element in has pre image in

Hence is onto.

Now, , replacing by and by ,

Function Exercise 2.4 Question 19

Bijection function should fulfil the injection and surjection condition.

Injectivity: let and , such that

So, is an injection.

Surjectivity: let , then

is real for all

is a surjection.

is a bijection. Hence, is invertible.

Let

Squaring on both sides

or

Function Exercise 2.4 Question 20

and

or exists only if or is a bijection.

is not one-one, sinceand have the same image under.

is not a bijection. So, does not exist.

Injectivity of:

So, is one-one.

Surjectivity of:

Range of

(Codomain of)

is onto.

is a bijection. So,exists.

Finding,

Function Exercise 2.4 Question 21

is not invertible

If the function is invertible, that should fulfil the one-one, onto condition.

Let be two elements in the domain

So, we can’t say

For example,

So, and

So, and have the same image

is not one-one, bijection, surjection.

Hence, is not invertible

Function Exercise 2.4 Question 22

Bijection

Bijection should fulfil the one-one, onto condition.

Clearly, all these are bijections because they are one-one and onto

Function Exercise 2.4 Question 23

Bijective

be two sets, each with a finite number of elements. Prove that bijection.

Bijection should fulfil the surjective injective condition.

are two non-empty sets.

Let be a function from to .

It is given that there is injective map from to .

That means is one-one.

It also given that there is injective map from to .

That means is onto

is bijective.

Function Exercise 2.4 Question 24 (i)

is an injection.

Injection means every domain has a distinct range.

Let be two elements of the domain such that

[As, is one-one]

[As, is one-one]

So, is an injection.

RD Sharma Class 12th Exercise 2.4 deals with the chapter ‘Functions.’ First, you will learn about different types and properties of functions followed by examples. There are a total of 24 - level one questions in this exercise. They have relatively low complexity and can be solved by learning the fundamentals.

The following section contains 16 - level one questions and 8 - level two questions. Career360 has provided the best solutions on their website. The RD Sharma Class 12th Exercise 2.4 material has been developed with the specific needs of Class 12 students in mind. These solutions will give you a solid conceptual foundation for dealing with any queries.

The chapter has four exercises totaling more than 100 questions. For a better comprehension of our solutions, the formulae and questions have been broken down into steps. For students to cover the entire syllabus, they have to practice every question from the book. This is not possible as there are hundreds of questions per chapter. As there is a significant time constraint for students, they should refer to RD Sharma Class 12th Exercise 2.4 by Career360 to reduce their preparation time and quickly solve all the problems.

**Benefits of ****RD Sharma Solutions**** for Class 12**

Highly qualified specialists with years of experience have developed RD Sharma Class 12 Chapter 2 Exercise 2.4 - ‘Functions.'

Experts have provided these solutions in a simple and easy-to-understand manner to enable students to learn properly.

RD Sharma Class 12th Exercise 2.4 solutions are beneficial for completing homework and studying for CBSE examinations as they comply with the CBSE syllabus.

RD Sharma Class 12th Exercise 2.4 solutions also aid in the clear understanding of ideas and preparing for not only 12th boards but also upcoming competitive exams such as BITSAT, JEE, and NEET.

The questions and answers assist you in revising the material and scoring well on your tests. In addition, solved questions save a lot of time and effort, which is why students can breeze through this chapter after understanding the concepts.

The material is updated to the latest version, which means you will be up to date with the book’s standards.

The following are the fundamentals of RD Sharma Class 12 Chapter 2 Exercise 2.4

This exercise focuses on questions that are a little more advanced and complex.

Apart from that, read the chapter summary to gain a better understanding. It will assist you in quickly reviewing all of the major themes and formulas discussed in this chapter.

This chapter focuses on the different types of functions and their properties.

Class 12 RD Sharma Chapter 2 Exercise 2.4 Solution is presented clearly to allow students to complete the exercises without difficulty.

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Download E-book**Q1. How do I get the solution of RD Sharma chapter 2 exercise 2.4 for class 12?**

Downloading the Class 12 RD Sharma Chapter 2 Exercise 2.4 Solution is fairly straightforward. In addition, you can go to Career360's website and get the book in PDF format for free.

**Q2. Do the RD Sharma Relations Ex 2.4 class 12 solutions follow the most recent syllabus?**

The RD Sharma class 12 solution Chapter 2 Exercise 2.4 Solution is always updated with the latest syllabus. With each new edition of the NCERT book, the pdf of the RD Sharma Class 12 Solutions Chapter 2 Ex 2.4 is updated so that students can get answers to all of the problems in the book.

**Q3. Is the content of the RD Sharma solutions up to date with the most recent syllabus?**

RD Sharma book is updated each year to reflect the most up-to-date CBSE class 12 syllabus. As a result, you can be confident that the book will provide you with all of the knowledge you need.

**Q4. Define a Function.**

A function is the criteria that define the relationship between two or more variables. For example, if A is a variable, its function is defined as f(A). To learn more about functions, check RD Sharma Class 12 Solutions Chapter 2 Ex 2.4.

**Q5. What are the different types of functions?**

The different types of functions are

Injective function

Surjective function

Many to one function

Polynomial function

Identical function

Quadratic function etc.

To know more about this, check out RD Sharma Class 12 Solutions Functions Ex 2.4.**Chapter-wise RD Sharma Class 12 Solutions**

- Chapter 1 - Relations
- Chapter 2 - Functions
- Chapter 3 - Inverse Trigonometric Functions
- Chapter 4 - Algebra of Matrices
- Chapter 5 - Determinants
- Chapter 6 - Adjoint and Inverse of a Matrix
- Chapter 7 - Solution of Simultaneous Linear Equations
- Chapter 8 - Continuity
- Chapter 9 - Differentiability
- Chapter 10 - Differentiation
- Chapter 11 - Higher Order Derivatives
- Chapter 12 - Derivative as a Rate Measurer
- Chapter 13 - Differentials, Errors and Approximations
- Chapter 14 - Mean Value Theorems
- Chapter 15 - Tangents and Normals
- Chapter 16 - Increasing and Decreasing Functions
- Chapter 17 - Maxima and Minima
- Chapter 18 - Indefinite Integrals
- Chapter 19 - Definite Integrals
- Chapter 20 - Areas of Bounded Regions
- Chapter 21 - Differential Equations
- Chapter 22 - Algebra of Vectors
- Chapter 23 - Scalar Or Dot Product
- Chapter 24 - Vector or Cross Product
- Chapter 25 - Scalar Triple Product
- Chapter 26 - Direction Cosines and Direction Ratios
- Chapter 27 - Straight Line in Space
- Chapter 28 - The Plane
- Chapter 29 - Linear programming
- Chapter 30- Probability
- Chapter 31 - Mean and Variance of a Random Variable

1. How do I get the solution of RD Sharma chapter 2 exercise 2.4 for class 12?

Downloading the Class 12 RD Sharma Chapter 2 Exercise 2.4 Solution is fairly straightforward. In addition, you can go to Career360's website and get the book in PDF format for free.

2. Do the RD Sharma Relations Ex 2.4 class 12 solutions follow the most recent syllabus?

The RD Sharma class 12 solution Chapter 2 Exercise 2.4 Solution is always updated with the latest syllabus. With each new edition of the NCERT book, the pdf of the RD Sharma Class 12 Solutions Chapter 2 Ex 2.4 is updated so that students can get answers to all of the problems in the book.

3. . Is the content of the RD Sharma solutions up to date with the most recent syllabus?

RD Sharma book is updated each year to reflect the most up-to-date CBSE class 12 syllabus. As a result, you can be confident that the book will provide you with all of the knowledge you need.

4. Define a Function.

A function is the criteria that define the relationship between two or more variables. For example, if A is a variable, its function is defined as f(A). To learn more about functions, check RD Sharma Class 12 Solutions Chapter 2 Ex 2.4.

5. What are the different types of functions?

The different types of functions are

Injective function

Surjective function

Many to one function

Polynomial function

Identical function

Quadratic function etc.

To know more about this, check out RD Sharma Class 12 Solutions Functions Ex 2.4.

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