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Edited By Kuldeep Maurya | Updated on Jan 20, 2022 10:43 AM IST

RD Sharma books are no less than a Math bible for students as they contain concepts and chapters that help students understand the subject. Similarly, RD Sharma Class 12th Exercise 2.3 is specifically designed for Class 12 students to prepare for their exams. Therefore, students will find this material very helpful for their preparation as they are preparing for their Class 12 exams and the entrance exams that will be conducted in the future.

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Functions Exercise 2.3 Question 1 (i).

And dom dom f

Here we have find out and

Here first we will find out

Here we have,

Now, we find out , we have ,

Functions Exercise 2.3 Question 1 (ii).

Range of

Range of g

Range of f domain of g

exists

and Range of g domain of f

exists

Here first we will find out

We have,

and again for

Functions Exercise 2.3 Question 1 (iii).

exists

And Range of domain R

exists

Here we have to find out and

First , we will find out

Again, we find out

Functions Exercise 2.3 Question 1 (iv).

exists

Here we have to find out

First, we will find out

We have,

And again, we will find out

Functions Exercise 2.3 Question 1 (v).

Here, we have to compute

Here we have to compute :

Now, we have to find out :

Clearly, the range of f is subset of the domain of g.

Functions Exercise 2.3 Question 1 (vi).

we have to compute and

First, we have to compute

Clearly, the range of g is a subset of the domain f.

Set of the domain of f.

Now, we will compute

Functions Exercise 2.3 Question 1 (vii).

Now, we have to compute and

First, we will compute

Here,

Clearly the range of g is a subset of the domain of f.

Now, we will compute

Clearly the range of f is subset of the domain of g

Hence,

Functions Exercise 2.3 Question 1 (viii).

Here we have to compute and .

First , we will compute the

We have,

Now we will compute

Clearly the range of f is a subset of the domain of g .

Functions Exercise 2.3 Question 1 (ix).

For domain of

Domain of

For range of g

Range of

First, we compute

Here clearly the range of g is subset of domain of f.

Now, we will compute

Clearly the range of f is a subset of the domain of g

Functions Exercise 2.3 Question 2.

First, we have to compute and

For

...(i)

And now for

... (ii)

From enq. (1) and (2) we proved that

Functions Exercise 2.3 Question 3.

Here we have to compute

We have ,

So,

Hence,

Functions Exercise 2.3 Question 4 (i).

Here we have to compute

Since and are polynomials.

Hence,

Functions Exercise 2.3 Question 4 (ii).

Here we have to compute

Since and are polynomials.

Hence,

Functions Exercise 2.3 Question 4 (iii).

Here we have to compute

Since and are polynomials

Hence,

Functions Exercise 2.3 Question 4 (iv).

Here we have to find out

Since is a polynomial.

In this question we have to prove that

Then first we have to find out

For

...(1)

Again we will compute

Since

...(2)

From (1) and (2) we have

Hence proved

Function Exercise 2.3 Question 5.

Answer : and they are not equal functions.

Given : Here given that

Here we have to compute and

Hint : First we will compute

we know that

Clearly the range of f is a subset of the domain of g.

Second, we will compute

Clearly the range of g is a subset of the domain of f.

Solution :

....(1)

....(2)

From (1) and (2) clearly

Hence, they are not equal functions.

Function Exercise 2.3 Question 6.

We know that

Clearly, the range of g is a subset of the domain of f .

Now,

Domain of h is R.

Since range of

Range of

So,

Clearly the range of fh is a subset of g

Domain of are the same gain

So,

And again ,

Hence,

Functions Exercise 2.3 Question 7.

Take, L.H.S.

Hence,

Functions Exercise 2.3 Question 8.

Range of

Range of

Clearly Range domain of g

exists

And Range domain of f

Again, we will compute

Functions Exercise 2.3 Question 9.

Range of domain of

First, we compute

We know that

Let

Again we will compute

Functions Exercise 2.3 Question 10.

Here we have to find out and

First, we will compute

Again, we will compute

Hence,

Functions Exercise 2.3 Question 11 (i) .

Clearly, the range of is not a subset of domain of f.

Here we have to compute

We know that

Domain of

Now,

Hence,

Functions Exercise 2.3 Question 11 (ii) .

We have to compute

Clearly the range f is not subset of domain of f.

Clearly the range of

Now, we will compute

First we compute

Now,

Hence,

Functions Exercise 2.3 Question 11 (iii) .

Now, we will compute

Clearly the range is not a subset of domain of

Clearly range of

Now, we will compute

Again we will compute

Now,

Hence,

Functions Exercise 2.3 Question 11 (iv) .

Now we have to compute and

First, we will compute

Clearly, Domain

We observe that the range is not subset of domain of f

Now, we will compute

and again for

Here we see that

Hence,

Functions Exercise 2.3 Question 12 .

Here we have find out

We have,

Range of

Functions Exercise 2.3 Question 13 .

Answer :Hint : We know about modulus function,

Given : Let f, be two function defined as :

Here, we have to find out

Solution :

Here,

Thus for

For

Thus for

For

Again,

Then,

RD Sharma Class 12th Exercise 2.3 covers many essential topics related to functions and will help students quickly understand many fun and exciting concepts. Here you will learn about different kinds of functions and solve fundamental problems involving them. As this is an important chapter, students can expect many questions from it in entrance exams.

In RD Sharma Class 12th Exercise 2.3, you will learn basic definitions of identity function, modulus function, fractional part function, signum function, etc. Then, you will learn some theories on relations between functions like 'one to one,' 'onto' function, etc., followed by problems. Next, the numerical are divided into different levels to gradually build their problem-solving skills and move on to complex problems.

RD Sharma Class 12th Exercise 2.3 has 20 - Level 1 and 6 - Level 2 example questions that students can use to build their basic problem-solving skills. Level 1 sums contain all the introductory concepts you have covered first in this chapter, followed by Level 2, which has more advanced questions. The Level 2 questions carry more marks but have more complexity, so students have to be perfect in Level 1 sums first.

There are many ways to solve a problem, you don't need to follow the textbook's steps. Students are free to use any method they like to solve a problem as long as it is valid. Getting a thorough knowledge of this material can help students find alternative ways to solve a problem. This saves time and allows students to reduce complexity for their sums.

The following material: RD Sharma Class 12 Solutions Chapter 2 Ex.2.3 is designed by experts to cover all important questions and help students get different alternatives to finding a problem. As these materials are widely used, there might be good chances that their questions can appear in your exams. Therefore, experts suggest that students should practice this material regularly to understand the subject.

RD Sharma Class 12th Exercise 2.3 on Career360 provides the best materials on RD Sharma books free of cost. Never before have these solutions been available so conveniently. Students can take advantage of the informative material and score well in exams. A lot of students have chosen Career360 as the best source for their RD Sharma solutions. People who haven’t yet discovered it should undoubtedly refer to this to score good marks. You can find more expert-created answers to different books on the website, which are free of cost and accessible to everyone.

**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. Can I score good marks in exams by referring to RD Sharma books?

Yes, RD Sharma books are the best material students can refer to for preparation. This material will help them score good marks in exams and get a good hold on the subject. You will easily score good marks if you practice well using this material. RD Sharma Class 12 Chapter 2 Exercise 2.3 is made for students to understand the fundamentals of Functions and prepare well for exams.

2. Do these solutions contain important questions?

Yes, the solutions provided by Career360 contain important questions that are helpful for easy preparation before exams. Students can refer to RD Sharma Class 12 Chapter 2 Exercise 2.3 to get a good idea about the chapter ‘Functions’ and its important questions.

3. What are the advantages of these solutions?

The Class 12 RD Sharma Chapter 2 Exercise 2.3 solution is made by a team of experts who help students prepare well for their exams. Their advantages are:

Solutions are easy to understand.

Important questions are discussed.

Material is updated to the latest version.

Solutions are free of cost.

4. What is a Function?

Elements that are arranged in a group and have certain fixed values are called sets. A function is a set of rules or expressions that define a relationship between two or more sets. You can refer to Class 12 RD Sharma Chapter 2 Exercise 2.3 solution for thorough information on this topic.

5. . What are the different types of Functions?

The different types of functions are:

One-one function or Injection

Many-one function

Onto function or Surjection

Bijection

You can learn more about these with examples using RD Sharma Class 12 functions Ex 2.3

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