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The RD Sharma Solutions for Class 12 Mathematics help the students to understand and work out the problems efficiently. It is essential to possess the right guide while working on homework or assignments, especially on complex topics like the RD Sharma Class 12th Exercise 2.2. This single compilation is adequate for the CBSE students who are preparing for their board exams.

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Functions Exercise 2.2 Question 1 (i) .

means function is in function

Now first we find

Now we find

Hence,

Then we find

Hence, and

Functions Exercise 2.2 Question 1 (ii) .

means function is in funcion

First, we find

Then we find

Hence, and

Functions Exercise 2.2 Question 1 (iii) .

means function is in function

First, we find

Thus,

Similarly,

Hence,

Functions Exercise 2.2 Question 1 (iv) .

means function is in function

Since

Now,

Similarly ,

Hence,

Functions Exercise 2.2 Question 1 (v) .

means function is in function

Since

Now,

Hence,

Functions Exercise 2.2 Question 1 (vi).

**Hint :** means function is in function

means function is in function

**Given :**

**Solution :**

Since

Now,

Similarly,

Hence,

means function is in function

Since

Now,

Similarly,

Hence,

Functions Exercise 2.2 Question 2

**Prove :** and are both defined**Solution :**

Co-domain of f is a subset of the domain g

So, exist and

Co-domain of g is subset of the domain of f.

So, exist and

Hence proved, and are both defined.

and

Functions Exercise 2.2 Question 3.

Now, Domain of

Range of

Domain of

Range of

Clearly, Range of f = domain of g

But Range of domain of f

Now,

Hence proved, is defined but is mot defined.

Functions Exercise 2.2 Question 4.

f and g be two functions from A to B and from B to A ;

For both f and g, Different elements of the domain have different images.

f and f are onto

Again, for each element in co-domain of f and g, there in a preimage in domain

f and f are onto

Thus, f and g are bijectives.

Now,

Hence prove, f and g both are bijections

Functions Exercise 2.2 Question 5.

means function is in function

First, we find

Similarly,

Hence,

Function Exercise 2.2 Question 6.

means function is in function

defined by

Similarly,

Thus

Hence proved functions are equal.

Function Exercise 2.2 Question 7.

means function is in function

defined by

......(i)

Similarly,

.....(ii)

Now we have to make a comparison between eq. (i) and (ii)

Hence proved

Function Exercise 2.2 Question 8.

means function is in function

.....(i)

Similarly,

..... (ii)

From equation (i) and (ii)

Hence proved :

Function Exercise 2.2 Question 9.

means function is in function

For associative property

So, both have same domains

.....(ii)

Now,

....[From (i)]

....[From (ii)]

...[since ]

....[From (ii)]

So,

Hence, the associative property has benn verified.

Function Exercise 2.2 Question 10.

Answer :means function is in function

Given : defined as

Solution :

Similarly,

This shows,

Functions Exercise 2.2 Question 11.

Let

and

We will first show that f is not onto

Checking f is not onto

for

But 0 is not a natural number

f is not onto.

Finding

Since

So,

For

Since,

So,

Let

So,

Here, y is a natural number. As

So, x is also a natural number.

Hence, is onto.

Functions Exercise 2.2 Question 12.

where

Let

Checking injective (one-one)

Since, different elements 1, -1 have the same image 1

g is not injective (one -one).

Checking for injective (one-one)

So, x is always a natural number.

Here will always be a natural number.

So, has a unique image

is injective.

Hence, is injective but g is not injective

Functions Exercise 2.2 Question 13.

[ g is one - one ]

[ f is one - one]

is a one - one function

Functions Exercise 2.2 Question 14.

Let then

....(i)

Since, g is onto, for each element in C, then exists a pre-image in B.

.....(ii)

From (i) and (ii)

Since f is onto, for each element in B, there exist a pre image in A

....(iii)

From (ii) and (iii) we can conclude that for each , there exist a preimage in A

Such that

is onto.

Mathematics may seem complicated for some students, but with the help of RD Sharma Solutions for Class 12 Mathematics, they can figure out an easier way to solve the problems.

Not every child could understand a standard way of finding the answer; hence, this book gives multiple formats of arriving at mathematics solutions. And the most significant advantage is that you can find this book free of cost, without wanting you to pay even a penny.

The 2nd chapter in RD Sharma Class 12th Exercise 2.2 is regarding Functions in mathematics. This chapter focuses on the various kinds of functions, compositions of functions, and inverse of functions. Again, this is a specific topic to be grasped by students compared to the other chapters in the book. The first exercise, 2.1, is a bit essential in the mathematical functions. The second exercise, 2.2, dives deeply into the concepts of fog and gof, injective, surjective, and bijective.

The Class 12 RD Sharma Chapter 2 Exercise 2.2 Solution will lend the students a helping hand. Making the students understand the concept of functions is essential as the upcoming chapters are based on it. For the convenience of the students, the solutions for RD Sharma Class 12th Exercise 2.2 are given in the same order as present in the textbooks.

RD Sharma Class 12th Exercise 2.2 is divided into two levels, Level 1 and Level 2, according to the questions' complexity and difficulty. Level 1 has 21 questions, and level 2 has a couple of questions. Even though numerous examples are provided in the book, solutions elaborately make the students understand the concept more easily and effectively. Therefore, every answer provided in the RD Sharma Class 12 Solutions Chapter 2 Ex 2.2 is given in a more manageable format for the welfare of the students.

With the help of this best solutions book, the students would find the Function chapter easier than before. The RD Sharma Class 12 Solutions Function Ex 2.2 makes the students connect more effortlessly with the concepts than expected. Moreover, once they start using it to prepare their homework and assignments, facing tests and exams in this chapter would become easier based on the NCERT syllabus. Hence, if you are a CBSE board class 12 student, it is better to possess the RD Sharma Class 12th Exercise 2.2 Solutions as there are many chances of questions being asked from this book on your board exam.

**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

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Download E-book1. Are the solutions provided in RD Sharma Solutions book updated?

Every solution provided in the RD Sharma Solutions books is updated frequently according to the change in the syllabus.

2. Which RD Sharma solutions book is highly recommended to learn Chapter 2 of Class 12 Mathematics?

The RD Sharma Class 12 Chapter 2 part is enough for the students to prepare the second chapter easily. They can download it from websites like Career360 for free of cost.

3. Is RD Sharma Solutions available according to particular exercises in Class 12 mathematics?

The students can refer according to the specific exercises they require. For instance, the RD Sharma Class 12 Chapter 2 Exercise 2.2 part is enough to work out the sums in exercise 2.2.

4. Are the solutions provided in the RD Sharma books prepared from the exam point of view?

The solutions provided in the RD Sharma book are prepared by experts and are constructed from the exam point of view.

5. Is the RD Sharma Solutions book enough for the CBSE board students to prepare for their public exams?

The RD Sharma Solutions books are more than enough for the class 12 students to prepare for their board exams. In addition, these books provide a deeper insight into the concepts to the students hence making it easier for them to learn.

Sep 11, 2024

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