NCERT Solutions for Class 11 – Mathematics – Chapter 2 Relations and Functions – Exercise 2.3
Exercise 2.3 deals with analyzing different types of functions and their properties. Key aspects include:
- Domain and Range: Identifying the set of possible input values (domain) and output values (range) for a function.
- Function Characteristics: Determining if a relation is a function using the vertical line test and understanding one-to-one (injective) and onto (surjective) functions.
Table of Contents
Exercise 2.3
1. Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.
(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}
(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}
(iii) {(1, 3), (1, 5), (2, 5)}
Solution:
(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}
As 2, 5, 8, 11, 14, and 17 are the elements of the domain of the given relation having their unique images, this relation can be called a function.
Here, domain = {2, 5, 8, 11, 14, 17} and range = {1}
(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}
As 2, 4, 6, 8, 10, 12, and 14 are the elements of the domain of the given relation having their unique images, this relation can be called a function.
Here, domain = {2, 4, 6, 8, 10, 12, 14} and range = {1, 2, 3, 4, 5, 6, 7}
(iii) {(1, 3), (1, 5), (2, 5)}
It’s seen that the same first element, i.e., 1, corresponds to two different images, i.e., 3 and 5; this relation cannot be called a function.
2. Find the domain and range of the following real function:
(i) f(x) = –|x|
(ii) f(x) = √(9 – x2)
Solution:
(i) Given,
f(x) = –|x|, x ∈ R
We know that, |x| =
| x | if x >= 0 |
| -x | if x < 0 |
Here f(x) = -x =
| -x | x >= 0 |
| x | x < 0 |
As f(x) is defined for x ∈ R, the domain of f is R.
It is also seen that the range of f(x) = –|x| is all real numbers except positive real numbers.
Therefore, the range of f is given by (–∞, 0].
(ii) f(x) = √(9 – x2)
As √(9 – x2) is defined for all real numbers that are greater than or equal to –3 and less than or equal to 3, for 9 – x2 ≥ 0.
So, the domain of f(x) is {x: –3 ≤ x ≤ 3} or [–3, 3].
Now, For any value of x in the range [–3, 3], the value of f(x) will lie between 0 and 3.
Therefore, the range of f(x) is {x: 0 ≤ x ≤ 3} or [0, 3].
3. A function f is defined by f(x) = 2x – 5. Write down the values of
(i) f(0), (ii) f(7), (iii) f(–3)
Solution:
Given,
Function, f(x) = 2x – 5
Therefore,
(i) f(0) = 2 × 0 – 5 = 0 – 5 = –5
(ii) f(7) = 2 × 7 – 5 = 14 – 5 = 9
(iii) f(–3) = 2 × (–3) – 5 = – 6 – 5 = –11
4. The function ‘t’, which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C) = 9C/5 + 32.
Find (i) t (0) (ii) t (28) (iii) t (–10) (iv) The value of C, when t(C) = 212
Solution:
Here in question , it is given that :
t(C) = 9C / 5 +32
So, (i) t(0) = 9(0) / 5 + 32
= 0 + 32
= 32
(ii) t(28) = 9(28) / 5 + 32
Taking LCM and solving ,
= ( 252 +160 ) / 5
= 412 / 5
(iii) t(-10) = 9(-10) / 5 + 32
= -18 + 32
= 14
(iv) Here , in this ques we have to find the value of C.
Given that , t(C) = 212,
9C / 5 + 32 = 212
9C / 5 = 180
9C = 180 X 5
C = 100
The value of C is 100.
5. Find the range of each of the following functions:
(i) f(x) = 2 – 3x, x ∈ R, x > 0
(ii) f(x) = x2 + 2, x is a real number
(iii) f(x) = x, x is a real number
Solution:
(i) Given f (x) = 2 – 3x, x ∈ R, x > 0
∵ x > 0 ⇒ -3x < 0 (Multiplying both sides by -3)
⇒ 2 – 3x < 2 + 0 ⇒ f (x) < 2
∴ Hence, The range of f (x) is (-∞ , 2).
(ii) Given f (x) = x2+ 2, x is a real number
We know x2≥ 0 ⇒ x2+ 2 ≥ 0 + 2
⇒ x2 + 2 > 2 ∴ f (x) ≥ 2
∴ Hence, The range of f (x) is [2, ∞).
(iii) Given f (x) = x, x is a real number.
Let y = f (x) = x ⇒ y = x
∴ Range of f (x) = Domain of f (x)
∴ Hence, Range of f (x) is R. (f(x) takes all real values)
NCERT Solutions for Class 11 Maths Chapter 2 – Relations and Functions
The following ideas from Chapter 2 Relations and Functions for Class 11, are given elaborately.
2.1 Introduction
This section introduces the concepts covered in the chapter Relations and Functions.
The combination of the register number of the student and their corresponding height is a relationship, which can be written as a set of ordered-pair numbers. Ordered-pair numbers are expressed as (x, y). The set of all elements of x is called the domain of the relation, and the set of all elements of y is called the range of the relation.
2.2 Cartesian Product of Sets
This section defines the Cartesian product and ordered pairs by giving a real-life model, its representation, and some worked examples.
Lindt chocolates come in five shapes, three flavours and six colours.
C :={circle, triangle, rectangle, rhombus, square}
N :={orange, vanilla, peach}
S :={red, blue, pink, white, yellow, purple}
C:={circle, triangle, rectangle, rhombus, square}, N:={orange, vanilla, peach}, S:={red, blue, pink, white, yellow, purple}
be the five shapes, three flavours and six colours, respectively. Then the set of all chocolates to be manufactured in the triple Cartesian product C×N×S and consists of 5⋅3⋅6=90 elements. As a manager, to sell this set of chocolates would have to make room for 90 heaps.
2.3 Relations
This section explains the mapping of set A to set B with a few solved problems. Definitions of domain and codomain are also introduced.
The idea of mapping a particular phone number to the respective person to whom the number belongs. That’s a relation – from phone number to person.
2.4 Functions
This section covers functions, the visualisation of functions, and how a relation is said to be a function, with a few examples. Meaning of image and preimage.
The height of a person can be determined by the length of his femur bone. Hence, it is an example of a function.
2.4.1 Some functions and their graphs
This section talks about different types of functions and their graphical representations. Some of the types of functions are listed below.
- Identity function
- Constant function
- Polynomial function
- Rational functions
- The Modulus function
- Signum function
- Greatest integer function
2.4.2 Algebra of real functions
This section includes the algebraic operations on functions.
- Addition of two real functions
- Subtraction of a real function from another
- Multiplication by a scalar
- Multiplication of two real functions
- The quotient of two real functions
Key Features of NCERT Solutions for Class 11 Maths Chapter 2 – Relations and Functions
- The two elements grouped in a particular order are called an ordered pair.
- Cartesian product A × B of two sets A and B is given by A × B = {(a, b): a ∈ A, b ∈ B}
- Relation R from a set A to a set B is a subset of the cartesian product A × B obtained by explaining the relationship between the first element x and the second element y of the ordered pairs in A × B.
- The image of an element x under a relation R is given by y, where (x, y) ∈ R.
- The domain of R is the set of all first elements of the ordered pairs in a relation R.
- The range of a relation R is the set of all second elements of the ordered pairs in a relation R.
- Function A from a set A to a set B is a specific type of relation for which every element x of set A has one and only one image y in set B. We write f: A→B, where f(x) = y.
- The range of the function is the set of images.
- A real function has a set of real numbers or one of its subsets both as to its domain and as its range.
Basic concepts covered in the NCERT Solutions for Class 11 Maths modules aid students in moving ahead in their studies. The latest update of the CBSE Syllabus ensures that the content covered is apt for the students to move ahead in their respective streams in the future. A student needs to understand the concept of Relations and Functions as it covers the main part of the question paper. Before solving real-world applications and problems, the concept has to be learned thoroughly.
Frequently Asked Questions on NCERT Solutions for Class 11 Maths Chapter 2
Q1
How to find which relation is a function in Chapter 2 of NCERT Solutions for Class 11 Maths?
According to the definition, a function can relate every element which is present in a domain to only one element, which is found in the range. It means that any vertical line drawn by a student on a graph can pass through the x-axis only once. A relation from a function can be found by using vertical line tests or with the help of different formulas.
Q2
Explain the basic steps for the Cartesian product of sets in NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions.
In order to understand the basic steps for solving a question regarding the Cartesian product of sets, students must comprehend the first exercise of the chapter thoroughly. Students are provided with solved examples before each exercise-wise problem to help them understand the method of solving problems in a shorter duration. By solving the problems from the NCERT textbook, students will improve their conceptual understanding, which is necessary to perform well in the exams.
Q3
What is the meaning of relations in Chapter 2 of NCERT Solutions for Class 11 Maths?
Relations are nothing but the collection of ordered pairs which has one object from every set. A function can also be considered as a relation, but the conceptual ideas of both of them are completely different. The NCERT Solutions for Class 11 Maths Chapter 2 provides the students with a proper definition and analysis of relations as per the CBSE Syllabus 2023-24. Several examples present in the solutions will help students solve problems related to relations without difficulty.
Relations and Functions NCERT Solutions for Class 11 Maths Chapter 2 Free PDF Download
NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions are solved in detail in the PDF given below. All the solutions to the problems in the exercises are created in such a way that it enables the students to prepare for the exam and ace it. The NCERT Solutions are prepared by the most experienced teachers in the education space, making the explanation of each solution simple, understandable, and according to the latest CBSE Syllabus. The solution helps Class 11 students to master the concept of Relations and Functions.
The solutions provide a good understanding of the fundamental concepts before they solve the equations. Through regular practice, students will know the difference between relations and functions, which are included under the syllabus, and become well-versed in its concepts. Numerous examples are present in the textbook before the exercise questions to help them understand the methodologies to be followed while solving the problems. Referring to the NCERT Class 11 Solutions PDF, students can get a glimpse of the important concepts before facing their final exams.
