Chapter 2 of the Class 11 NCERT Mathematics textbook, "Relations and Functions," covers the foundational concepts of relations and functions, including their definitions, types, and properties. Exercise 2.3 focuses on problems related to various types of functions and their properties.
NCERT Solutions for Class 11 - Mathematics - Chapter 2 Relations and Functions - Exercise 2.3
This section provides detailed solutions for Exercise 2.3 from Chapter 2 of the Class 11 NCERT Mathematics textbook. The exercise involves problems related to identifying and analyzing different types of functions, including domain, range, and function composition.
Question 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)}
Here, each element in domain is having unique/distinct image. So, the given relation is a function.
Domain = {2, 5, 8, 11, 14, 17}
Range of the function = {1}
(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}
Here, each element in domain is having unique/distinct image. So, the given relation is a function.
Domain = {2, 4, 6, 8, 10, 12, 14}
Range of function = {1, 2, 3, 4, 5, 6, 7}
(iii) {(1, 3), (1, 5), (2, 5)}
This relation is not a function since an element 1 corresponds to two elements/images i.e, 3 and 5.
Hence, this relation is not a function.
Question 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.
|x| <=3
So, the domain of f(x) is {x: –3 ≤ x ≤ 3} or Domain of f = [–3, 3].
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 we can say Range of f = [0, 3].
Question 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.
(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
Question 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 ques , 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.
Question 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)
Summary
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.