NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions Exercise 2.3

NCERT Solutions for Class 11 Maths Chapter 2 Exercise 2.3 help students explore more advanced concepts of relations and functions, including composition of functions and inverse of a function. This exercise focuses on applying the concepts of functions and their operations to solve problems systematically.

Designed according to the latest CBSE Class 11 Maths syllabus, Exercise 2.3 focuses on the composition of functions and finding the inverse of a function, which are essential concepts for higher mathematics, particularly in calculus. Students will learn how to combine multiple functions and solve problems involving the composition and inverse functions.

NCERT Solutions for Class 11 Maths Chapter 2 Relations and Functions Exercise 2.3

Relations and Functions

Medium

Q.

The Cartesian product A × A has 9 elements among which are found (–1, 0) and (0, 1).
Find the set A and the remaining elements of A × A.

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Relations and Functions

Easy

Q.

$\begin{array}{l}\text{The function 't' which maps temperature in degree Celsius\hspace{0.33em}}\mathrm{into}\\ \text{temperature in degree Fahrenheit is defined by t}\left(\text{C}\right)=\frac{\text{9C}}{\text{5}}\text{+32.}\\ \text{Find}\\ \text{(i) \hspace{0.33em}t(0)}\\ \text{(ii)\hspace{0.33em}t(28)}\\ \text{(iii)\hspace{0.33em}t(–10)}\\ \text{(iv)\hspace{0.33em}The value of C, when t(C) = 212.}\end{array}$

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Relations and Functions

Medium

Q.

 $\begin{array}{l}{Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from }\\ {Z to Z defined by f(x) = ax + b, for some integers a, b.}\\ {Determine a, b.}\end{array}$ 

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Relations and Functions

Easy

Q.

State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.

(i) If P = {m, n} and Q = {n, m}, then P × Q = {(m, n), (n, m)}.
(ii) If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (x, y) such that x ∈ A and y ∈ B.
(iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ ∅) = ∅ .

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Relations and Functions

Difficult

Q.

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that

(i) A × (B ∩ C) = (A × B) ∩ (A × C).
(ii) A × C is a subset of B × D.

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Relations and Functions

Difficult

Q.

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.

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Relations and Functions

Difficult

Q.

Let A = {9, 10, 11, 12, 13} and let f: A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

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The solutions are explained in a clear, step-by-step format so students can develop a solid understanding of how functions can be combined and inverted, which is crucial for solving more complex problems in exams.

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)}
  • Answer: This relation is a function.
  • Reason: All elements of the domain have unique images.
  • Domain: {2, 5, 8, 11, 14, 17}.
  • Range: {1}.
• (ii) {(2,1), (4,2), (6,3), (8,4), (10,5), (12,6), (14,7)}
  • Answer: This relation is a function.
  • Reason: All domain elements have unique images.
  • Domain: {2, 4, 6, 8, 10, 12, 14}.
  • Range: {1, 2, 3, 4, 5, 6, 7}.
• (iii) {(1,3), (1,5), (2,5)}
  • Answer: This relation is not a function.
  • Reason: The element 1 corresponds to two different images (3 and 5).

Question 2: Find the domain and range of the following real function:

• (i) f(x) = -|x|
  • Domain: R (all real numbers).
  • Range: (-infinity, 0] (all non-positive real numbers).
• (ii) f(x) = square root of (9 - x^2)
  • Domain: {x : -3 <= x <= 3} or [-3, 3].
  • Range: {x : 0 <= x <= 3} or [0, 3].

Question 3: A function f is defined by f(x) = 2x - 5. Find:

• (i) f(0): 2(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' maps Celsius (C) to Fahrenheit (F) by t(C) = 9C/5 + 32. Find:

• (i) t(0): (9*0)/5 + 32 = 32.
• (ii) t(28): (9*28)/5 + 32 = 252/5 + 32 = 50.4 + 32 = 82.4.
• (iii) t(-10): (9*-10)/5 + 32 = -18 + 32 = 14.
• (iv) The value of C when t(C) = 212:
  • 212 = 9C/5 + 32
  • 180 = 9C/5
  • C = (180 * 5) / 9 = 100.

Question 5: Find the range of each of the following functions:

• (i) f(x) = 2 - 3x, x belongs to R, x > 0
  • Range: (-infinity, 2).
• (ii) f(x) = x^2 + 2, x is a real number
  • Range: [2, infinity).
• (iii) f(x) = x, x is a real number
  • Range: R (all real numbers).

FAQs – Class 11 Maths Chapter 2 Exercise 2.3 Relations and Functions

Q1. What is the focus of Exercise 2.3?
Exercise 2.3 focuses on understanding the composition of functions and finding the inverse of a function. These concepts help students work with multiple functions and their operations.

Q2. What is the composition of functions?
The composition of two functions

$f$

f and

$g$

g, denoted as

$f \circ g$

f∘g, is defined as:

$$(f \circ g)(x) = f(g(x))$$

(f∘g)(x)=f(g(x))

This means applying the function

$g$

g first, then applying the function

$f$

f to the result.

Q3. How do you find the inverse of a function?
To find the inverse of a function

$f(x)$

f(x), denoted as

$f^{-1}(x)$

f−1(x), follow these steps:

1. Replace

   $f(x)$f(x) with

   $y$y.

2. Swap

   $x$x and

   $y$y in the equation.

3. Solve for

   $y$y.

4. Replace

   $y$y with

   $f^{-1}(x)$f−1(x).

Q4. Why is Exercise 2.3 important for board exams?
This exercise is important because it deals with composition and inverse functions, which are frequently tested in board exams. These concepts are foundational for calculus and understanding how to work with multiple functions simultaneously.

Q5. How can students prepare effectively for Exercise 2.3?
Students should:

• Understand the definition and process of function composition.

• Practice finding the inverse of functions by following systematic steps.

• Solve a variety of problems involving function composition and inversion to gain confidence.