Draw the graph of the real function f x = x 2 - 4 x - 2 . Hence find their range.

Given f x = x 2 - 4 x - 2 Since the function is defined for every x ∈ R except at x = 2 Hence, the domain of the function is D f = R - 2 Let y = f x = x 2 - 4 x - 2 = x + 2 , which is the first degree equation in x , y and hence it represents a straight line. Two points are sufficient to determine a straight line uniquely. The table of values are x - 2 0 y 0 2 Therefore, the graph of the given function is shown in figure and it is clear that y takes all real values except 4 Hence, the range of function is R - 4 .

Draw the graph of the real function f x = x + 3 , x < 2 2 x + 1 , x ≥ 2 . Hence find their range.

Given f x = x + 3 , x < 2 2 x + 1 , x ≥ 2 Since the function is defined for every x ∈ R . Hence, the domain of the function is D f = R Let y = f x = x + 3 , x < 2 2 x + 1 , x ≥ 2 , When x < 2 , y = x + 3 Which is the first degree equation in x , y and hence it represents a straight line. Two points are sufficient to determine a straight line uniquely. The table of values are x 1 - 1 y 4 2 The graph is a part of a straight line. When x ≥ 2 , y = 2 x + 1 , which is also a first degree equation in x , y The table of values are x 2 3 y 5 7 The graph is part of a straight line. The graph of the function is shown in figure. From the graph it is cleat that, the range of function is R .

Draw the graph of the real function f x = x - 1 , x < 2 2 x + 3 , x ≥ 2 . Hence find their range.

Given f x = x - 1 , x < 2 2 x + 3 , x ≥ 2 Since the function is defined for every x ∈ R . Hence, the domain of the function is D f = R Let y = f x = x - 1 , x < 2 2 x + 3 , x ≥ 2 , When x < 2 , y = x - 1 Which is the first degree equation in x , y and hence it represents a straight line. Two points are sufficient to determine a straight line uniquely. The table of values are x 1 - 1 y 0 - 2 The graph is a part of a straight line. Since, x < 2 ⇒ x - 1 < 2 - 1 ⇒ f x < 1 Hence, the range of this interval is ( - ∞ , 1 ) ......(i) When x ≥ 2 , y = 2 x + 3 , which is also a first degree equation in x , y The table of values are x 2 3 y 7 9 The graph is part of a straight line. Since, x ≥ 2 ⇒ 2 x ≥ 4 ⇒ 2 x + 3 ≥ 4 + 3 ⇒ f x ≥ 7 Hence, the range of this interval is [ 7 , ∞ ) ......(ii) The graph of the function is shown in figure. Hence, from (i) and (ii), we get the range is ( - ∞ , 1 ) ∪ [ 7 , ∞ ) .

Draw the graph of the real function f x = - 1 , x ≤ - 1 x , - 1 < x < 1 1 , x ≥ 1 . Hence find their range.

Given f x = - 1 , x ≤ - 1 x , - 1 < x < 1 1 , x ≥ 1 Since the function is defined for every x ∈ R . Hence, the domain of the function is D f = R . Let y = f x = - 1 , x ≤ - 1 x , - 1 < x < 1 1 , x ≥ 1 , When x ≤ - 1 , y = - 1 , Which is straight line parallel to x - axis . The table of values are x - 1 - 2 y - 1 - 1 The graph is a part of a straight line. When - 1 < x < 1 , y = x , Which a straight line passing through origin. The table of values are x 0 . 1 0 . 2 y 0 . 1 0 . 2 The graph is part of a straight line. When x ≥ 1 , y = 1 , Which is also straight line parallel to x - axis . The table of values are x 1 2 y 1 1 The graph of the function is shown in figure. Hence, from the graph we have range is [ - 1 , 1 ] .

Draw the graph of the real function f x = 1 2 x 2 . Hence find their range.

Given f ( x ) = 1 2 x 2 Since the function is defined for every x ∈ R . Hence, the domain of the function is D f = R Let y = f ( x ) = 1 2 x 2 The table of values x . . . - 3 - 2 - 1 0 1 2 3 . . . y . . . 9 2 2 1 2 0 1 2 2 9 2 . . . Plot all the points and join these points by free hand. Hence, the graph is shown in the figure. From the graph, it is clear that y takes all non- negative real values. Hence, the range of function is [ 0 , ∞ ) .

E M B I B E

Understanding ISC Mathematics Class 11 Volume 1>Chapter 2 - Relations and Functions>EXERCISE 2.5

M L Aggarwal Solutions for Chapter: Relations and Functions, Exercise 5: EXERCISE 2.5

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Relations and Functions, Exercise 5: EXERCISE 2.5

Attempt the practice questions on Chapter 2: Relations and Functions, Exercise 5: EXERCISE 2.5 with hints and solutions to strengthen your understanding. Understanding ISC Mathematics Class 11 Volume 1 solutions are prepared by Experienced Embibe Experts.

Questions from M L Aggarwal Solutions for Chapter: Relations and Functions, Exercise 5: EXERCISE 2.5 with Hints & Solutions

EASY

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 2 - Relations and Functions>EXERCISE 2.5>Q 1

Draw the graph of the real function $f (x) = x + 10$ . Hence find their range.

EASY

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 2 - Relations and Functions>EXERCISE 2.5>Q 1

Draw the graph of the real function $f (x) = 1 - 2 x$ . Hence find their range.

EASY

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 2 - Relations and Functions>EXERCISE 2.5>Q 2

Draw the graph of the real function $f (x) = \frac{x^{2} - 4}{x - 2}$ . Hence find their range.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 2 - Relations and Functions>EXERCISE 2.5>Q 2

Draw the graph of the real function $f (x) = \{\begin{cases} x + 3 & , x < 2 \\ 2 x + 1 & , x \geq 2 \end{cases}$ . Hence find their range.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 2 - Relations and Functions>EXERCISE 2.5>Q 3

Draw the graph of the real function $f (x) = \{\begin{cases} x - 1 & , x < 2 \\ 2 x + 3 & , x \geq 2 \end{cases}$ . Hence find their range.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 2 - Relations and Functions>EXERCISE 2.5>Q 3

Draw the graph of the real function $f (x) = \{\begin{cases} - 1 & , x \leq - 1 \\ x & , - 1 < x < 1 \\ 1 & , x \geq 1 \end{cases}$ . Hence find their range.