Find the domain of the function given by: f ( x ) = 1 1 - cos x .

Given the function f ( x ) = 1 1 - cos x We know that - 1 ≤ cos x ≤ 1 ⇒ 1 ≥ - cos x ≥ - 1 ⇒ 1 + 1 ≥ 1 - cos x ≥ - 1 + 1 ⇒ 2 ≥ 1 - cos x ≥ 0 ⇒ 0 ≤ 1 - cos x ≤ 2 For real value of domain 1 - cos x = 0 ⇒ cos x = 1 ⇒ cos x = cos 0 ⇒ x = 2 n π ∀ n ∈ Z ∵ cos x = cos α ⇒ x = 2 n π ± α The above part should be removed from real numbers Hence, the domain of f = R - { 2 n π , n ∈ Z } .

Find the domain of the function given by: f ( x ) = x 3 - x + 3 x 2 - 1 .

Given the function f ( x ) = x 3 - x + 3 x 2 - 1 Here, f ( x ) is only defined if denominator is not equal to zero i.e., x 2 - 1 ≠ 0 ⇒ ( x - 1 ) ( x + 1 ) ≠ 0 ∴ x ≠ 1 , x ≠ - 1 Hence, the domain of f = R - { - 1 , 1 } .

Find the domain of the function given by: f ( x ) = 3 x 2 x - 8 .

Given the function f ( x ) = 3 x 2 x - 8 Here, f ( x ) is only defined if denominator is not equal to zero i.e., 2 x - 8 ≠ 0 ⇒ 2 x ≠ 8 ⇒ x ≠ 4 Hence, the domain f ( x ) = 3 x 2 x - 8 is R - { 4 } .

Find the domain and range of the function f ( x ) = 1 x - 5 .

It is given that f ( x ) = 1 x - 5 We have to find the domain and range of f ( x ) Domain of f ( x ) is set of values x where f ( x ) is well-defined. For x = 5 , f ( x ) = 1 5 - 5 = 1 0 ⇒ f ( x ) is not well-defined at x = 5 Clearly that f ( x ) is real when x - 5 > 0 i.e., x > 5 Thus, the domain of the given function is ( 5 , ∞ ) . Range of f ( x ) is the set of values y such that f ( x ) = y Consider, f ( x ) = y = 1 x - 5 ⇒ x - 5 = 1 y ⇒ x - 5 = 1 y 2 ⇒ x = 1 y 2 + 5 For x ∈ ( 5 , ∞ ) , we have y ∈ R + . Hence, the range of f ( x ) is R + .

E M B I B E

NCERT Exemplar Mathematics - Class 11>Chapter 2 - Relations and Functions>EXERCISE

NCERT Solutions for Chapter: Relations and Functions, Exercise 1: EXERCISE

Author:NCERT

NCERT Mathematics Solutions for Exercise - NCERT Solutions for Chapter: Relations and Functions, Exercise 1: EXERCISE

Attempt the free practice questions on Chapter 2: Relations and Functions, Exercise 1: EXERCISE with hints and solutions to strengthen your understanding. NCERT Exemplar Mathematics - Class 11 solutions are prepared by Experienced Embibe Experts.

Questions from NCERT Solutions for Chapter: Relations and Functions, Exercise 1: EXERCISE with Hints & Solutions

EASY

11th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 11>Chapter 2 - Relations and Functions>EXERCISE>Q 12

Let $f$ and $g$ be real functions defined by $f (x) = 2 x + 1$ and $g (x) = 4 x - 7$ .

For what real numbers $x$ , $f (x) < g (x)$ ?

EASY

11th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 11>Chapter 2 - Relations and Functions>EXERCISE>Q 15

Find the value of $x$ for which the functions $f (x) = 3 x^{2} - 1$ and $g (x) = 3 + x$ are equal?

MEDIUM

11th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 11>Chapter 2 - Relations and Functions>EXERCISE>Q 16

Is $g = {(1, 1), (2, 3), (3, 5), (4, 7)}$ a function? Justify. If this is described by the relation, $g (x) = α x + β$ then what values should be assigned to $α$ and $β$ ?