# The Range of F ( X ) = 1 1 − 2 Cos X is (A) [1/3, 1] (B) [−1, 1/3] (C) (−∞, −1) ∪ [1/3, ∞) (D) [−1/3, 1] - Mathematics

MCQ

The range of  $f\left( x \right) = \frac{1}{1 - 2\cos x}$ is

#### Options

• (a) [1/3, 1]

•   (b) [−1, 1/3]

•   (c) (−∞, −1) ∪ [1/3, ∞)

•    (d) [−1/3, 1]

#### Solution

We know that −1 ≤ cosx ≤ 1 for all x ∈ R.

Now,

$- 1 \leq \cos x \leq 1$
$\Rightarrow - 1 \leq - \cos x \leq 1$
$\Rightarrow - 2 \leq - 2\cos x \leq 2$
$\Rightarrow - 1 \leq 1 - 2\cos x \leq 3 \left( \text{ Adding 1 to each term } \right)$

But,

$\cos x \neq \frac{1}{2}$
$\Rightarrow 1 - 2\cos x \in \left[ - 1, 3 \right] - \left\{ 0 \right\}$
$\Rightarrow \frac{1}{1 - 2\cos x} \in ( - \infty , - 1] \cup [\frac{1}{3}, \infty )$
∴ Range of f(x) = (−∞, −1] ∪[ $\frac{1}{3}$

Disclaimer: The range of the function does not matches with either of the given options. The range matches with option (c) if it is given as "(−∞, −1] ∪ [1/3, ∞)".

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 3 Functions
Q 45 | Page 45