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


A.\left [ \frac{1}{3},1 \right ]

B.\left [-1, \frac{1}{3} \right ]

C. \left ( -\infty ,-1 \right ]\cup \left [ -\frac{1}{3},1 \right )

D.\left [ -\frac{1}{3},1 \right ]

Answers (1)

Given data: f\left ( x \right )=\frac{1}{1-2\cos x}

 

Now, we know that,

-1 \geq \cos x \geq1

Thus, -1\leq cosx \leq 1

-2 \leq -2cosx \leq 2

\\ -1 \leq 1-2 \cos x<0 \text { or } 0<1-2 \cos x \leq 3 \\\\ -1 \geq \frac{1}{1-2 \cos x}>-\infty \text { or } \infty>\frac{1}{1-2 \cos x} \geq \frac{1}{3} \\\\\ \frac{1}{1-2 \cos x} \in(-\infty,-1] \cup\left[\frac{1}{3}, \infty\right)

Therefore, (c) is the correct option.
 

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infoexpert21

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