Question

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

 

विकल्प

• (a) [1/3, 1]    

•   (b) [−1, 1/3]    

•   (c) (−∞, −1) ∪ [1/3, ∞)   

•    (d) [−1/3, 1]   

Answer 1

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, ∞)".