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Question
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,
\[ \Rightarrow \frac{1}{1 - 2\cos x} \in ( - \infty , - 1] \cup [\frac{1}{3}, \infty )\]
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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