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]

Advertisement Remove all ads

#### Solution

We know that −1 ≤ cos*x* ≤ 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,

\[\cos x \neq \frac{1}{2}\]

\[\Rightarrow 1 - 2\cos x \in \left[ - 1, 3 \right] - \left\{ 0 \right\}\]

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

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

∴ 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, ∞)".

Concept: Concept of Functions

Is there an error in this question or solution?

Advertisement Remove all ads

#### APPEARS IN

Advertisement Remove all ads

Advertisement Remove all ads