Question

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

(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,

$$\begin{gathered} -1\le \cos x\le 1 \\ \Rightarrow -1\le -\cos x\le 1 \\ \Rightarrow -2\le -2\cos x\le 2 \\ \Rightarrow -1\le 1-2\cos x\le 3\left(\mathrm{Adding}1\mathrm{to}\mathrm{each}\mathrm{term}\right) \end{gathered}$$

But,

$\cos x\ne \frac{1}{2}$
$$\begin{gathered} \Rightarrow 1-2\cos x\in \left[-1,3\right]-\left\{0\right\} \\ \Rightarrow \frac{1}{1-2\cos x}\in (-\infty ,-1]\cup [\frac{1}{3},\infty ) \end{gathered}$$

∴ 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, ∞)".