The range of f(x)=

Question:

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

(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,

$-1 \leq \cos x \leq 1$

$\Rightarrow-1 \leq-\cos x \leq 1$

$\Rightarrow-2 \leq-2 \cos x \leq 2$

$\Rightarrow-1 \leq 1-2 \cos x \leq 3$     (Adding 1 to each term)

But,

$\cos x \neq \frac{1}{2}$

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

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

$\therefore$ Range of $f(x)=(-\infty,-1] \cup\left[\frac{1}{3}, \infty\right)$

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, ∞)".