Question

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

• A. $\left[\frac{1}{3}, 1\right]$
• B. $\left[-1, \frac{1}{3}\right]$
• C. $\left(-\infty, -1\right)\cup\left[ \frac{1}{3}, \infty\right]$
• D. $\left[- \frac{1}{3}, 1\right]$

Answer 1

Answer: (C)

Given, $f\left(x\right) = \frac{1}{1-2\,cos\,x}$
We know that, $-1 \le cos\, x \le 1$
$\Rightarrow -2 \le 2cos\, x \le 2$
$\Rightarrow -2 \le - 2\,cos\, x \le 2$
$\Rightarrow 1-2 \le 1-2\,cos\, x \le 2+1$
$\Rightarrow -1 \le 1 -2 \,cos\, x \le 3$