Range of f(x) = (1/1-2 cos x) is

Question

Range of f(x) = (1/1-2 cos x) is (A) [(1/3), 1] (B) [-1, (1/3)] (C) (-∞, -1)∪[ (1/3), ∞] (D) [- (1/3), 1]

Tardigrade Admin · Accepted Answer

Given, f(x) = (1/1-2 cos x) We know that, -1 le cos x le 1 ⇒ -2 le 2cos x le 2 ⇒ -2 le - 2 cos x le 2 ⇒ 1-2 le 1-2 cos x le 2+1 ⇒ -1 le 1 -2 cos x le 3

Q. Range of $f (x) = \frac{1}{1 - 2 cos x}$ is

$[\frac{1}{3}, 1]$

$[- 1, \frac{1}{3}]$

$(- \infty, - 1) \cup [\frac{1}{3}, \infty]$

$[- \frac{1}{3}, 1]$

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

Q. Range of f(x)=1−2cosx1​ is

[31​,1]

[−1,31​]

(−∞,−1)∪[31​,∞]

[−31​,1]