Choose the correct answers: Range of f(x)=1 1-2 is. - Vidyadip

MCQ

Choose the correct answers:
Range of $\text{f(x)}=\frac{1}{1-2\cos\text{x}}$ is.

A
$\Big[\frac{1}{3}, 1\Big]$
B
$\Big[-1, \frac{1}{3}\Big]$
✓
$(-\infty, -1]\cup\Big[\frac{1}{3},\infty\Big)$
D
$\Big[-\frac{1}{3}, 1\Big]$

✓

Answer

Correct option: C.

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

We know that, $-1\leq-\cos\text{x}\leq1$
$\Rightarrow-1\leq-\cos\text{x}\leq1$
$\Rightarrow-2\leq-2\cos\text{x}\leq2$
$\Rightarrow-1\leq-2\cos\text{x}\leq3$
Now $\text{f(x)}=\frac{1}{1-2\cos\text{x}}$ is defined if
$-1\leq-2\cos\text{x}\leq0$ or $0<1-2\cos\text{x}\leq3$
$\Rightarrow-1\geq\frac{1}{1-2\cos\text{x}}>-\infty$ or $\infty>\frac{1}{1-2\cos\text{x}}\geq\frac{1}{3}$
$\Rightarrow\frac{1}{1-2\cos\text{x}}\in(-\infty, -1]\cup\Big[\frac{1}{3},\infty\Big)$

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