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Relations and Functions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsRelations and Functions1 Mark
MCQ
The range of $\text{f(x)}=\frac{1}{1-2\cos\text{x}}$ is:
  • A
    $\Big[\frac{1}{3},1\Big]$
  • ✓
    $\Big[-1,\frac{1}{3}\Big]$
  • C
    $\big(-\infty,-1\big)\cup\Big[\frac{1}{3},\infty\Big)$
  • D
    $\Big[-\frac{1}{3},1\Big]$
✓

Answer

Correct option: B.
$\Big[-1,\frac{1}{3}\Big]$
We know that $-1\leq\cos\text{x}\leq1$ for all $\text{x}\in\text{R}$
Now,
$-1\leq\cos\text{x}\leq1$
$\Rightarrow-1\leq\cos\text{x}\leq1$
$\Rightarrow-2\leq-2\cos\text{x}\leq2$
$\Rightarrow-1\leq1-2\cos\text{x}\leq3 ($Adding $1$ ro each term$)$
But,
$\cos\text{x}\neq\frac{1}{2}$
$\Rightarrow1-2\cos\text{x}\in\big[-1,3\big]-\{0\}$
$\Rightarrow\frac{1}{1-2\cos\text{x}}\in\big(-\infty,-1\big]\cap\Big[\frac{1}{3},\infty\Big)$
$\therefore\ \text{Range of }\text{f(x)}=\big(-\infty,-1\big]\cap\Big[\frac{1}{3},\infty\Big)$
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 $\big(-\infty,-1\big]\cap\Big[\frac{1}{3},\infty\Big)$

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