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RELATIONS AND FUNCTIONS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSRELATIONS AND FUNCTIONS1 Mark
MCQ
The domain of definition of the function $\text{f(x)}=\sqrt{\frac{\text{x}-2}{\text{x}+2}}+\sqrt{\frac{1-\text{x}}{1+\text{x}}}$ is:
  • A
    $\big(-\infty,-2\big]\cap\big[2,-\infty\big)$
  • B
    $\big[-1,1\big]$
  • ✓
    $\phi$
  • D
    None of these.
✓

Answer

Correct option: C.
$\phi$
$\text{f(x)}=\sqrt{\frac{\text{x}-2}{\text{x}+2}}+\sqrt{\frac{1-\text{x}}{1+\text{x}}}$
For f(x) to be defined,
$\text{x}+2\neq0$
$\Rightarrow\text{x}\neq-2\ ...(\text{i})$
And $1+\text{x}\neq0$
$\Rightarrow\text{x}\neq-1\ ...(\text{ii})$
Also, $\frac{\text{x}-2}{\text{x}+2}\geq0$
$\Rightarrow\frac{(\text{x}-2)(\text{x}-2)}{(\text{x}-2)^2}\ge0$
$\Rightarrow(\text{x}-2)(\text{x}+2)\geq0$
$\Rightarrow\text{x}\in(\infty,-2)\cup\big[2,\infty\big)\ ...(\text{iii})$
And $\frac{1-\text{x}}{1+\text{x}}\geq0$
$\Rightarrow\frac{(1-\text{x})(1+\text{x})}{(1+\text{x})^2}\geq0$
$\Rightarrow(1-\text{x})(1+\text{x})\geq0$
$\Rightarrow\text{ x}\in\big(-\infty,-1\big)\cup\big[1,\infty\big)\ ...(\text{iv})$
From (i), (ii), (iii) and (iv) we get
$\text{x }\in\phi$
Thus, domain $(\text{f(x)})=\phi$

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