The domain of the function f(x)= ^ -1 [2 x^ 2 -3 ]+ _ 2 ( _ 1 2 (… - Vidyadip

MCQ

The domain of the function $f(x)=\sin ^{-1}\left[2 x^{2}-3\right]+\log _{2}\left(\log _{\frac{1}{2}}\left(x^{2}-5 x+5\right)\right)$ where $[ t ]$ is the greatest integer function, is.

A
$\left(-\sqrt{\frac{5}{2}}, \frac{5-\sqrt{5}}{2}\right)$
B
$\left(\frac{5-\sqrt{5}}{2}, \frac{5+\sqrt{5}}{2}\right)$
✓
$\left(1, \frac{5-\sqrt{5}}{2}\right)$
D
$\left[1, \frac{5+\sqrt{5}}{2}\right)$

✓

Answer

Correct option: C.

$\left(1, \frac{5-\sqrt{5}}{2}\right)$

c
$f(x)=\sin ^{-1}\left[2 x^{2}-3\right]+\log _{2}\left(\log _{\frac{1}{2}}\left(x^{2}-5 x+5\right)\right)$

$P_{1}:-1 \leq\left[2 x^{2}-3\right]<1$

$\Rightarrow-1 \leq 2 x ^{2}-3<2$

$\Rightarrow 2<2 x ^{2}<5$

$\Rightarrow 1< x ^{2}<\frac{5}{2}$

$\Rightarrow P _{1}: x \in\left(-\sqrt{\frac{5}{2}},-1\right) \cup\left(1, \sqrt{\frac{5}{2}}\right)$

$P_{2}: x^{2}-5 x+5>0$

$\Rightarrow\left( x -\left(\frac{5-\sqrt{5}}{2}\right)\right)\left( x -\left(\frac{5+\sqrt{5}}{2}\right)\right)>0$

$P_{3}: \log _{\frac{1}{2}}\left(x^{2}-5 x+5\right)>0$

$x ^{2}-5 x -5<1$

$x ^{2}-5 x +4<0$

$P _{3}: x \in(1,4)$

So, $P _{1} \cap P _{2} \cap P _{3}=\left(1, \frac{5-\sqrt{5}}{2}\right)$

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