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  • What is the domain of the function <img alt="\mathrm{f(x)=\sin^{-1}\left ( \log_{4}x^2 \right )}" src="http://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bf%28x%29%3D%5Csin%5E%7B-1%7D%5

What is the domain of the function \mathrm{f(x)=\sin^{-1}\left ( \log_{4}x^2 \right )} ?

Option: 1

[-2,-\frac{1}{2}]\cup[\frac{1}{2},2]


Option: 2

[-2,-\frac{1}{2}]


Option: 3

[\frac{1}{2},2]


Option: 4

[-2,2]


Answers (1)

As we have learnt in

Inverse Trigonometric Function

y = sin-1(x)

\mathrm{Domain\;is\;[-1,1]\;\;and\;\;Range\;\;is\;\;\left [ \frac{-\pi}{2},\frac{\pi}{2} \right ]}

 

Now,

For f(x) to be defined

 -1\leq\log_4x^2\leq1     ........(i)         and         x2 > 0      .........(ii)

 From (ii); x\in\mathbb{R}-\left\{0\right\}

From (i), we have

\\4^{-1}\leq x^2 \leq 4^1\\\Rightarrow \frac{1}{4}\leq x^2\leq4\\\Rightarrow -2\leq x\leq\frac{-1}{2} \;\;\;or\;\;\;\frac{1}{2}\le \:x\le \:2

 

Posted by

Sumit Saini

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