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Find the value of  \sin ^{-1}e^x

Option: 1

[0,\frac{\pi}{2}]


Option: 2

(0,\frac{\pi}{2}]


Option: 3

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


Option: 4

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


Answers (1)

best_answer

 

 

Domain and range of Inverse Trigonometric Function (Part 1) -

Domain and range of Inverse Trigonometric Function (Part 1)

y = sin-1(x)

The function y = sin(x) is many one so it is not invertible. Now consider the small portion of the function \mathrm{y=\sin x,\;x\in\left [ -\frac{\pi}{2},\frac{\pi}{2}\right ]\;\;and\;\;y=[-1,1]}

 

             


 

Which is strictly increasing, Hence, one-one and inverse is y = sin-1(x)

 

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



 

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e^x >0\\ \text{for }\sin ^{-1}e^x \text{ to be defined }, e^x \leq1\\ 0<e^x\leq 1\\ 0<\sin ^{-1}e^x\leq \frac{\pi}{2}

Posted by

Ritika Harsh

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