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  • Find the value of x for which is a well-defined function.&nbsp

Find the value of x for which \sin ^{-1}(\log_e{x})is a well-defined function. 

Option: 1

(e^{-1}, e)


Option: 2

[e^{-1}, e]


Option: 3

[e^{-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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\frac{-\pi}{2}\leq \sin ^{-1}(\log_e{x})\leq\frac{\pi}{2}\\ -1\leq\log_e{x}\leq1\\ e^{-1} \leq x \leq e

Posted by

manish painkra

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