The largest interval lying in \left ( \frac{-\pi }{2} ,\frac{\pi }{2}\right ) for which the function,

f\left ( x \right )= 4^{-x^{2}}+\cos ^{-1}\left ( \frac{x}{2} -1\right )+\log \left ( \cos x \right ), is defined, is

  • Option 1)

  • Option 2)

  • Option 3)

    \left [ 0,\pi \right ]

  • Option 4)

    \left ( -\frac{\pi}{2} ,\frac{\pi}{2}\right )

 

Answers (1)

As we learnt in 

Domains and Ranges of Inverse Trigonometric Functions -

For \cos ^{-1}x

Domain \epsilon \left [ -1, 1 \right ]

Range \epsilon \left [ 0, \pi \right ]

-

 

 f\left ( x \right ) = 4^{-x^2} + \cos ^{-1}\left ( \frac{x}{2}-1 \right ) + log \cos x

Now 4^-^{x^2} always differed

\cos ^{-1}\left ( \frac{x}{2}-1 \right )   is  differed if  -1 \leq \frac{x}{2}-1\leq 1

\Rightarrow 0\leq \frac{x}{2}\leq 2

\Rightarrow 0\leq x\leq 4 ...........\left ( 1 \right )

log  \left ( \cos x \right )  is difred if \cos x> 0

x\varepsilon \left ( -\frac{\pi }{2}, \frac{\pi }{2} \right )................\left ( 2 \right )

Combining \left ( 1 \right ) and \left ( \left 2 \right \right )

x\varepsilon \left [ 0, \frac{\pi }{2} \right ]


Option 1)

Incorrect

Option 2)

Correct

Option 3)

\left [ 0,\pi \right ]

Incorrect

Option 4)

\left ( -\frac{\pi}{2} ,\frac{\pi}{2}\right )

Incorrect

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