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

D Divya Saini

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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