The largest interval lying in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ for which the function $\left[f(x)=4^{-x^{2}}+\cos^{-1}\left(\frac{x}{2}-1\right)+\log (\cos x)\right]$ is defined, is

(D) $\left[0, \frac{\pi}{2}\right)$

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

(C) $\left[-\frac{\pi}{4}, \frac{\pi}{2}\right)$

JEE Main · 2007 · 29 Apr 2007 · Offline · Sets Relations and Functions

The largest interval lying in

(- \frac{π}{2}, \frac{π}{2})

for which the function

[f (x) = 4^{- x^{2}} + cos^{- 1} (\frac{x}{2} - 1) + lo g (cos x)]

is defined, is

Answer: (D)

Every piece of the function must be defined. The factor

4^{- x^{2}}

is fine for all

x

; the term

cos^{- 1} (\frac{x}{2} - 1)

needs

- 1 \leq \frac{x}{2} - 1 \leq 1

, i.e.

0 \leq x \leq 4

; and

lo g (cos x)

needs

cos x > 0

, i.e.

- \frac{π}{2} < x < \frac{π}{2}

. Intersecting these within

(- \frac{π}{2}, \frac{π}{2})

leaves

[0, \frac{π}{2})

as the largest interval on which

f

is defined.

JEE Main · 2007 · 29 Apr 2007 · Offline · Single correct option