Question

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

• A. $[0,\pi ]$
• B. $(\frac{-\pi }{2},\frac{\pi }{2})$
• C. $[-\frac{\pi }{4},\frac{\pi }{2})$
• D. $[0,\frac{\pi }{2})$

Answer 1

Answer: (D)

$[0,\frac{\pi }{2})$

For given function to be defined $-1\le \frac{x}{2}-1\le 1$ and $\cos x>0$
$\Rightarrow 0\le \frac{x}{2}\le 2$ and $\frac{-\pi }{2}\le x<\frac{\pi }{2}$
$\Rightarrow 0\le x\le 4$ and $\frac{-\pi }{2}\le x<\frac{\pi }{2}$
Taking common of both we get $x\in [0,\frac{\pi }{2})$