Question

The domain of the function $f(x)={\sin }^{-1}\left[{\log }_4\left(\frac{x}{4}\right)\right]+\sqrt{17x-x^2-16}$ is

• A. $[-1,1]$
• B. $[1,4]$
• C. $(0,16]$
• D. $[1,16]$

Answer 1

Answer: (D)

We have,
$f(x)={\sin }^{-1}\left[{\log }_4\left(\frac{x}{4}\right)\right]+\sqrt{17x-x^2-16}$
$f(x)$ is defined of
${\log }_4\left(\frac{x}{4}\right)\in [-1,1]$ and $17x-x^2-16\ge 0$
$\therefore 4^{-1}\le \frac{x}{4}\le 4$ and $x^2-17x+16\le 0$
$\Rightarrow 1\le x\le 16$ and $(x-16)(x-1)\le 0$
$\Rightarrow 1\le x\le 16$ and $1\le x\le 16$
$\therefore$ Domain of $f(x)$ is $[1,16]$