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Q.
The domain of the function $f(x)=\sin ^{-1}\left[\log _{4}\left(\frac{x}{4}\right)\right]+\sqrt{17 x-x^{2}-16}$ is
TS EAMCET 2019
Solution:
We have,
$f(x)=\sin ^{-1}\left[\log _{4}\left(\frac{x}{4}\right)\right]+\sqrt{17 x-x^{2}-16}$
$f(x)$ is defined of
$\log _{4}\left(\frac{x}{4}\right) \in[-1,1]$ and $17 x-x^{2}-16 \geq 0$
$\therefore 4^{-1} \leq \frac{x}{4} \leq 4$ and $x^{2}-17 x+16 \leq 0$
$\Rightarrow \,1 \leq x \leq 16$ and $(x-16)(x-1) \leq 0$
$\Rightarrow \,1 \leq x \leq 16$ and $1 \leq x \leq 16$
$\therefore $ Domain of $f(x)$ is $[1,16] $