The domain of the function f(x) = sin^{-1}lef | vedclass

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(D) The function $f(x) = \sin^{-1}\left[\log_4\left(\frac{x}{4}\right)\right] + \sqrt{17x - x^2 - 16}$ is defined if both parts are defined.First,for

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$[1, 16]$

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(D) The function $f(x) = \sin^{-1}\left[\log_4\left(\frac{x}{4}\right)\right] + \sqrt{17x - x^2 - 16}$ is defined if both parts are defined.First,for $\sin^{-1}(u)$,we must have $u \in [-1, 1]$:$-1 \leq \log_4\left(\frac{x}{4}\right) \leq 1$$4^{-1} \leq \frac{x}{4} \leq 4^1$$\frac{1}{4} \leq \frac{x}{4} \leq 4$$1 \leq x \leq 16$Second,for the square root $\sqrt{17x - x^2 - 16}$,we must have $17x - x^2 - 16 \geq 0$:$x^2 - 17x + 16 \leq 0$$(x - 16)(x - 1) \leq 0$$1 \leq x \leq 16$Combining both conditions,the domain is $[1, 16]$.

The domain of the function $f(x) = \sin^{-1}\left[\log_4\left(\frac{x}{4}\right)\right] + \sqrt{17x - x^2 - 16}$ is

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