Discussion
The domain of the function \(f(x) = \left(\sqrt{8x - x^{2} - 7}\right)^{3/2}\) is
Answer: A
The domain is \([1, 7]\).
The square root needs a non-negative argument:
\[ \begin{aligned} 8x - x^{2} - 7 &\ge 0 \\ x^{2} - 8x + 7 &\le 0 \\ (x - 1)(x - 7) &\le 0 \;\Rightarrow\; 1 \le x \le 7 \end{aligned} \]
Raising the root to the power \(\frac{3}{2}\) keeps it real, since the root is already non-negative. \([3, 7]\) and \([1, 4]\) are only parts of the domain, \([-7, -1]\) has the signs reversed, and \([-3, 3]\) is unrelated.
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