Question

The domain of the function \( f(x) = \left(\sqrt{8x - x^2 - 7}\right)^{\frac{3}{2}} \) is:

• A. [1,7]
• B. [-3,3]
• C. [-7,-1]
• D. [3,7]
• E. [1,4]

Hint:

For \((\sqrt{g(x)})^{3/2}\), the condition is \(g(x) \geq 0\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For the function to be defined, the expression inside the square root must be non-negative.

Step 2: Detailed Explanation:
\[ 8x - x^2 - 7 \geq 0 \implies -x^2 + 8x - 7 \geq 0 \implies x^2 - 8x + 7 \leq 0 \] \[ (x - 1)(x - 7) \leq 0 \implies 1 \leq x \leq 7 \] Thus, domain = \([1, 7]\).

Step 3: Final Answer:
The domain is \([1, 7]\).