Question

The domain of the real-valued function \( f(x) = \sqrt{9 - \sqrt{x^2 - 144}} \) is

• A. \([-15, -12] \cup [12, 15]\)
• B. \((-\infty, -12] \cup [12, \infty)\)
• C. \([-15, 15]\)
• D. \([-12, 12]\)

Hint:

For functions with nested square roots, work step by step, ensuring each expression inside a square root is non-negative.

Answer 1

The Correct Option is A

Step 1: Identifying conditions for real values For \( f(x) \) to be defined, the expression inside the outer square root must be non-negative: \[ 9 - \sqrt{x^2 - 144} \geq 0 \] which simplifies to: \[ \sqrt{x^2 - 144} \leq 9 \]
Step 2: Squaring both sides Squaring both sides results in: \[ x^2 - 144 \leq 81 \] Rearranging: \[ x^2 \leq 225 \]
Step 3: Determining \( x \) Taking square roots: \[ -15 \leq x \leq 15 \]
Step 4: Inner Square Root Constraint The inner square root requires: \[ x^2 - 144 \geq 0 \] which means: \[ x^2 \geq 144 \] This gives: \[ x \leq -12 \quad \text{or} \quad x \geq 12 \]
Final Answer: The intersection of these conditions results in: \[ [-15, -12] \cup [12, 15] \]