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: \[ [-15, -12] \cup [12, 15] \]

Answer 2

To determine the domain of the function \( f(x) = \sqrt{9 - \sqrt{x^2 - 144}} \), we must ensure that the expression under both square roots is non-negative, as square roots of negative numbers are undefined in the real number system.

The innermost expression, \(\sqrt{x^2 - 144}\), must satisfy \(9 - \sqrt{x^2 - 144} \geq 0\). This implies:

\(\sqrt{x^2 - 144} \leq 9\)

Squaring both sides gives:

\(x^2 - 144 \leq 81\)

Simplifying gives:

\(x^2 \leq 225\)

The solution to the inequality \(-15 \leq x \leq 15\) comes from the fact that \(x^2 \leq 225\) leads to finding the square root of both sides which gives us the interval \([-15, 15]\).

Moreover, for \(f(x)\) to be defined, the expression under the square root must be non-negative:

\(9 - \sqrt{x^2 - 144} \geq 0\)

\(\sqrt{x^2 - 144} \leq 9\) needs to hold true, leading to \(-15 \leq x \leq 15\).

Additionally, we recognize that \(\sqrt{x^2 - 144}\) is defined if \(x^2 - 144 \geq 0\) which gives \(x^2 \geq 144\).

Solving \(x^2 \geq 144\) gives \(|x| \geq 12\), meaning \((-∞, -12] \cup [12, ∞)\).

The overlap of \([-15, 15]\) and \((-∞, -12] \cup [12, ∞)\) is:

\([-15, -12] \cup [12, 15]\).

Thus, the domain of the function is \([-15, -12] \cup [12, 15]\).