<p>Find the domain of the function \(f(x) = \dfrac{1}{\sqrt{|x|^2 - |x| - 6}}\).</p>
Step-by-Step Solution
Key Concept: For the function to be defined, the expression under the square root must be positive (not just non-negative). Substitute t = |x| to reduce |x|² - |x| - 6 > 0 to a quadratic inequality, then backtrack to find x values.
<p><strong>Step 1:</strong> For f(x) to be defined, the expression under the square root must be strictly positive:</p><p>|x|² - |x| - 6 > 0</p><p><strong>Step 2:</strong> Since |x|² = x², substitute t = |x| where t ≥ 0:</p><p>t² - t - 6 > 0</p><p><strong>Step 3:</strong> Factor the quadratic:</p><p>(t - 3)(t + 2) > 0</p><p><strong>Step 4:</strong> Solve the inequality. The roots are t = 3 and t = -2. Since t ≥ 0, we analyze the sign for t ∈ [0, ∞):</p><p>• For t ∈ [0, 3): (t - 3) < 0 and (t + 2) > 0, so product < 0 ✗</p><p>• For t > 3: (t - 3) > 0 and (t + 2) > 0, so product > 0 ✓</p><p>Therefore: t > 3, which means |x| > 3</p><p><strong>Step 5:</strong> Convert back to x:</p><p>|x| > 3 implies x > 3 or x < -3</p><p>∴ Answer: (-∞, -3) ∪ (3, +∞)</p><p><em>Note: The given answer (-∞, -2) ∪ (4, +∞) appears to be from a different problem. The correct domain for this function is (-∞, -3) ∪ (3, +∞).</em></p>
Correct Answer: (-∞, -2) ∪ (4, +∞)