<p>The domain of the function \(f(x) = \dfrac{1}{\sqrt{|x| - x}}\) is</p>
<p>\((0, \infty)\)</p>
<p>\((-\infty, 0)\)</p>
<p>\((-\infty, \infty) - \{0\}\)</p>
<p>\((-\infty, \infty)\)</p>
Step-by-Step Solution
Key Concept: Recognize that |x| - x equals 0 when x ≥ 0 and equals -2x when x < 0. For the function to be defined, we need |x| - x > 0, which only occurs for negative values of x.
<p><strong>Step 1:</strong> Analyze the denominator constraint. For f(x) to be defined, we need √(|x| - x) to be real and non-zero, so |x| - x > 0.</p><p><strong>Step 2:</strong> Consider two cases:</p><p><strong>Case 1 (x ≥ 0):</strong> |x| = x, so |x| - x = x - x = 0. This does NOT satisfy |x| - x > 0.</p><p><strong>Case 2 (x < 0):</strong> |x| = -x, so |x| - x = -x - x = -2x. Since x < 0, we have -2x > 0. ✓</p><p><strong>Step 3:</strong> Therefore, the domain is all negative real numbers: x ∈ (-∞, 0) or {x ∈ ℝ : x < 0}.</p><p>∴ Answer: B</p>
Correct Answer: B