<p><strong>934.</strong> Find the number of integers not in the domain of \(f(x) = \cos^{-1}\!\left(\dfrac{2-x}{2x}\right)\).</p>
Step-by-Step Solution
Key Concept: The domain of cos⁻¹(u) requires -1 ≤ u ≤ 1. We must solve the compound inequality -1 ≤ (2-x)/(2x) ≤ 1 carefully, considering the sign of the denominator 2x.
<p><strong>Step 1:</strong> For f(x) to be defined, we need: -1 ≤ (2-x)/(2x) ≤ 1 and x ≠ 0</p><p><strong>Step 2:</strong> <u>Left inequality:</u> -1 ≤ (2-x)/(2x)</p><p>Rearrange: (2-x)/(2x) + 1 ≥ 0 → (2-x+2x)/(2x) ≥ 0 → (2+x)/(2x) ≥ 0</p><p>Sign analysis: positive when x ∈ (-∞, -2] ∪ (0, ∞)</p><p><strong>Step 3:</strong> <u>Right inequality:</u> (2-x)/(2x) ≤ 1</p><p>Rearrange: (2-x)/(2x) - 1 ≤ 0 → (2-x-2x)/(2x) ≤ 0 → (2-3x)/(2x) ≤ 0</p><p>Sign analysis: non-positive when x ∈ (-∞, 0) ∪ [2/3, ∞)</p><p><strong>Step 4:</strong> Find intersection of both conditions:</p><p>[(-∞, -2] ∪ (0, ∞)] ∩ [(-∞, 0) ∪ [2/3, ∞)] = (-∞, -2] ∪ [2/3, ∞)</p><p><strong>Step 5:</strong> Integers NOT in domain are those in (-2, 2/3) = {-1, 0}</p><p>∴ <strong>Answer: 2</strong></p>
Correct Answer: 2