<p>Find the number of elements contained in the range of the function <strong>f(x) = \left\lfloor \frac{x}{6} \right\rfloor \left\lfloor \frac{−6}{x} \right\rfloor</strong> for <strong>x ∈ (0, 30]</strong> (where <strong>[·]</strong> denotes greatest integer function)</p>
Step-by-Step Solution
Key Concept: We need to analyze the function by breaking the interval (0, 30] into regions where ⌊x/6⌋ and ⌊-6/x⌋ take constant integer values, then compute the product for each region.
<p><strong>Step 1: Analyze ⌊x/6⌋ on (0, 30]</strong></p><p>For x ∈ (0, 30]:<br/>• x ∈ (0, 6): ⌊x/6⌋ = 0<br/>• x ∈ [6, 12): ⌊x/6⌋ = 1<br/>• x ∈ [12, 18): ⌊x/6⌋ = 2<br/>• x ∈ [18, 24): ⌊x/6⌋ = 3<br/>• x ∈ [24, 30]: ⌊x/6⌋ = 4</p><p><strong>Step 2: Analyze ⌊-6/x⌋ on (0, 30]</strong></p><p>Since -6/x is negative for x > 0, we need ⌊-6/x⌋:<br/>• x ∈ (0, 1]: -6/x ∈ [-∞, -6), so ⌊-6/x⌋ ≤ -6<br/>• x ∈ (1, 2]: -6/x ∈ [-6, -3), so ⌊-6/x⌋ ∈ {-6, -5, -4}<br/>• x ∈ (2, 3]: -6/x ∈ [-3, -2), so ⌊-6/x⌋ ∈ {-3, -2}<br/>• x ∈ (3, 6]: -6/x ∈ [-2, -1), so ⌊-6/x⌋ = -2<br/>• x ∈ (6, 30]: -6/x ∈ [-1, 0), so ⌊-6/x⌋ = -1</p><p><strong>Step 3: Calculate f(x) for each region</strong></p><p>• x ∈ (0, 1]: ⌊x/6⌋ = 0, so f(x) = 0·⌊-6/x⌋ = 0</p><p>• x ∈ (1, 2]: ⌊x/6⌋ = 0, so f(x) = 0</p><p>• x ∈ (2, 3]: ⌊x/6⌋ = 0, so f(x) = 0</p><p>• x ∈ (3, 6]: ⌊x/6⌋ = 0, so f(x) = 0</p><p>• x ∈ (6, 12): ⌊x/6⌋ = 1, ⌊-6/x⌋ = -1, so f(x) = 1·(-1) = -1</p><p>• x ∈ [12, 18): ⌊x/6⌋ = 2, ⌊-6/x⌋ = -1, so f(x) = 2·(-1) = -2</p><p>• x ∈ [18, 24): ⌊x/6⌋ = 3, ⌊-6/x⌋ = -1, so f(x) = 3·(-1) = -3</p><p>• x ∈ [24, 30]: ⌊x/6⌋ = 4, ⌊-6/x⌋ = -1, so f(x) = 4·(-1) = -4</p><p><strong>Step 4: Identify the range</strong></p><p>The function takes values: {0, -1, -2, -3, -4}</p><p>Therefore, the range contains exactly 5 distinct elements.</p><p><strong>∴ Answer: 5</strong></p>
Correct Answer: 5