<p>First draw the graph of \([x] + [y] = 3\), where \([\cdot]\) denotes the greatest integer function. Then find the area of the graph of \(\lfloor |x| \rfloor + \lfloor |y| \rfloor = 3\).</p>
Step-by-Step Solution
Key Concept: The equation [|x|] + [|y|] = 3 creates a symmetric region across both axes. By analyzing integer pairs (a,b) where a + b = 3 with a,b ≥ 0, each pair defines a rectangular strip, and total area equals 4 times the sum of widths due to four-quadrant symmetry.
<p><strong>Step 1:</strong> Understand the equation [|x|] + [|y|] = 3. For non-negative integers a and b where a + b = 3, we have a ∈ {0,1,2,3} and correspondingly b ∈ {3,2,1,0}.</p><p><strong>Step 2:</strong> For each pair (a,b), the condition [|x|] = a means a ≤ |x| < a+1, creating a vertical strip of width 1 (from |x| = a to |x| = a+1). Similarly [|y|] = b creates a horizontal strip of height 1.</p><p><strong>Step 3:</strong> Due to symmetry with |x| and |y|, analyze the first quadrant (x ≥ 0, y ≥ 0):<br/>• (0,3): 0 ≤ x < 1, 3 ≤ y < 4 → area = 1 × 1 = 1<br/>• (1,2): 1 ≤ x < 2, 2 ≤ y < 3 → area = 1 × 1 = 1<br/>• (2,1): 2 ≤ x < 3, 1 ≤ y < 2 → area = 1 × 1 = 1<br/>• (3,0): 3 ≤ x < 4, 0 ≤ y < 1 → area = 1 × 1 = 1<br/>Total in first quadrant = 4</p><p><strong>Step 4:</strong> By four-fold symmetry across both axes, total area = 4 × 4 = <strong>16</strong></p>
Correct Answer: 16