If a function is defined by an implicit equation \(2|x| + 2|y| = 2\), then -
(A) Domain of function is singleton
(B) Range of function is singleton
(C) graph of the function intersects the line <span class="math-inline">y = x</span>
(D) maximum value of function is <span class="math-inline">2</span>
Step-by-Step Solution
Key Concept: The equation 2|x| + 2|y| = 2 represents a geometric figure (diamond/square) in the coordinate plane. We need to analyze whether this defines a function and examine its domain, range, and properties.
<p><strong>Step 1: Simplify the equation</strong></p><p>2|x| + 2|y| = 2</p><p>Dividing by 2: |x| + |y| = 1</p><p><strong>Step 2: Analyze the curve in different quadrants</strong></p><p>• First quadrant (x ≥ 0, y ≥ 0): x + y = 1, so y = 1 - x</p><p>• Second quadrant (x ≤ 0, y ≥ 0): -x + y = 1, so y = 1 + x</p><p>• Third quadrant (x ≤ 0, y ≤ 0): -x - y = 1, so y = -1 - x</p><p>• Fourth quadrant (x ≥ 0, y ≤ 0): x - y = 1, so y = x - 1</p><p><strong>Step 3: Determine domain and range</strong></p><p>The curve forms a diamond (square rotated 45°) with vertices at (1,0), (0,1), (-1,0), (0,-1).</p><p>Domain: [-1, 1] (not singleton)</p><p>Range: [-1, 1] (not singleton)</p><p>This rules out options A and B.</p><p><strong>Step 4: Check if the graph intersects y = x</strong></p><p>The line y = x passes through the origin with slope 1.</p><p>In the first quadrant: y = 1 - x intersects y = x when 1 - x = x, giving x = 1/2, y = 1/2. ✓</p><p>In the third quadrant: y = -1 - x intersects y = x when -1 - x = x, giving x = -1/2, y = -1/2. ✓</p><p>The graph intersects y = x at points (1/2, 1/2) and (-1/2, -1/2).</p><p><strong>Step 5: Check maximum value</strong></p><p>The maximum value of y is 1 (at vertex), not 2. This rules out option D.</p><p><strong>∴ Answer:</strong> C</p>
Correct Answer: C