<p>Find \(f(x) = \frac{1}{2}|x+2| - 1\bigg|\frac{1}{2}(x-2)\bigg| + 1\). Hence find the range of \(f(x)\).</p>
Step-by-Step Solution
Key Concept: Break the absolute value function into piecewise parts by analyzing critical points x = -2 and x = 2, then evaluate f(x) on each interval to find minimum and maximum values.
<p><strong>Step 1: Identify critical points</strong></p><p>The function has critical points where expressions inside absolute values equal zero: x = -2 and x = 2.</p><p><strong>Step 2: Construct piecewise function</strong></p><p>For <strong>x < -2:</strong> |x+2| = -(x+2) and |x-2| = -(x-2)</p><p>f(x) = ½(-(x+2)) - 1·(½(-(x-2))) + 1 = -½x - 1 + ½x - 1 + 1 = <strong>-1</strong></p><p>For <strong>-2 ≤ x < 2:</strong> |x+2| = (x+2) and |x-2| = -(x-2)</p><p>f(x) = ½(x+2) - 1·(½(-(x-2))) + 1 = ½x + 1 + ½x - 1 + 1 = <strong>x + 1</strong></p><p>For <strong>x ≥ 2:</strong> |x+2| = (x+2) and |x-2| = (x-2)</p><p>f(x) = ½(x+2) - 1·(½(x-2)) + 1 = ½x + 1 - ½x + 1 + 1 = <strong>3</strong></p><p><strong>Step 3: Analyze the piecewise function</strong></p><p>• For x < -2: f(x) = -1 (constant)</p><p>• For -2 ≤ x < 2: f(x) = x + 1 is linear, increasing from f(-2) = -1 to f(2⁻) = 3</p><p>• For x ≥ 2: f(x) = 3 (constant)</p><p><strong>Step 4: Determine range</strong></p><p>The minimum value is -1 (achieved for x ≤ -2).</p><p>The maximum value is 3 (achieved for x ≥ 2).</p><p>∴ <strong>Range of f(x) = [-1, 3]</strong></p>
Correct Answer: Range of f(x) = [-1, 3]