Sets, Relations & Functions
Range of functions involving modulus
Grade 11
<p>Find all values of \(f(x)\) for which \(f(x) = x + \sqrt{x^2}\).</p>
Step-by-Step Solution
Key Concept: Recognize that √(x²) = |x|, not x. The function becomes f(x) = x + |x|, which equals 2x when x ≥ 0 and 0 when x < 0. The range is all non-negative real numbers.
<p><strong>Step 1:</strong> Simplify √(x²) using the definition: √(x²) = |x|</p><p><strong>Step 2:</strong> Write f(x) = x + |x|</p><p><strong>Step 3:</strong> Case analysis on the sign of x:</p><ul><li><strong>When x ≥ 0:</strong> |x| = x, so f(x) = x + x = 2x</li><li><strong>When x < 0:</strong> |x| = -x, so f(x) = x + (-x) = 0</li></ul><p><strong>Step 4:</strong> Determine the range:</p><ul><li>For x ≥ 0: f(x) = 2x ranges from 2(0) to 2(∞), giving [0, ∞)</li><li>For x < 0: f(x) = 0, which contributes only the value 0</li></ul><p><strong>Step 5:</strong> Combine both cases to get the complete range.</p><p>∴ Answer: [0, ∞)</p>
Correct Answer: [0, ∞)