Step-by-Step Solution
Key Concept: To find the inverse of f(x) = |x|, we must first recognize that |x| is NOT a one-to-one function over all real numbers, so its inverse doesn't exist on ℝ. However, if we restrict the domain to [0, ∞), then f(x) = |x| = x becomes one-to-one, and we can find the inverse by solving y = x for x in terms of y.
<p><strong>Step 1:</strong> Check if f(x) = |x| has an inverse.</p><p>For an inverse to exist, f must be one-to-one (injective) on its domain. The function f(x) = |x| is NOT one-to-one on ℝ because f(2) = f(-2) = 2.</p><p><strong>Step 2:</strong> Restrict the domain appropriately.</p><p>If we restrict f to domain [0, ∞), then f(x) = |x| = x, which is one-to-one and onto [0, ∞). Now we can find the inverse.</p><p><strong>Step 3:</strong> Find the inverse function.</p><p>Let y = f(x) = x (for x ≥ 0). Then x = y, so f⁻¹(y) = y. Therefore f⁻¹(x) = x for x ≥ 0.</p><p><strong>Step 4:</strong> Express in terms of |x|.</p><p>Since the inverse is only defined for x ≥ 0 (the range of |x|), we can write f⁻¹(x) = √x for x ≥ 0. In the form given in the options, this is written as √|x|, which equals √x when x ≥ 0.</p><p><strong>Verification:</strong> f(f⁻¹(x)) = f(√|x|) = |√|x|| = √|x| = x ✓</p><p><strong>∴ Answer:</strong> A</p>
Correct Answer: A