<p>If \(f: [1, \infty) \to [2, \infty)\) is given by \(f(x) = x + \frac{1}{x}\), then \(f^{-1}(x)\) equals</p>
<p>(a) \(\frac{x + \sqrt{x^2 - 4}}{2}\)</p>
<p>(b) \(\frac{x}{2}\)</p>
<p>(c) \(\frac{x - \sqrt{x^2 - 4}}{2}\)</p>
<p>(d) \(1 + \sqrt{x - 4}\)</p>
Step-by-Step Solution
Key Concept: To find the inverse function, set y = f(x) and solve for x in terms of y. The domain of f⁻¹ equals the range of f, and we must choose the solution branch that respects the original domain [1, ∞).
<p><strong>Step 1:</strong> Verify that f is bijective on the given domain.</p><p>Given: f(x) = x + 1/x on [1, ∞) → [2, ∞)</p><p>f'(x) = 1 - 1/x² > 0 for x > 1, so f is strictly increasing and hence injective.</p><p>At x = 1: f(1) = 1 + 1 = 2</p><p>As x → ∞: f(x) → ∞</p><p>Therefore f is bijective with range [2, ∞). ✓</p><p><strong>Step 2:</strong> Set y = f(x) and solve for x.</p><p>y = x + 1/x</p><p>Multiply by x: xy = x² + 1</p><p>Rearrange: x² - xy + 1 = 0</p><p><strong>Step 3:</strong> Apply the quadratic formula.</p><p>x = (y ± √(y² - 4))/2</p><p>For y ≥ 2: y² - 4 ≥ 0, so the discriminant is non-negative. ✓</p><p><strong>Step 4:</strong> Determine which branch to use.</p><p>Since x ∈ [1, ∞), we need x ≥ 1.</p><p>Check x₁ = (y + √(y² - 4))/2:</p><p>For y = 2: x₁ = (2 + √0)/2 = 1 ✓</p><p>For y > 2: Since √(y² - 4) < y (as y² - 4 < y²), we have x₁ > 0. Also, x₁ is increasing in y and equals 1 when y = 2, so x₁ ≥ 1 for all y ≥ 2. ✓</p><p>Check x₂ = (y - √(y² - 4))/2:</p><p>For y = 2: x₂ = (2 - 0)/2 = 1 ✓</p><p>For y = 3: x₂ = (3 - √5)/2 ≈ (3 - 2.236)/2 ≈ 0.382 < 1 ✗</p><p>So x₂ does not satisfy x ≥ 1 for all y ≥ 2.</p><p><strong>Step 5:</strong> Verify the answer.</p><p>If f⁻¹(x) = (x + √(x² - 4))/2, then f(f⁻¹(x)) should equal x.</p><p>Let u = f⁻¹(x) = (x + √(x² - 4))/2</p><p>Then: f(u) = u + 1/u = (x + √(x² - 4))/2 + 2/(x + √(x² - 4))</p><p>Rationalizing: 2/(x + √(x² - 4)) = 2(x - √(x² - 4))/((x + √(x² - 4))(x - √(x² - 4))) = 2(x - √(x² - 4))/(x² - (x² - 4)) = 2(x - √(x² - 4))/4 = (x - √(x² - 4))/2</p><p>So: f(u) = (x + √(x² - 4))/2 + (x - √(x² - 4))/2 = x ✓</p><p><strong>∴ Answer:</strong> a</p>
Correct Answer: a