Sets, Relations & Functions
Inverse functions
Grade 11
Question:
<p>Let \(f(x) = (x+2)^2 - 2,\ x \geq -2\). If \(g(x)\) is a function whose graph is reflection of the graph of \(y = f(x)\) in the line \(y = x\), then \(g(x)\) is equal to:</p>
<p>(a) \(-\sqrt{2+x} - 2\)</p>
<p>(b) \(\sqrt{2+x} + 2\)</p>
<p>(c) \(\sqrt{2+x} - 2\)</p>
<p>(d) \(-\sqrt{2+x} + 2\)</p>
Step-by-Step Solution
Key Concept: To find g(x) as the reflection of f(x) in the line y=x, you must find the inverse function f⁻¹(x). The reflection of any function in y=x is precisely its inverse function.
<p><strong>Step 1:</strong> Find the range of f(x). Since f(x) = (x+2)² - 2 with x ≥ -2, the minimum occurs at x = -2: f(-2) = 0 - 2 = -2. As x → ∞, f(x) → ∞. Thus range of f is [-2, ∞).</p><p><strong>Step 2:</strong> To find g(x) = f⁻¹(x), set y = (x+2)² - 2 and solve for x in terms of y:</p><p>y = (x+2)² - 2</p><p>y + 2 = (x+2)²</p><p>±√(y+2) = x + 2</p><p>Since x ≥ -2, we take the positive root: x + 2 = √(y+2)</p><p>x = √(y+2) - 2</p><p><strong>Step 3:</strong> Swap x and y to get g(x) = √(x+2) - 2, where x ≥ -2 (the range of f becomes the domain of g).</p><p><strong>Verification:</strong> f(g(x)) = f(√(x+2) - 2) = ((√(x+2) - 2 + 2)² - 2) = (√(x+2))² - 2 = x + 2 - 2 = x ✓</p><p>∴ Answer: g(x) = √(x+2) - 2, x ≥ -2</p>
Correct Answer: C