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 need to find the inverse function f⁻¹(x). The reflection of a function in y = x is precisely its inverse function.
<p><strong>Step 1:</strong> Recognize that reflection of y = f(x) in the line y = x gives y = f⁻¹(x), so g(x) = f⁻¹(x).</p><p><strong>Step 2:</strong> Find the inverse of f(x) = (x+2)² - 2 for x ≥ -2. Let y = (x+2)² - 2, then swap and solve for y:</p><p>x = (y+2)² - 2</p><p>x + 2 = (y+2)²</p><p>√(x+2) = |y+2|</p><p><strong>Step 3:</strong> Since x ≥ -2 in the original function, the range of f is y ≥ -2. Thus y + 2 ≥ 0, so |y+2| = y+2.</p><p>√(x+2) = y + 2</p><p>y = √(x+2) - 2</p><p><strong>Step 4:</strong> Therefore, g(x) = √(x+2) - 2 with domain x ≥ -2 (the original range of f).</p><p>∴ Answer: C</p>
Correct Answer: C