Sets, Relations & Functions
Composite functions
Grade 11
Question:
<p>If \(f(x) = \sin^2 x + \sin^2\left(x + \frac{\pi}{3}\right) + \cos x \cdot \cos\left(x + \frac{\pi}{3}\right)\) and \(g\left(\frac{5}{4}\right) = 1\), then \((g \circ f)(x)\) is</p>
<p>(a) a polynomial of degree 2</p>
<p>(b) a constant function</p>
<p>(c) an odd function</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: Simplify f(x) using trigonometric identities (product-to-sum and sum formulas) to reduce it to a constant value, then evaluate the composition g(f(x)) using the given condition on g.
<p><strong>Step 1: Simplify f(x) using trigonometric identities</strong></p><p>Let f(x) = sin²x + sin²(x+π/3) + cos·x·cos(x+π/3)</p><p>Expand sin(x+π/3) = sin x cos(π/3) + cos x sin(π/3) = (1/2)sin x + (√3/2)cos x</p><p>So sin²(x+π/3) = (1/4)sin²x + (√3/2)sin x cos x + (3/4)cos²x</p><p><strong>Step 2: Expand cos(x+π/3)</strong></p><p>cos(x+π/3) = cos x cos(π/3) - sin x sin(π/3) = (1/2)cos x - (√3/2)sin x</p><p>So cos x·cos(x+π/3) = (1/2)cos²x - (√3/2)sin x cos x</p><p><strong>Step 3: Combine all terms</strong></p><p>f(x) = sin²x + (1/4)sin²x + (√3/2)sin x cos x + (3/4)cos²x + (1/2)cos²x - (√3/2)sin x cos x</p><p>= (5/4)sin²x + (5/4)cos²x</p><p>= (5/4)(sin²x + cos²x)</p><p>= <strong>5/4</strong></p><p><strong>Step 4: Evaluate (g∘f)(x)</strong></p><p>Since f(x) = 5/4 for all x, we have (g∘f)(x) = g(f(x)) = g(5/4) = 1</p><p>∴ Answer: <strong>B</strong> (the constant function 1)</p>
Correct Answer: B