<p>If \(f : R \to S\), defined by \(f(x) = \sin x - \sqrt{3}\cos x + 1\), is onto, then the interval of \(S\) is</p>
Step-by-Step Solution
Key Concept: For an onto function, the codomain S must equal the range of f. Find the range of sin x - √3cos x + 1 by converting to single sinusoidal form using R sin(x + φ) technique.
<p><strong>Step 1:</strong> Express f(x) = sin x - √3 cos x + 1 in the form R sin(x + φ) + 1</p><p>For a sin x + b cos x, we use R = √(a² + b²)</p><p>Here a = 1, b = -√3, so R = √(1 + 3) = 2</p><p><strong>Step 2:</strong> Rewrite: sin x - √3 cos x = 2[½ sin x - (√3/2) cos x]</p><p>= 2 sin(x - π/3)</p><p><strong>Step 3:</strong> Therefore: f(x) = 2 sin(x - π/3) + 1</p><p><strong>Step 4:</strong> Since -1 ≤ sin(x - π/3) ≤ 1, we have:</p><p>-2 ≤ 2 sin(x - π/3) ≤ 2</p><p>-1 ≤ 2 sin(x - π/3) + 1 ≤ 3</p><p><strong>Step 5:</strong> Range of f = [-1, 3]. For f to be onto, S = [-1, 3]</p><p>∴ Answer: S = [-1, 3] (or equivalent interval notation)</p>
Correct Answer: D