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Find the range of f(x) = sgn[(sin² x + 2 sin x + 4)/(sin² x + 2 sin x + 3)], where sgn denotes the signum function.

1. (A) {-1, 0, 1}
2. (B) {-1, 0}
3. (C) {1}
4. (D) {0, 1}

Correct answer: (C) {1}

Solution

Let t = sin x. Numerator = (t+1)² + 3 > 0 and denominator = (t+1)² + 2 > 0 for all x. Their ratio is always positive, so the signum is always +1.

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