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Determine the range of each of the following functions: (i) f: R -> R, f(x) = 5 - 4*sin(x); (ii) f: R -> R, f(x) = 5 - 4*cos²(x); (iii) f: R -> R, f(x) = sin²(x) - 3*sin(x) + 8.

1. (i) [1, 9] (ii) [1, 5] (iii) [6, 12]
2. (i) [-4, 4] (ii) [0, 5] (iii) [5, 11]
3. (i) [1, 9] (ii) [1, 9] (iii) [6, 18]
4. (i) [0, 9] (ii) [1, 5] (iii) [6, 12]

Correct answer: (i) [1, 9] (ii) [1, 5] (iii) [6, 12]

Solution

Each function is built from a bounded trig term. Apply the known bounds and, for the quadratic-in-sin case, examine monotonicity over t in [-1,1].

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