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A function g(x) meets all of these conditions: (i) its domain is (-inf, inf); (ii) its range is [-1, 7]; (iii) it is periodic with period pi; (iv) g(2) = 3. Which of the following could be a valid form of g(x)?

1. g(x) = 3 + 4 sin(n*pi + 2x - 4), n in I
2. g(x) = piecewise: 3 if x = n*pi; 3 + 4 sin x if x != n*pi
3. g(x) = 3 + 4 cos(n*pi + 2x - 4), n in I
4. g(x) = 3 - 8 sin(n*pi + 2x - 4), n in I

Correct answer: g(x) = 3 + 4 sin(n*pi + 2x - 4), n in I

Solution

Range [-1,7] means midline 3 and amplitude 4, so the form 3 + 4 sin(...) fits. Period pi requires the coefficient of x to be 2 (since 2 pi / 2 = pi). At x = 2: argument = n*pi + 4 - 4 = n*pi, sin(n*pi) = 0, so g(2) = 3. All conditions are satisfied by the first option. The cos option fails g(2) = 3 (cos(n*pi) = +-1), and the -8 sin option gives the wrong range.

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