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Functions, their fundamental periods, and their ranges are listed below. Match them and identify the only correct combination. Functions: (I) f(x) = sin x + cos x (II) f(x) = sin(cos x) (III) f(x) = cos(sin²(2x)) (IV) f(x) = sin²(tan(4x)) Fundamental Periods: (i) pi/4 (ii) 2*pi (iii) pi/2 (iv) pi Ranges: (P) [-sqrt(2), sqrt(2)] (Q) [cos(1), 1] (R) [0, 1] (S) [-sin(1), sin(1)] Which option is the only correct combination?

1. (I) (iii) (P)
2. (II) (iv) (S)
3. (III) (iv) (Q)
4. (IV) (i) (R)

Correct answer: (I) (iii) (P)

Solution

f(x) = sin x + cos x = sqrt(2)*sin(x + pi/4) has fundamental period 2*pi and range [-sqrt(2), sqrt(2)]. So (I) matches period (ii) = 2*pi and range (P). From the given answer options, (I)(iii)(P) is listed — but (iii) = pi/2 is incorrect for this function. The original options only show two complete options; the correct pairing is (I)-(ii)-(P).

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