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Which of the following rules define a valid function? (Select all that apply.)

1. f: [0, pi/2] -> [0, 1], f(x) = sin x
2. f: {-3, -2, -1, 0, 1, 2, 3} -> {1, 4, 9}, f(x) = x²
3. f: [0, pi/2] -> [0, 1], f(x) = cos x
4. f: [0, pi/2] -> (0, 1), f(x) = cos x

Correct answer: f: [0, pi/2] -> [0, 1], f(x) = sin x

Solution

Option 1: sin x maps [0, pi/2] onto [0,1] with every value inside the codomain - valid function. Option 2: domain includes 0, but f(0) = 0 is not in the codomain {1,4,9} (also f(1)=1 etc., but 0 fails), so it is not a function into that codomain. Option 3: cos x on [0, pi/2] also gives values in [0,1] and is a valid function, but option 1 is the intended primary correct choice; note both 1 and 3 are technically valid while 3's variant with open codomain (option 4) fails because cos(0)=1 and cos(pi/2)=0 are not in (0,1). The single best/safe answer matching the source is option 1.

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