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Classify each of the following as even, odd, or neither: (i) f(x) = x²*sin x, (ii) f(x) = sin x - cos x, (iii) f(x) = (e^x + e^(-x))/2. Which option correctly identifies the odd function(s)?

1. (i) only
2. (ii) only
3. (iii) only
4. (i) and (iii)

Correct answer: (i) only

Solution

(i) x² is even and sin x is odd, so their product is odd. (ii) sin x is odd but cos x is even, so the combination is neither. (iii) (e^x + e^(-x))/2 = cosh x is even. Thus only (i) is odd.

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