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Find the fundamental period (if the function is periodic) of each: (i) f(x) = e^(ln(cos x) + tan³ x); (ii) f(x) = e^(x - [x]) where [.] is the greatest integer function; (iii) f(x) = |sin(x/2)| + |cos(x/2)|.

1. (i) 2*pi; (ii) 1; (iii) pi
2. (i) pi; (ii) 1; (iii) 2*pi
3. (i) 2*pi; (ii) 2; (iii) pi/2
4. (i) not periodic; (ii) 1; (iii) 2*pi

Correct answer: (i) 2*pi; (ii) 1; (iii) pi

Solution

Simplify each expression first. The fundamental period is the smallest positive T with f(x+T)=f(x) for all valid x.

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