We first need to define a periodic function. A function is called periodic with period p if f(x + p) = f(x), for all x, even if f is not defined everywhere. A simple example is the function f(x) =\mathop{ sin}\nolimits (bx) which is periodic with period (2π)∕b. Of course it is also periodic with periodic (4π)∕b. In general a function with period p is periodic with period 2p,3p,\mathop{\mathop{…}}. This can easily be seen using the definition of periodicity, which subtracts p from the argument
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f(x + 3p) = f(x + 2p) = f(x + p) = f(x).
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The smallest positive value of p for which f is periodic is called the (primitive) period of f.
Question: What is the primitive period of \mathop{sin}\nolimits (4x)?
Answer: π∕2.