# 4.3: Periodic Functions

- Page ID
- 8355

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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 \(f(x)=\sin(bx)\) 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 2*p*3*p*…. This can easily be seen using the definition of periodicity, which subtracts *p* from the argument

\[ f(x+3p) = f(x+2p) = f(x+p) = f(x).\]

The smallest positive value of *p* for which *f* is periodic is called the (primitive) period of *f*.

Exercise \(\PageIndex{1}\)

What is the primitive period of \(\sin(4x)\)?

**Answer**-
\(\frac{π}{2}\).