# Even and odd periodic extensions

#### msegling

Consider the piecewise function,

$f(x)=\left\{\begin{array}{r|r}x+4 & 0 < x < 2\\0 & 2 < x < 3\\\end{array}\right\}$

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The odd periodic extension $F^O (x)$ of $f(x)$ is defined as:

$F^O (x)=\left\{\begin{array}{r|r}0 & -3 < x < -2\\x-4 & -2 < x < 0\\x+4 & 0 < x < 2\\0 & 2 < x < 3\\\end{array}\right\}$

where $F^O (x)$ has a period of 6.

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The odd periodic extension $F^O (x)$ of $f(x)$ is defined as:

$F^e (x)=\left\{\begin{array}{r|r}0 & -3 < x < -2\\-x+4 & -2 < x < 0\\x+4 & 0 < x < 2\\0 & 2 < x < 3\\\end{array}\right\}$

where $F^e (x)$ has a period of 6.

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Does the notation used expressions $F^O (x)$ and $F^e (x)$ represent periodic extensions of $f (x)$ that repeats indefinitely every six units?

What other notation can be used to present a periodic function of a piecewise function?

#### mathman

Forum Staff
By definition a periodic expression of period 6 repeats every 6 units. The notation you use is as good as any.

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