Consider the piecewise function,
. . . . .\(\displaystyle f(x)\, =\, \begin{cases}x\, +\, 4&\mbox{for }\, 0\, <\, x\, <\, 2\\0&\mbox{for }\, 2\, <\, x\, <\, 3 \end{cases}\)
The odd periodic extension FO (x) of f (x) is defined as:
. . . . .\(\displaystyle F^O(x)\, =\, \begin{cases} 0&\mbox{for }\, -3\, <\, x\, <\, -2 \\ x\, -\, 4&\mbox{for }\, -2\, <\, x\, <\, 0 \\ x\, +\, 4&\mbox{for }\, 0\, <\, x\, <\, 2 \\ 0&\mbox{for }\, 2\, <\, x\, <\, 3 \end{cases}\)
where FO(x) has a period of 6.
The even periodic extension F e (x) of f (x) is defined as:
. . . . .\(\displaystyle F^e(x)\, =\, \begin{cases}0\, &\mbox{for }\, -3\, <\, x\, <\, -2\\-x\, +\, 4&\mbox{for }\, -2\, <\, x\, <\, 0\\x\, +\, 4&\mbox{for }\, 0\, <\, x\, <\, 2\\0&\mbox{for }\, 2\, <\, x\, <\, 3 \end{cases}\)
where F e(x) has a period of 6.
Does the notation used expressions FO (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?
