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November 9th, 2016, 12:27 PM  #1 
Newbie Joined: Nov 2016 From: South Africa Posts: 2 Thanks: 0  Even and odd periodic extensions
Consider the piecewise function, $f(x)=\left\{\begin{array}{rr}x+4 & 0 < x < 2\\0 & 2 < x < 3\\\end{array}\right\}$  The odd periodic extension $F^O (x)$ of $f(x)$ is defined as: $F^O (x)=\left\{\begin{array}{rr}0 & 3 < x < 2\\x4 & 2 < x < 0\\x+4 & 0 < x < 2\\0 & 2 < x < 3\\\end{array}\right\}$ where $F^O (x)$ has a period of 6.  The odd periodic extension $F^O (x)$ of $f(x)$ is defined as: $F^e (x)=\left\{\begin{array}{rr}0 & 3 < x < 2\\x+4 & 2 < x < 0\\x+4 & 0 < x < 2\\0 & 2 < x < 3\\\end{array}\right\}$ where $F^e (x)$ has a period of 6.  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? 
November 9th, 2016, 02:49 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,177 Thanks: 477 
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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extensions, odd, periodic 
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