Expert: Josh - 11/16/2006

Question
so if it asks a graph for function of period 2pi which is defined by f(x)=x in o<x<pi and is odd in intervals -3pi<x<3pi and ask the same question and says is given what is their difference on the graph?
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-----Question-----
If it says that a function f(x) is periodic with period 2 and is defined by f(x)=x in 0<x< , and is odd. And asks us to sketch the function in interval -3<x<3 how is that sketch; because it gives me the same question but the only difference is that it says that the function is even and asks about a different sketch. Can you explain me the difference in the two sketches please? And if you can show me one of them so that I can compare? I searched in many books but I didn't found any similar
-----Answer-----
Hello,

In the interval 0<=x<0.5*T, the function is described by f(x)=x. Here, T=2 is the period. If it is an odd function, f(-x)=-f(x). i.e., f(x)=x for -0.5*T<=x<0. Graphically, we have a straight line which extends from (-1,-1), passing through (0,0) up to but not including the point (1,1). By periodic extension, the pattern repeats itself in intervals of (n-0.5)*T<=x<(n+0.5)*T. We get a sawtooth waveform. |/|/|/|

If it's an even function, by definition, f(-x)=f(x). i.e., it is symmetrical about the vertical axis, x=0. In this case, f(x)=-x in the interval -0.5*T<=x<0. viz., a straight line with a negative slope of 1 in the negative portion. If we consider a single period between x=-1 and x=1, we have a V-shape graph. This pattern repeats itself periodically. Thus, we get VVV in -3<=x<3.

Hope this makes sense. Let me know if you need further explanation.

Answer
I plotted the odd and even functions. Take a look at the pdf file (http://www.geocities.com/joshcameron_ae/Graph/plot.pdf).

Please note the horizontal and vertical axes are both measured in units of "pi". So, where it says 2, it corresponds to 2*pi.

Cheers.