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Even and Odd functions

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  Even and Odd functions yitzhakbg <yitzhakbg@gmail.com> - 2011-05-31 11:48 +0000

#2824 — Even and Odd functions

Applying makeper makes the piecewise functions x and y periodic, yet they don't appear as even and odd functions whereas the Sin and Cos functions do show as odd and even.
xx and yy are the functions which fail the even/odd test yet show as even and odd according to the plots.
I couldn't determine why. Help appreciated.


ClearAll[x, y, xx, yy, makeper]

x[t_] := Piecewise[{{-1, -1/2 <= t < 0}, {1, 0 <= t < 1/2}}]
y[t_] := Piecewise[{{-t - 1/4, -1/2 <= t < 0}, {t - 1/4, 0 <= t < 1/2}}]

makeper[f_, p_, d_][t_] := f[Mod[t, p, d]]
(* Plot[x@t,{t,-5,5},PlotRange->{-1,1}] *)

xx[t_] := makeper[x, 1, -1/2]@t
yy[t_] := makeper[y, 1, -1/2]@t
Plot[xx[t], {t, -5, 5}, PlotRange -> {-1, 1}]
Plot[yy[t], {t, -5, 5}, PlotRange -> {-1, 1}]
OddFunctionQ[f_] := Module[{x}, f[x] + f[-x] === 0]
EvenFunctionQ[f_] := Module[{x}, f[x] - f[-x] === 0]
fs = {# &, #^3 &, Sin, Cos, xx, yy}
OddFunctionQ /@ fs
{True, True, True, False, False, False}
EvenFunctionQ /@ fs
{False, False, False, True, False, False}

Best,
Yitzhak

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