Re: plotting even function

[Gunnar]

On Sunday 20 March 2005 18.09, Geraint Paul Bevan wrote:
> John B. Thoo wrote:
> > Also, why is  (x^(1/3))^2  not even?  If  f(x) = (x^(1/3))^2,  then isn't
> >   f(-x) = ((-x)^(1/3))^2 = (- x^(1/3))^2 = (x^(1/3))^2 = f(x)
>
> It is not true to say that (-x)^(1/3) is the same as -(x^(1/3)). That
> is one possibility but there are two more. If n is an integer, x^n has
> a unique value but x^(1/n) has n possible values i.e. the square root
> x^(1/2) has two possible values, the cube root x^(1/3) has three
> possible values, etc.
Isn't that just a matter of definition and our frames of references? 

What do we mean when we talk about a^(1/3) ? In simple real analysis it means 
the positive solution, x,  to the equation x^3=a for a>0, and when a<0 it is 
then  -x where x is the positive solution to x^3= -a. At this level x^(2/3) 
is even. 

It's not even when you have studied complex analysis, but of course, then you 
know how to make it an even function, and you can also make it a not-even, 
and not-odd function.

I must say that I enjoyed reading this discussion.
B.t.w. Maple does not plot anything for negative x-values for the function 
x^(2/3). It also does not simplify (-1)^(2/3) to -1.

Cheers,
Gunnar.



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