Even and Odd functions... x^(1/2) is neither, right?

[Jakotheshadows]

Problem: Consider the functions:

. . .E(x)= [f(x) + f(-x)] / 2 (which is supposed to be even), and

. . .O(x)= [f(x) - f(-x)] / 2 (which is supposed to be odd)

Part A) Show that f(x) = E(x) + O(x).

This means that every function can be expressed as the sum of an even and an odd function-

Work: f(x) = [f(x) + f(-x) + f(x) - f(-x)] / 2 -----> f(-x) cancels out and I'm left with 2f(x) / 2 = f(x)

Part B) (This is where I'm having an issue...) Let f(x) = 4x^3 - 11x^2 + x^(1/2) - 10. Express f as a sum of an even function and an odd function...

I'm stuck right there because of x^(1/2), which is neither even nor odd since its domain must be greater than or equal to zero; it seems to me as if any function with an even indexed radical variable can not be even or odd. So if I follow the model given above, I'm dealing with imaginary numbers right?

. . .f(x) = 4x^3 - 11x^2 + x^(1/2) - 10

. . .f(-x) = 4(-x)^3 - 11(-x)^2 + (-x)^(1/2) - 10 ---> -4x^3 - 11x^2 + (-x)^(1/2) - 10

. . .So E(x) = [(4x^3 - 11x^2 + x^(1/2) - 10) + (-4x^3 - 11x^2 + (-x)^(1/2) - 10)] / 2

Which comes to -22x^2 + x^(1/2) + ix^(1/2) - 20)] / 2 ... So that is supposed to be an even function? I don't understand how a function with imaginary numbers in the terms can be even or odd. Can anyone shed some light on my confusion?

 

[skeeter]

It's been a long time, but if I remember correctly the function f(x) = E(x) + O(x) must have a symmetrical domain over the reals and be real valued.

f(x) = 4x[sup:1ijvv04l]3[/sup:1ijvv04l] - 11x[sup:1ijvv04l]2[/sup:1ijvv04l] + x[sup:1ijvv04l]1/2[/sup:1ijvv04l] - 10 is only defined for x > 0, and hence does not have the required symmetrical domain about 0, which leads to one of two conclusions ... either you were to deduce this fact for yourself, or there is some way it will work over the complete set of complex numbers. Sorry I can't comment on that due to my lack of knowledge of function maps from C -> C.

 

[Jakotheshadows]

Yeah, I just got an email back from my professor who told me that the answer he found in the back of the book was as inadequate as I thought it was, and that the scope of the proof that the problem is asking for is much larger than the "back-of-the-book" answer. If anyone feels up to the task of demonstrating why it is true in this case, however, I would greatly appreciate the effort.