Question

How do you know if `f(s) = 4s^(3/2)` is an even or odd function?

Answer 1

Find that `f(s)` satisfies neither the condition for an even function nor the condition for an odd function, so it is neither.

Explanation:

An even function satisfies `f(-x) = f(x)` for any `x` in the domain.

An odd function satisfies `f(-x) = -f(x)` for any `x` in the domain.

If `f(s)` is considered as a Real valued function, then it's not defined for `s < 0`.

If `f(s)` is considered as a Complex valued function, then we have to think about how `s^(3/2)` is defined for negative values of `s`.

I have changed my mind on this one. I thought it was not well defined, but I think that if `s < 0` then `s^(3/2) = -i (-s)^(3/2)`.

You can arrive at this definition by considering the polar representation of negative numbers `s = (-s, pi)`.

Hence `s^(3/2) = ((-s)^(3/2), (3 pi)/2) = -i(-s)^(3/2)`

Hence we find: `f(-s) = -i f(s)`

For example, `f(-1) = 4 (-1)^(3/2) = -4i 1^(3/2) = -4i = -i f(1)`

so `f(s)` is neither even nor odd.