Let $f:(-\pi,\pi)\rightarrow[0,1], f(x)=\cos(x)$
Clearly this function is even as $f(x)=f(-x)$ for all $x\in (-\pi,\pi)$
Suppose if we defined $f$ as $f:[0,\pi)\rightarrow[0,1], f(x)=\cos(x)$
How do we now classify $f$ as even or odd since in this given domain we will not be able to calculate $f(-x)$ for all $x\in[0,\pi)$?
I think that since we can't check for condition of $f$ being even or odd , we should report our answer as $f$ is neither even nor odd in this domain.
Am I right?
It's kind of silly to ask about function if it's odd or even when the whole domain included on one "side" of the real line. It's like asking about function being injective where the domain is formed by 1 point. I think that logicly speaking, your claim is correct, but it has no real information we can get about the function.