Determining whether a Function is Even or Odd given a specific Domain

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I have learned that,to determine if a function is even or odd, I should use the following Formula: if f(-x)=f(x) ---> f(x) is Even. if f(-x)=-f(x)---> f(x) is Odd. if neither, well then neither.

But, I have stumbled a cross different functions that at first seem for example Odd, but they are actully neither because of its Domain.

Example:Fourier Series with

f(t) = 2pit , t in [0,2*pi).

This function is neither odd nor even because of the given domain and i can not understand why ?

Does the non-existence of negative numbers in the domain mean that the function is neither odd nor even? Since there can't be any symmetry ?

Can we plot negative numbers even if no negative numbers in the domain exist ?

This is how the Tutor drew the function to explain why is it neither odd nor even and i am confused how are we even allowed to plot negative numbers, because of the domain...

Thank you in advance