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So, for example, domain is $A=\{2,1,0,-1\}$ and codomain is $B=\{5,3,-2,-3\}$, and we have with $f:A\to B$ be a function these relations: $$f(2)=5,\\ f(1)=-3,\\ f(0)=-2,\\f(-1)=3.$$ How do I know if the function is even or odd?

A6SE
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    I closed formula need not exist. A function can be defined the way you did on the question. – Git Gud Sep 01 '13 at 17:48
  • You can make a fourth degree polynomial function passing through these points? Is that what you are looking for, A6Tech? Otherwise I wouldn't really know... – imranfat Sep 01 '13 at 17:52
  • Actually, I want to find out if that function is even or odd. The only way I know how to do it is when I know the formula. – A6SE Sep 01 '13 at 17:53
  • @A6Tech Add that information to the question and you'll promptly get an answer. – Git Gud Sep 01 '13 at 17:54

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To check if a function $g\colon A\to B$ is even or odd, you first need for its domain to be symmetric, (check the first line in this section), that is, $(\forall a\in A)(-a\in A)$.

This is not the case in $D$, (note that $2\in D$, but $-2\not \in D$) and so it doesn't make sense to talk about evenness or oddness of your function.

Even if you were to look for a function defined on a symmetric set such that it would coincide with $f$ on $D$, it could be the case that the extended function is neither even or odd. As an example consider the function $x\overset {\Bbb R}{\mapsto}\dfrac{5x^3+12x^2-23x-12}{6}$ which coincides with $f$ on $D$, it fails to be odd and it fails to be even.

Git Gud
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