Hint :USE permutation or combination.Thank you
This isn't my area, but since there aren't any other answers yet ...
It seems you want to count the functions where -(f(x)) = f(-x) for each x. So you have 5 choices for f(-2)--it could = -2, -1, 0, 1, or 2, but that choice also determines f(2) since it has to be the opposite of f(-2). Does f(0) have more than one choice? Can you go from there to find the answer?
Not sure I've got the right idea but perhaps it will help.