Guess this is more of a math question...
Like Jasonrox I don't know what you mean I would advise you making
it very clear exactly what the question says because it does not look
anything like any other question I have seen teaching at uni or school.
is F(-2) = 1, F(-1) =-2 ,F(0)=0,F(1) =-1. F(2)=2 ?
If you have something like F(x)=F(y) when x is not y
then the function is not one to one.
eg if F(-1) = 2 and F(2)=2 then the function F is not one to one.
the comment about odd functions is a common question I have taught at uni and school if f(x) = x^3 then f(-x) = -f(x) so x^3 is odd and symmetrical about the origin.
A function is NOT "one-to-one" if f(x)= f(y) for different x and y.
A function is "symmetric about the origin" if f(-x)= -f(x) (A line drawn through the points (x, f(x)) and (-x,-f(x)) has the origin at its center).
I'm not crazy about problems where you are asked to decide something like that given only some of the values (strictly speaking that's impossible!) but assuming that the values given are sufficient, I note that G(-2)= 3= G(2) so G is NOT one to one. Also, H(-2)= 1 and H(2)= -1, H(-1)= -3 and H(1)= 3 so, at least for the values shown, H is symmetric about the origin.
Separate names with a comma.