the function is f(x,y)=2xy/(x^2+y^2)
the domain is R-{(x,y)=(0,0)}
I managed to know that the lower bound of the function is -1 because:
2xy=(x+y)^2-x^2-y^2
so z=((x+y)^2+(x^2-y^2))/(x^2+y^2)
which simplifies to (x+y)^2/(x^2+y^2) -1
(x+y)^2/(x^2+y^2) >=0 so (x+y)^2/(x^2+y^2) -1>=-1
but I didn't manage to know the upper bound of the domain. Thank you for helping me.
Yes you're right that the range is [-1,1], and to prove that I was trying to prove why (x+y)^2/(x^2+y^2)<=2 (you can try with different number combinations and you will always see that the results are less or equal to 2 but there's no way to prove why I think).