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.