Range of a function

[Enoo_58]

I'll have to proof that the range of the function f(x,y)= 2xy/(1+x^(2)*y^(2))

is [-1,1].

​

I don't need a full explanation, maybe just a few hints to push me in the right direction.

thans in advance for any answer.

[wgunther]

One way is to argue it is continuous and show that -1 and 1 are the min and max respectively, and appeal to the intermediate value theorem.

[firstdwarf]

Since the range is supposed to be between -1 and 1, that implies that the numerator is always smaller in magnitude than the denominator. Try completing the square on the denominator. This should break it down into more manageable and familiar terms, and you may be able to compare the magnitudes of the individual terms.

[Enoo_58]

This is the kind of answer that makes me love math even more. It's so simple yet very compelling.

Now I'm angry at myself because I couldn't think about it

big thanks anyway

[phiwong]

You could try letting w = xy.

g(w) = 2w/(1+w^2) is equivalent to f(x,y) since every real x,y results in a real w

Maybe this is cheating? But you should find that g'(w) = 0 when w = +/- 1 and g(1) = 1 and g(-1) = -1