As Steve pointed out y=-x does always work. What remains is to show that for any given x, y=-x is the only thing that works.
For many values of x, the function of y is strictly increasing, but I'm finding this hard to prove. And for some values it isn't.
Another line of reasoning is that because we are looking for a product that is positive it is clear that if x is positive, y must be negative. Unfortunately a negative x does not imply a positive y.
Musing on possible lines of attack
