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Range of Quadratic Function

  1. Jul 31, 2012 #1
    Consider the map [tex]f : \mathbb{R}^2 \rightarrow \mathbb{R}^2[/tex]
    defined by
    [tex](x,y) \mapsto (xy-x^2, xy-y^2)[/tex]

    I'm interested in figuring out the range of this function, but I keep thinking myself in circles. What would be a systematic method for approaching something like this?
     
  2. jcsd
  3. Jul 31, 2012 #2

    haruspex

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    Writing (u, v) for the new co-ordinates, you could look at a linear combination of these, u+m.v say, and find the extremal points as a function of m. This will give you a parametric equation describing the boundary.
    In the present case, u+v is interesting.
     
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