Preimage of Subset under Mapping/Examples/Preimages of f(x, y) = (x^2 + y^2, x y)

Example of Preimage of Subset under Mapping

 * Example-x2+y2-and-xy.png

Let $g: \R^2 \to \R^2$ be the mapping defined as:
 * $\forall \tuple {x, y} \in \R^2: \map g {x, y} = \tuple {x^2 + y^2, x y}$

Let the following subset of $\R^2$ be defined:


 * $S = g^{-1} \sqbrk {\openint 0 3 \times \openint 0 1}$

Then:
 * $S = \set {\tuple {x, y} \in \R^2: x y < 1} \cap \set {\tuple {x, y} \in \R^2: 0 < x^2 + y^2 < 3}$