Question: How could one visulaize the region which the multivariable function f maps

Here is a solve problem based on theoritical analytic approach, http://math.stackexchange.com/q/460365/8581. May I ask make me hints in which I can visualze the region f maps. In the question we are speaking about $f(E)$, so I am thinking about a plot illustaring $f(E)$. Thanks for the time and any hints.

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