Problem

Compare the unrestricted and physical input domains of \(A(x)=x(10-x)\) for a nondegenerate rectangle. State the polynomial domain, solve \(x>0\) and \(10-x>0\), intersect the constraints, and justify the open endpoints in length units.

Neutral domain-comparison card. It contains no graph, contextual interval, endpoint status, or answer information.

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algebraic/unrestricted domain=(-∞,∞); contextual constraints=x≥0 and10−x≥0; intersection=0≤x≤10; contextual domain=[0,10] length units; endpoint status=both included as zero-area rectangles; retained graph=closed segment between the intercepts

algebraic/unrestricted domain=(-∞,∞); contextual constraints=x>0 and10−x>0; intersection=0<x<10; contextual domain=(0,10) length units; endpoint status=0,10 excluded as degenerate zero-length cases

algebraic/unrestricted domain=(-∞,∞); contextual constraints=x>0 only; intersection=x>0; contextual domain=(0,∞) length units; endpoint status=0 excluded and no upper endpoint; retained graph=all points with positive x

algebraic/unrestricted domain=(0,10); contextual constraints=x>0 and10−x>0; intersection=0<x<10; contextual domain=(0,10) length units; endpoint status=both excluded; retained graph=entire parabola

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Compare the unrestricted and physical input domains of \(A(x)=x(10-x)\) for a nondegenerate rectangle. State the polynomial domain, solve \(x>0\) and \(10-x>0\), intersect the constraints, and justify the open endpoints in length units.

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