The domain of a Function
$$y=x(18−2x)(13−2x) $$ $y$ is a function of $x$ , what is the contextual domain of this function (i.e. considering the meanings of the quantities $x$ and $y$ relative to the open-top box)? I've put this answer: $(-\infty,\infty).$
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$\begingroup$If the problem were asking only for the domain and range for the given equation where $y$ is a function of $x$, then both the domain and range would be all real numbers, i.e., $(-\infty,\infty)$. However, if $y$ represents the volume of an open box created from a 13" x 18" piece of paper, then each factor represents a side length of the box. So, in this context, we need the volume $y$ to be positive and we need each side to have positive length. Therefore, we require $x >0$, $18-2x>0$, and $13-2x>0$. All three inequalities will be satisfied when $0<x<\frac{13}{2}$.
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