Question

f(-4)=-76, meaning when the width of the rectangular area is -4 ft, the area would be -76 ft². this interpretation does not make sense in the context of the problem.
f(11.5)=40.25, meaning when the width of the rectangular area is 11.5 ft, the area would be 40.25 ft². this interpretation makes sense in the context of the problem.
f(16)=-16, meaning when the width of the rectangular area is 16 ft, the area would be -16 ft². this interpretation does not make sense in the context of the problem.
based on the observations above, it is clear that an appropriate domain for the function is

Explanation:

Step1: Analyze the context

In the context of the width of a rectangular area, width cannot be negative and area cannot be negative as well.

Step2: Determine the domain

Since width should be non - negative and we want non - negative areas, the appropriate domain for the function (where the function makes sense in the context of the problem) is the set of non - negative real numbers for which the function gives non - negative output values. So the domain should be \(x\geq0\) such that \(f(x)\geq0\). In interval notation, it could be \([0, a]\) where \(a\) is the value of \(x\) for which \(f(x)\) starts to become non - positive. From the given values, we can see that for positive values of \(x\) we want to avoid getting negative \(f(x)\) values. So the domain is \(x\in[0,11.5]\) (since \(f(11.5) = 40.25>0\) and \(f(16)=- 16<0\)).

Answer:

\([0,11.5]\)