Question

Mr. MacArthur’s art class plans to paint a square mural on the side of the art building. The monthly art budget allows for no more than 6 quarts of paint to be purchased. The situation can be modeled by the square root function s(x), the maximum side length of the square mural if x quarts of paint are used to paint the mural. Which statement best compares and contrasts the domain and range of the mathematical function, s(x), to the reason?

a) The domain of s(x) is the set of positive real numbers, while the range is also the set of positive real numbers.
b) The domain of s(x) is the set of non-negative real numbers, while the range is the set of positive real numbers.
c) The domain of s(x) is the set of integers, while the range is the set of rational numbers.
d) The domain of s(x) is the set of negative real numbers, while the range is the set of non-negative real numbers.

Answer 1

The correct statement regarding the domain and range of the function s(x), which models the maximum side length of a square mural painted with x quarts of paint, is (b). The domain is the set of non-negative real numbers, and the range is the set of positive real numbers, including zero for the mural side length.

Step-by-step explanation:

The student is inquiring about the domain and range of a square root function, s(x), which models the maximum side length of a square mural that can be painted with x quarts of paint. Given the constraint that no more than 6 quarts can be purchased, we can deduce the domain and range of the function s(x).

The domain of s(x) refers to all possible values of x that can be input into the function. Since you cannot have a negative amount of paint, the domain is all non-negative real numbers. This includes zero, because you could have a scenario where no paint is used, corresponding to a mural of side length zero. Thus, the domain includes 0 to 6 quarts of paint.

Regarding the range of s(x), since the square root of any non-negative number is also non-negative, and considering that a mural with side length zero is possible if no paint is used, the range of s(x) is all non-negative real numbers as well. However, we must be aware that the upper limit of the range is determined by the maximum amount of paint available (6 quarts).

Accordingly, the best choice that compares and contrasts the domain and range is option (b), which states: 'The domain of s(x) is the set of non-negative real numbers, while the range is the set of positive real numbers.'