A clothing company's profit \( P \) (in dollars) is modeled by the function \( P(x)=-5 x^{2}+150 x-1000 \), where \( x \) is the number of shorts sold. 1. Find the domain and range. Write your answers in interval notation. It's okay to use a graphing calculator like desmos.com/calculator to help you visualize the graph. a. Find the mathematical domain and range of the function (as though it had no meaning in real life). Mathematical Domain: b. Find the logical domain and logical range of the function (only the values that would make sense for this real-life model). Logical Domain:

To find the domain and range of the function \( P(x)=-5x^{2}+150x-1000 \), we need to consider the nature of the function and the real-life context of the problem. 1. Mathematical Domain: The mathematical domain of a function is the set of all possible input values (in this case, the number of shorts sold) for which the function is defined. Since the function is a quadratic function, it is defined for all real numbers. Therefore, the mathematical domain is all real numbers. 2. Logical Domain: The logical domain of a function in a real-life context is the set of input values that make sense in the context of the problem. In this case, the number of shorts sold cannot be negative, as you cannot sell a negative number of items. Therefore, the logical domain is all non-negative real numbers. 3. Mathematical Range: The mathematical range of a function is the set of all possible output values (in this case, the profit) that the function can produce. Since the function is a quadratic function, it can produce any real number as an output. Therefore, the mathematical range is all real numbers. 4. Logical Range: The logical range of a function in a real-life context is the set of output values that make sense in the context of the problem. In this case, the profit cannot be negative, as you cannot have a negative profit. Therefore, the logical range is all non-negative real numbers. In summary: a. Mathematical Domain: All real numbers b. Logical Domain: All non-negative real numbers c. Mathematical Range: All real numbers d. Logical Range: All non-negative real numbers