Question

a company manufactures and sells shirts. the daily profit the company makes depends on how many shirts they sell. the profit, in dollars, when the company sells x shirts can be found using the function f(x)=7x - 30. find and interpret the given function values and determine an appropriate domain for the function. answer attempt 1 out of 2 f(-9)=, meaning if the company sells shirts, they would make a profit of dollars. this interpretation in the context of the problem. f(2)=, meaning if the company sells shirts, they would make a profit of dollars. this interpretation in the context of the problem. f(5.5)=, meaning if the company sells shirts, they would make a profit of dollars. this interpretation in the context of the problem. based on the observations above, it is clear that an appropriate domain for the function is

Explanation:

Step1: Calculate $f(-9)$

Substitute $x = - 9$ into $f(x)=7x - 30$. So $f(-9)=7\times(-9)-30=-63 - 30=-93$. This means if the company sells - 9 shirts (which is not possible in the real - world context of selling shirts), they would make a profit of - 93 dollars. This interpretation is not valid in the context of the problem.

Step2: Calculate $f(2)$

Substitute $x = 2$ into $f(x)=7x - 30$. So $f(2)=7\times2-30=14 - 30=-16$. This means if the company sells 2 shirts, they would make a profit of - 16 dollars. This interpretation is valid as it represents a loss when selling a small number of shirts.

Step3: Calculate $f(5.5)$

Substitute $x = 5.5$ into $f(x)=7x - 30$. So $f(5.5)=7\times5.5-30=38.5 - 30 = 8.5$. But selling 5.5 shirts is not possible in the real - world context. This interpretation is not valid in the context of the problem.

Step4: Determine the domain

Since the number of shirts sold $x$ must be a non - negative integer (you can't sell a negative or non - whole number of shirts in practice), the domain is the set of all non - negative integers, i.e., $\{x\in\mathbb{Z}:x\geq0\}$.

Answer:

$f(-9)=-93$, meaning if the company sells - 9 shirts, they would make a profit of - 93 dollars. This interpretation is not valid in the context of the problem.
$f(2)=-16$, meaning if the company sells 2 shirts, they would make a profit of - 16 dollars. This interpretation is valid in the context of the problem.
$f(5.5)=8.5$, meaning if the company sells 5.5 shirts, they would make a profit of 8.5 dollars. This interpretation is not valid in the context of the problem.
The domain is $\{x\in\mathbb{Z}:x\geq0\}$