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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 [tex]$x$[/tex] shirts can be found using the function [tex]$f(x) = 7x - 80$[/tex].

Find and interpret the given function values and determine an appropriate domain for the function.

1. [tex]f(-7) = \square[/tex], meaning if the company sells [tex]\square[/tex] shirts, they would make a profit of [tex]\square[/tex] dollars. This interpretation [tex]\square[/tex] in the context of the problem.

2. [tex]f(8) = \square[/tex], meaning if the company sells [tex]\square[/tex] shirts, they would make a profit of [tex]\square[/tex] dollars. This interpretation [tex]\square[/tex] in the context of the problem.

3. [tex]f(12.5) = \square[/tex], meaning if the company sells [tex]\square[/tex] shirts, they would make a profit of [tex]\square[/tex] dollars. This interpretation [tex]\square[/tex] in the context of the problem.

Based on the information above, it is clear that an appropriate domain for the function is [tex]\square[/tex].



Answer :

GinnyAnswer
Let's go through the solution step-by-step:

1. Given Function:
The profit function is given as [tex]\( f(x) = 7x - 80 \)[/tex], where [tex]\( x \)[/tex] is the number of shirts sold.

2. Interpret [tex]\( f(-7) \)[/tex]:
[tex]\[ f(-7) = -129 \][/tex]
So, if the company sells [tex]\(-7\)[/tex] shirts, they would make a profit of [tex]\(-129\)[/tex] dollars. This interpretation doesn't make sense in a real-world context since selling [tex]\(-7\)[/tex] shirts is not possible. Therefore, negative sales imply a nonsensical situation in this context.

3. Interpret [tex]\( f(8) \)[/tex]:
[tex]\[ f(8) = -24 \][/tex]
So, if the company sells [tex]\( 8 \)[/tex] shirts, they would make a profit of [tex]\(-24\)[/tex] dollars. This means that even after selling [tex]\(8\)[/tex] shirts, the company is still incurring a loss of [tex]\(24\)[/tex] dollars.

4. Interpret [tex]\( f(12.5) \)[/tex]:
[tex]\[ f(12.5) = 7.5 \][/tex]
So, if the company sells [tex]\( 12.5 \)[/tex] shirts, they would make a profit of [tex]\( 7.5 \)[/tex] dollars. This means that after selling [tex]\(12.5\)[/tex] shirts, the company starts making a profit, [tex]$7.5$[/tex] dollars in this case.

5. Determine the appropriate domain:

From the interpretation above, it's clear that selling a negative number of shirts doesn't make sense. The smallest number of shirts that can be sold is [tex]\(0\)[/tex]. Therefore, the starting point of our domain is [tex]\(0\)[/tex].

Additionally, as the company can theoretically sell an unlimited number of shirts, the domain can extend to infinity.

Thus, the appropriate domain for the function is [tex]\([0, \infty)\)[/tex].

### Summary:
1. [tex]\( f(-7) = -129 \)[/tex]:

-[tex]\( f(-7) \)[/tex]: If the company sells [tex]\(-7\)[/tex] shirts, they would make a profit of [tex]\(-129\)[/tex] dollars. However, this interpretation isn't realistic in the problem context.

2. [tex]\( f(8) = -24 \)[/tex]:

-[tex]\( f(8) \)[/tex]: If the company sells [tex]\( 8\)[/tex] shirts, they would make a profit of [tex]\(-24\)[/tex] dollars, indicating a loss.

3. [tex]\( f(12.5) = 7.5 \)[/tex]:

-[tex]\( f(12.5) \)[/tex]: If the company sells [tex]\( 12.5 \)[/tex] shirts, they would make a profit of [tex]\( 7.5 \)[/tex] dollars, indicating a positive profit.

4. Domain:
The appropriate domain for the function is [tex]\([0, \infty)\)[/tex].

These interpretations help understand the company's profit behavior based on the number of shirts sold and define the realistic scope of the function.