Let's analyze the profit function [tex]$f(x) = 7x - 80$[/tex] and interpret the results step-by-step.
### Step-by-Step Solution:
1. Calculate the function value [tex]$f(-7)$[/tex]:
- [tex]$f(-7) = 7(-7) - 80$[/tex]
- Simplifying, we get: [tex]$f(-7) = -49 - 80 = -129$[/tex]
Therefore, [tex]$f(-7) = -129$[/tex], meaning if the company sells -7 shirts, they would make a profit of -129 dollars. This interpretation does not make sense in the context of the problem because selling a negative number of shirts is not possible.
2. Calculate the function value [tex]$f(8)$[/tex]:
- [tex]$f(8) = 7(8) - 80$[/tex]
- Simplifying, we get: [tex]$f(8) = 56 - 80 = -24$[/tex]
Therefore, [tex]$f(8) = -24$[/tex], meaning if the company sells 8 shirts, they would make a profit of -24 dollars. This interpretation indicates that selling 8 shirts is not sufficient to cover the fixed cost of [tex]$80, leading to a loss of $[/tex]24.
3. Calculate the function value [tex]$f(12.5)$[/tex]:
- [tex]$f(12.5) = 7(12.5) - 80$[/tex]
- Simplifying, we get: [tex]$f(12.5) = 87.5 - 80 = 7.5$[/tex]
Therefore, [tex]$f(12.5) = 7.5$[/tex], meaning if the company sells 12.5 shirts, they would make a profit of [tex]$7.5. This interpretation shows that selling 12.5 shirts yields a small profit of $[/tex]7.5 after covering the fixed cost.
### Determine the Break-Even Point and Domain:
4. Determine the break-even point:
The break-even point occurs when [tex]$f(x) = 0$[/tex], meaning the revenue just covers the fixed cost.
- Set the function equal to zero:
[tex]$7x - 80 = 0$[/tex]
- Solve for [tex]$x$[/tex]:
[tex]$7x = 80$[/tex]
[tex]$x = \frac{80}{7} \approx 11.43$[/tex]
Therefore, the break-even point is at approximately 11.43 shirts. This means the company needs to sell at least 11.43 shirts to start making a profit.
5. Determine an appropriate domain for the function:
Since selling a negative number of shirts is not meaningful, the appropriate domain for this function is [tex]$x \geq 0$[/tex]. However, since the company starts making a profit only after selling 11.43 shirts, the realistic domain for profitability would be [tex]$x \geq \frac{80}{7}$[/tex] or approximately [tex]$x \geq 11.43$[/tex].
### Summary:
- [tex]$f(-7) = -129$[/tex], meaning if the company sells -7 shirts, they would make a profit of -129 dollars. This interpretation does not make sense in the context of the problem.
- [tex]$f(8) = -24$[/tex], meaning if the company sells 8 shirts, they would make a profit of -24 dollars. This interpretation indicates a loss of [tex]$24.
- $[/tex]f(12.5) = 7.5[tex]$, meaning if the company sells 12.5 shirts, they would make a profit of $[/tex]7.5.
- The break-even point is at approximately 11.43 shirts, meaning the company needs to sell at least 11.43 shirts to start making a profit.
- The appropriate domain for the function in the context of profitability is [tex]$x \geq \frac{80}{7}$[/tex] or approximately [tex]$x \geq 11.43$[/tex].
