The question involves determining an appropriate domain for the function $f(x) = 7x - 30$, which represents the daily profit of a company in dollars when they sell $x$ shirts.

Analyzing the Image:

1. Function Definition: $f(x) = 7x - 30$

   • This is a linear function where the profit depends on the number of shirts $x$ sold.
   • The profit is calculated by multiplying the number of shirts $x$ by 7, and then subtracting 30.

2. Given Evaluations:

   • $f(-9) = -93$: Selling $-9$ shirts results in a profit of $-93$ dollars. Selling a negative number of shirts doesn't make sense in the context of this problem.
   • $f(2) = -16$: Selling 2 shirts results in a profit of $-16$ dollars. While this is mathematically correct, a negative profit indicates a loss.
   • $f(5.5) = 8.5$: Selling 5.5 shirts results in a profit of 8.5 dollars. Selling half a shirt is impractical, making this interpretation nonsensical.

Appropriate Domain:

• Contextual Consideration:
  • Selling a negative or fractional number of shirts doesn't make sense in this context.
  • Therefore, $x$ should be a non-negative integer (whole number), representing possible quantities of shirts.

Conclusion:

• The appropriate domain for the function is whole numbers (0, 1, 2, ...).

Would you like to delve deeper into how domains are determined in different contexts, or do you have any other questions related to this problem?

Further Questions:

1. What happens to the function if the price per shirt is changed?
2. How would you interpret the function if the fixed cost (subtracted amount) were different?
3. What is the minimum number of shirts the company needs to sell to make a profit?
4. How does changing the slope (coefficient of $x$) affect the profit?
5. Could the function $f(x)$ represent any other scenario beyond shirts?

Tip: Always think about the real-world implications of your mathematical model when determining the domain of a function.