We are given the profit function for a company that manufactures and sells shirts, defined as:

$f(x) = 5x - 50$

where $x$ represents the number of shirts sold, and $f(x)$ represents the profit in dollars.

Let's find the required function values and interpret them.

1. $f(-2)$

$f(-2) = 5(-2) - 50 = -10 - 50 = -60$ Interpretation: If the company sells $-2$ shirts (which doesn't make practical sense since the number of shirts sold cannot be negative), the profit would be $-60$ dollars. This indicates a loss of 60 dollars. However, this situation is unrealistic, so $x = -2$ is not valid in the context of the problem.

2. $f(17)$

$f(17) = 5(17) - 50 = 85 - 50 = 35$ Interpretation: If the company sells 17 shirts, they would make a profit of 35 dollars. This value makes sense in the context of the problem.

3. $f(18.5)$

$f(18.5) = 5(18.5) - 50 = 92.5 - 50 = 42.5$ Interpretation: If the company sells 18.5 shirts (which is also not practical since the number of shirts must be a whole number), the company would theoretically make a profit of 42.5 dollars. However, this value suggests that $x$ must also be restricted to whole numbers.

Domain of the Function

The domain should reflect the number of shirts sold, which must be non-negative and whole numbers, because it doesn't make sense to sell a negative or fractional number of shirts. Therefore, the appropriate domain for the function is:

$\text{Domain}: x \in \{0, 1, 2, 3, \dots\}$

Conclusion

• $f(-2) = -60$, but it is not valid in the context of the problem.
• $f(17) = 35$, meaning if the company sells 17 shirts, they make a profit of 35 dollars.
• $f(18.5) = 42.5$, but fractional values are not practical, so it is not valid in the context of the problem.
• The domain is restricted to non-negative integers: $x \in \{0, 1, 2, 3, \dots\}$.

Would you like any additional details or clarifications?

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Relative Questions:

1. How would the profit function change if the company incurred a higher fixed cost?
2. What is the minimum number of shirts the company must sell to make a profit?
3. How does the slope $5$ in the function represent the relationship between the number of shirts sold and profit?
4. What is the significance of the constant term $-50$ in the function?
5. How would you graph this profit function and interpret its key features?

Tip:

When determining the domain of a function in real-life scenarios, always consider practical restrictions like non-negativity or whole numbers, depending on the context.