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 xx shirts can be found using the function f, of, x, equals, 5, x, minus, 50, .f(x)=5x−50. Find and interpret the given function values and determine an appropriate domain for the function.
Answer
Attempt 2 out of 2

f, of, minus, 2, equalsf(−2)= 
, meaning if the company sells 
 shirts, they would make a profit of 
 dollars. This interpretation 
 in the context of the problem.
f, of, 17, equalsf(17)= 
, meaning if the company sells 
 shirts, they would make a profit of 
 dollars. This interpretation 
 in the context of the problem.
f, of, 18, point, 5, equalsf(18.5)= 
, 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 
.

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.

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) = 5x - 50. Find and interpret the given function values and determine an appropriate domain for the function.

Learn how to analyze the linear profit function f(x) = 5x - 50 for a company that manufactures shirts. This problem covers key concepts like function interpretation, profit calculation, and determining the appropriate domain for real-life scenarios. Suitable for Grades 8-10.

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