A contractor is in charge of hiring people for a construction project. The number of days it would take to complete the project with $$ x $$ full time workers can be found using the function $$ f(x)=\frac{156}{x} $$. Assume once workers are assigned to the project, no workers may be added or taken off the project until it is finished. Find and interpret the given function values and determine an appropriate domain for the function. Answer Attempt out of a $$ f(-3)=\square $$, meaning with $$ \square $$ full time workers, the project would be complete in $$ \square $$ days. This interpretation $$ f(2)=\square $$, meaning with $$ \square $$ full time workers, the project would be complete in $$ \square $$ days. This interpretation $$ f(6.5)=\square $$, meaning with $$ \square $$ full time workers, the project would be complete in $$ \square $$ days. This interpretation in the context of the problem. Based on the observations above, it is clear that an appropriate

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Let's analyze the function $$ f(x) = \frac{156}{x} $$, which represents the number of days required to complete a construction project with $$ x $$ full-time workers. We'll evaluate the function at specific values, interpret the results, and determine the appropriate domain for the function.

### 1. Evaluating $$ f(-3) $$

$$
f(-3) = \frac{156}{-3} = -52
$$

**Interpretation:**  
$$-52$$ days would imply that the project is completed **negative** time, which is impossible in a real-world context.

**Conclusion:**  
Having $$-3$$ full-time workers is not feasible. **Negative workers do not make sense in this scenario**, so $$ x = -3 $$ is not a valid input for the function.

### 2. Evaluating $$ f(2) $$

$$
f(2) = \frac{156}{2} = 78
$$

**Interpretation:**  
With **2 full-time workers**, the project would be completed in **78 days**.

### 3. Evaluating $$ f(6.5) $$

$$
f(6.5) = \frac{156}{6.5} = 24
$$

**Interpretation:**  
With **6.5 full-time workers**, the project would be completed in **24 days**.  
*Note:* While it’s uncommon to have half a worker, this could represent part-time workers or the effective workforce.

### 4. Determining the Appropriate Domain

Based on the evaluations:

- **Non-Negative Workers:** The number of workers $$ x $$ cannot be negative, as it doesn't make sense to have a negative number of workers.
- **Non-Zero Workers:** If $$ x = 0 $$, the project would take an infinite amount of time, which is impractical.
- **Realistic Workforce:** While workers are typically whole numbers, the function mathematically allows for fractional workers, which can represent part-time positions or varying productivity levels.

**Therefore, the appropriate domain for $$ f(x) $$ is:**

$$
\boxed{\text{All positive real numbers } x \text{ such that } x > 0}
$$

This ensures that the number of workers is positive and real, making the function applicable and meaningful in the context of the construction project.
The function $$ f(x) = \frac{156}{x} $$ calculates the number of days needed to complete a project with $$ x $$ full-time workers. - **For 2 workers:** The project takes 78 days. - **For 6.5 workers:** The project takes 24 days. - **Negative workers** (like -3) are not possible. The function is valid for any positive number of workers greater than zero.

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