1. Gabe works for a packaging company for a wage of $15.00 per hour. He can work a maximum of 5 hours. The graph is continuous because Gabe is not limited to working whole hours, his earnings adjust to how many whole and partial hours he works.  

For Gabe's earnings in this situation, we can define his income $ E $ as a function of the hours $ h $ he works, where $ h $ can be any value from 0 to 5 (since he can work partial hours). The function is continuous because his earnings increase proportionally with each additional hour or fraction of an hour worked.

Given:
- Wage: $15 per hour
- Maximum work hours: 5

### Step-by-Step Analysis:
1. **Set Up the Function**:
   - Gabe's earnings $ E(h) $ can be represented as:
     \[
     E(h) = 15h
     \]
   where $ h $ is the number of hours worked, ranging from $ 0 \leq h \leq 5 $.

2. **Interpret the Domain and Range**:
   - **Domain**: $ h \in [0, 5] $
   - **Range**: Since Gabe earns $15 per hour, the range of earnings is from $0 (if he works 0 hours) to $75 (if he works the full 5 hours). Therefore, the range of $ E(h) $ is $ [0, 75] $.

3. **Graph Interpretation**:
   - On a graph where the x-axis represents hours $ h $ and the y-axis represents earnings $ E $, this function would appear as a straight line starting from the origin $(0, 0)$ and ending at the point $(5, 75)$.
   - The slope of the line is 15, reflecting his hourly wage.
  
### Example Calculations:
- If Gabe works 2.5 hours:
  \[
  E(2.5) = 15 \times 2.5 = 37.5
  \]
  Gabe earns $37.50.

- If Gabe works 4.75 hours:
  \[
  E(4.75) = 15 \times 4.75 = 71.25
  \]
  Gabe earns $71.25.

Let me know if you’d like further details on the graph or the function’s properties.

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1. What would Gabe’s earnings be if he worked 3.25 hours?
2. How would the function change if Gabe’s hourly wage increased to $18 per hour?
3. What is the interpretation of the slope of the line in this context?
4. How would the function and graph change if Gabe’s maximum work hours were limited to 8?
5. Could this type of continuous function be applied if Gabe only worked whole hours?

**Tip**: When interpreting wage functions, remember that the slope represents the hourly wage, which is the rate of earnings per hour.

Gabe works for a packaging company for a wage of $15.00 per hour. He can work a maximum of 5 hours. The graph is continuous because Gabe is not limited to working whole hours, his earnings adjust to how many whole and partial hours he works.

Explore how to calculate Gabe's earnings at $15 per hour with a continuous linear function that allows for whole and partial hours, up to a 5-hour limit. This solution breaks down domain, range, and graphing for high school math students in Grades 9-10.

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