EXAMPLE 9. A jeepney ride costs P8.00 for the first 4 kilometers, and each additional integer kilometer adds P1.50 to the fare. Use a piecewise function to represent the jeepney fare in terms ofthe distance d in kilometers.

Solution. The input value is distance and the output is the cost of the jeepney fare. If F(d) represents the fare as a function of distance, the function can be represented as follows

To represent the jeepney fare \( F(d) \) as a piecewise function based on the distance \( d \) in kilometers, we can define the function as follows:

\[
F(d) = 
\begin{cases} 
8.00 & \text{if } 0 \leq d \leq 4 \\
8.00 + 1.50 \cdot \lceil d - 4 \rceil & \text{if } d > 4 
\end{cases}
\]

### Explanation:
1. **First Case \((0 \leq d \leq 4)\):** 
   - For distances \( d \) up to and including 4 kilometers, the fare is a flat rate of P8.00.
   - This is because the first 4 kilometers have a fixed cost, regardless of the exact distance.

2. **Second Case \((d > 4)\):**
   - For distances greater than 4 kilometers, the fare is calculated by adding P1.50 for each additional kilometer.
   - \( \lceil d - 4 \rceil \) represents the number of additional kilometers beyond the first 4 kilometers. The ceiling function \( \lceil x \rceil \) is used because even if a part of a kilometer is traveled beyond 4 kilometers, it is counted as an additional kilometer in terms of fare calculation.

### Example Calculation:
- If \( d = 3 \) km, then \( F(3) = 8.00 \).
- If \( d = 6.5 \) km, then \( F(6.5) = 8.00 + 1.50 \times \lceil 6.5 - 4 \rceil = 8.00 + 1.50 \times 3 = 12.50 \).

This function effectively models the jeepney fare structure based on the given rules.

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Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How does the piecewise function change if the base fare was for the first 5 kilometers instead?
2. How can you model a situation where the fare increases by a different amount for every subsequent kilometer after the first 4 kilometers?
3. What happens to the fare if the rate per additional kilometer is reduced to P1.00?
4. How would the function change if the jeepney fare increased by P0.50 for every half-kilometer instead of every full kilometer?
5. Can you express the fare function if there's a discount for distances greater than 10 kilometers?
6. How would you calculate the total fare for a round trip given this piecewise function?
7. What impact does rounding distances have on fare calculations?
8. How would the fare calculation change if the initial fare was P10.00 instead of P8.00 for the first 4 kilometers?

**Tip:** Understanding how to set up piecewise functions is crucial for modeling real-world situations like this, where different rules apply over different intervals.

Learn how to model jeepney fare using piecewise functions in mathematics. Understand the structure where the fare is P8.00 for the first 4 kilometers and increases by P1.50 for each additional kilometer. Includes step-by-step examples and detailed explanations. Suitable for Grades 7-9 students.

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