To derive the fare function formula, let's break down the conditions and determine the appropriate expressions for each case.

### 1. Understanding the Conditions

The tricycle fare is determined by two conditions:

1. **Flat Rate for the First 5 km:**
   - The fare is a fixed ₱20.00 for any distance from 0 to 5 km.

2. **Additional Charge Beyond 5 km:**
   - For every kilometer beyond the first 5 km, there is an additional charge of ₱0.75 per kilometer.

### 2. Constructing the Fare Function \( f(x) \)

Let \( x \) represent the number of kilometers traveled.

#### Case 1: \( 0 \leq x \leq 5 \)

If the distance \( x \) is between 0 km and 5 km, the fare is constant at ₱20.00.

So, the fare function in this range is:

\[
f(x) = 20 \quad \text{for } 0 \leq x \leq 5
\]

#### Case 2: \( x > 5 \)

If the distance \( x \) is greater than 5 km, the fare has two components:

1. A base fare of ₱20.00 for the first 5 km.
2. An additional charge of ₱0.75 for each kilometer beyond 5 km.

To express the additional charge mathematically:

- Distance beyond 5 km is \( x - 5 \) km.
- The additional fare is \( 0.75(x - 5) \).

Thus, the fare function for distances greater than 5 km becomes:

\[
f(x) = 20 + 0.75(x - 5) \quad \text{for } x > 5
\]

### 3. Combining the Two Cases

We now combine both cases to form a piecewise function:

\[
f(x) = \begin{cases} 
20 & \text{if } 0 \leq x \leq 5 \\
20 + 0.75(x - 5) & \text{if } x > 5 
\end{cases}
\]

This is the formula for calculating the fare based on the distance traveled.

### Summary

The **fare function** \( f(x) \) is defined as:

\[
f(x) = \begin{cases} 
20 & \text{if } 0 \leq x \leq 5 \\
20 + 0.75(x - 5) & \text{if } x > 5 
\end{cases}
\]

This formula accounts for both the flat rate for the first 5 km and the additional charge for any kilometers beyond that.

Would you like more details, or do you have any questions?

### Related Questions

1. How would the fare change if the flat rate was for the first 10 km instead?
2. If the cost per additional kilometer was increased to ₱1.50, what would be the new fare function?
3. What is the fare for a distance of 7 km?
4. How does the fare function change if the flat rate is ₱25 instead of ₱20 for the first 5 km?
5. If a passenger has only ₱30, what is the maximum distance they can travel?

**Tip:** Understanding how to construct piecewise functions helps in solving real-world problems with different conditions or rates!

Explore how to calculate the tricycle fare formula based on distance traveled with a detailed solution. This problem involves constructing a piecewise function for calculating fares, including a flat rate and additional charges per kilometer. Suitable for high school students.

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