A proposed tricycle fare would charge ₱20.00 for the first 5 km of travel and ₱0.75 for each additional kilometer over the proposed fare.
1. Find the fare function f(x) where x represents the number of kilometers travelled.
2. How much is the proposed fare for distance of 3 km?
3. Find the proposed fare for distance of 55 km.

Let's address each part of the problem step-by-step:

### 1. Find the Fare Function \( f(x) \)

To construct the fare function \( f(x) \), we need to consider the two scenarios given:

- The fare is ₱20.00 for the first 5 kilometers.
- For every additional kilometer beyond 5 km, the fare increases by ₱0.75 per kilometer.

We can write the fare function \( f(x) \) based on these conditions:

- For \( 0 \leq x \leq 5 \): The fare is a flat rate of ₱20.00.
- For \( x > 5 \): The fare is ₱20.00 for the first 5 kilometers plus ₱0.75 for each kilometer beyond 5.

So, the fare function \( f(x) \) can be defined as:

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

### 2. Proposed Fare for a Distance of 3 km

Since 3 km is within the first 5 km, the fare is a flat rate of ₱20.00.

\[
f(3) = 20
\]

**Answer:** The proposed fare for a distance of 3 km is **₱20.00**.

### 3. Proposed Fare for a Distance of 55 km

For a distance of 55 km, which is more than 5 km, we use the second part of the fare function:

\[
f(x) = 20 + 0.75(x - 5)
\]

Substitute \( x = 55 \):

\[
f(55) = 20 + 0.75(55 - 5)
\]

Calculate the value:

\[
f(55) = 20 + 0.75 \times 50
\]

\[
f(55) = 20 + 37.5
\]

\[
f(55) = 57.5
\]

**Answer:** The proposed fare for a distance of 55 km is **₱57.50**.

### Summary

1. **Fare Function**: 

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

2. **Fare for 3 km**: ₱20.00.
3. **Fare for 55 km**: ₱57.50.

Would you like more details on any of these solutions, or do you have any questions?

### Related Questions

1. What would the fare be for a trip of 10 km?
2. How would the fare change if the cost per additional kilometer was increased to ₱1.00?
3. If a passenger has a budget of ₱50, what is the maximum distance they can travel?
4. How would the fare function change if the flat rate was only for the first 3 kilometers instead of 5?
5. How does the fare increase per kilometer affect the total fare for longer distances, like 100 km?

**Tip:** When dealing with piecewise functions, always carefully consider the conditions given for each part of the function.

Explore how to calculate the proposed tricycle fare function and costs based on distance traveled. Understand the piecewise function involved and compute fares for specific distances like 3 km and 55 km. Suitable for high school students and those interested in practical applications of piecewise and linear functions in fare calculation.

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