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:

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.