To represent the bus fare in terms of the distance $d$ in kilometers, we can define a piecewise function. Here's the breakdown of the fare calculation:

1. For the first 4 kilometers, the bus fare is a fixed 12 pesos.
2. For each additional kilometer beyond the initial 4 kilometers, 1 peso is added to the fare.

Given this information, the bus fare $F(d)$ in terms of distance $d$ can be represented as a piecewise function:

\begin{cases} 12 & \text{if } 0 \leq d \leq 4, \\ 12 + (d - 4) & \text{if } d > 4. \end{cases}$$ ### Explanation: - **Case 1:** When $$d \leq 4$$, the distance is within the first 4 kilometers, so the fare remains a fixed 12 pesos. - **Case 2:** When $$d > 4$$, we calculate the fare by adding 1 peso for each additional kilometer beyond 4 kilometers. The term $$(d - 4)$$ represents the number of kilometers beyond the initial 4, and adding this to the base fare of 12 gives the total fare. ### Simplified Piecewise Function: For $$d > 4$$, we can simplify the fare calculation: $$F(d) = d + 8 \quad \text{if } d > 4.$$ So the complete piecewise function can also be written as: $$F(d) = \begin{cases} 12 & \text{if } 0 \leq d \leq 4, \\ d + 8 & \text{if } d > 4. \end{cases}$$ This function now provides the fare based on the distance $$d$$. Would you like more details, or do you have any questions? Here are some related questions you might consider: 1. What is the fare if Leona travels 3 kilometers? 2. What is the fare if Leona travels 5 kilometers? 3. How would the function change if the fare increased by 2 pesos for each kilometer beyond the first 4? 4. How can we represent the fare in a different mathematical form, like a step function? 5. What would the fare be if the initial fixed distance was 6 kilometers instead of 4? **Tip:** Understanding piecewise functions is essential when modeling situations with different conditions or rates, such as transportation fares, tax brackets, or material costs.