A tricycle ride cost 10 for the first 2 kilometers and each additional kilometers adds 8 to the fare. Use piecewise function to the represent the tricycle fare in terms of the distance d in kilometers what happen is d≥ 3

To represent the tricycle fare as a piecewise function in terms of the distance \( d \) in kilometers, given that the ride costs 10 for the first 2 kilometers and each additional kilometer adds 8 to the fare, we can write the piecewise function as follows:

\[ f(d) = \begin{cases} 
10 & \text{if } 0 \leq d \leq 2 \\
10 + 8(d - 2) & \text{if } d > 2 
\end{cases} \]

Let's break down what happens when \( d \geq 3 \):

For \( d \geq 3 \):
\[ f(d) = 10 + 8(d - 2) \]

This equation calculates the fare as 10 for the first 2 kilometers, plus an additional 8 for each kilometer beyond the first 2 kilometers. Simplifying the equation for \( d \geq 3 \):

\[ f(d) = 10 + 8(d - 2) \]
\[ f(d) = 10 + 8d - 16 \]
\[ f(d) = 8d - 6 \]

Thus, when \( d \geq 3 \), the fare is given by the equation \( f(d) = 8d - 6 \).

Here is the complete piecewise function:

\[ f(d) = \begin{cases} 
10 & \text{if } 0 \leq d \leq 2 \\
8d - 6 & \text{if } d > 2 
\end{cases} \]

### Example Calculations:
- For \( d = 3 \):
  \[ f(3) = 8(3) - 6 = 24 - 6 = 18 \]
  The fare for a 3-kilometer ride is 18.

- For \( d = 4 \):
  \[ f(4) = 8(4) - 6 = 32 - 6 = 26 \]
  The fare for a 4-kilometer ride is 26.

- For \( d = 5 \):
  \[ f(5) = 8(5) - 6 = 40 - 6 = 34 \]
  The fare for a 5-kilometer ride is 34.

Would you like more details or have any questions about this piecewise function?

Here are some related questions to expand your understanding:

1. How would the piecewise function change if the initial fare was for 3 kilometers instead of 2?
2. How can you graph this piecewise function?
3. What is the fare for a 2.5-kilometer ride?
4. How does the function change if the cost per additional kilometer was 10 instead of 8?
5. What is the fare for a 2-kilometer ride according to this function?
6. What are the domain and range of this piecewise function?
7. How do you write the piecewise function if the base fare changes to 12 for the first 2 kilometers and 9 for each additional kilometer?
8. How would the fare function change if there were a fixed booking fee in addition to the per kilometer fare?

**Tip:** When creating piecewise functions, ensure that each piece is properly defined for its interval, and check for continuity at the boundaries where the function's definition changes.

Understand how to calculate tricycle fare using a piecewise function based on distance. This mathematical problem covers piecewise functions and linear equations. Suitable for Grades 9-12.

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