Example 5: Piecewise function for Greenville Utilities' electricity charges based on kWh usage.

Problem Analysis:

We are tasked with writing a piecewise function to represent the pricing structure of Greenville Utilities. The pricing structure is as follows:
1. A basic customer charge of \$10.99.
2. For usage up to 400 kWh, the cost is \$0.1260 per kWh.
3. For usage exceeding 400 kWh, the cost for all kWh in excess of 400 kWh is \$0.1151 per kWh.

Step-by-Step Solution:

#### 1. Define Variables:
Let:
- \( C(x) \): The total cost for using \( x \) kWh of electricity.
- \( x \): The total kWh used by the customer.

#### 2. Break Down the Pricing Structure:
The pricing structure can be divided into two cases based on the value of \( x \):
- Case 1: When \( x \leq 400 \) (usage is 400 kWh or less).
- Case 2: When \( x > 400 \) (usage exceeds 400 kWh).

#### 3. Write the Function for Each Case:

##### Case 1: \( x \leq 400 \)
- The customer pays a basic charge of \$10.99.
- Additionally, they pay \$0.1260 per kWh for all \( x \) kWh used.
- Therefore, the total cost is:
\[
C(x) = 10.99 + 0.1260x
\]

##### Case 2: \( x > 400 \)
- The customer still pays the basic charge of \$10.99.
- For the first 400 kWh, they pay \$0.1260 per kWh:
\[
\text{Cost for first 400 kWh} = 0.1260 \times 400 = 50.40
\]
- For the additional kWh used beyond 400 kWh, they pay \$0.1151 per kWh. The number of additional kWh is \( x - 400 \).
- Therefore, the total cost is:
\[
C(x) = 10.99 + 50.40 + 0.1151(x - 400)
\]
Simplifying this:
\[
C(x) = 10.99 + 50.40 + 0.1151x - 46.04
\]
\[
C(x) = 15.35 + 0.1151x
\]

#### 4. Combine the Cases into a Piecewise Function:
The piecewise function can now be written as:
\[
C(x) =
\begin{cases}
10.99 + 0.1260x & \text{if } x \leq 400 \\
15.35 + 0.1151x & \text{if } x > 400
\end{cases}
\]

Final Answer:

\[
\boxed{C(x) =
\begin{cases}
10.99 + 0.1260x & \text{if } x \leq 400 \\
15.35 + 0.1151x & \text{if } x > 400
\end{cases}}
\]