To express the monthly cost $E(x)$ as a piecewise defined function of the number of kilowatt-hours $x$ used, we consider the different rates applied for different levels of usage.

Given:

• Base rate: $7.00 per month
• Rate for the first 300 kWh: 15¢ per kWh
• Rate for usage over 300 kWh: 4¢ per kWh

Case 1: $0 \leq x \leq 300$

If the customer uses $x$ kilowatt-hours where $x$ is less than or equal to 300 kWh, then the cost is simply the base rate plus 15¢ per kWh for all $x$ kWh.

$E(x) = 7.00 + 0.15x$

Case 2: $x > 300$

If the customer uses more than 300 kWh, then the cost is the base rate plus 15¢ per kWh for the first 300 kWh, and 4¢ per kWh for the remaining $x - 300$ kWh.

$E(x) = 7.00 + 0.15(300) + 0.04(x - 300)$

Simplifying the second case:

$E(x) = 7.00 + 45 + 0.04(x - 300)$ $E(x) = 52.00 + 0.04x - 12.00$ $E(x) = 40.00 + 0.04x$

Piecewise Function

Thus, the piecewise defined function for the monthly cost $E(x)$ is:

\begin{cases} 7.00 + 0.15x & \text{if } 0 \leq x \leq 300 \\ 40.00 + 0.04x & \text{if } x > 300 \end{cases}$$ This expression accounts for both the base rate and the varying cost per kilowatt-hour based on usage. --- Would you like more details or have any questions? Here are some related questions for further exploration: 1. What would be the cost for a customer who uses 250 kWh in a month? 2. Calculate the monthly bill for a customer who uses 400 kWh. 3. How would the piecewise function change if the base rate were increased to $10.00? 4. Determine the break-even point where the cost shifts from the first to the second rate. 5. How much more would a customer pay if they used 350 kWh instead of 300 kWh? **Tip:** Piecewise functions are useful for representing real-world scenarios where different rules apply to different conditions. Understanding them helps in analyzing such situations accurately.