Westside Energy charges its electric customers a base rate of $7.00 per month, plus 15¢ per kilowatt-hour (kWh) for the first 300 kWh used and 4¢ per kWh for all usage over 300 kWh. Suppose a customer uses x kWh of electricity in one month.
(a) Express the monthly cost E as a piecewise defined function of x. (Assume E is measured in dollars.)

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:

\[
E(x) = 
\begin{cases} 
7.00 + 0.15x & \text{if } 0 \leq x \leq 300 \\
40.00 + 0.04x & \text{if } x > 300 
\end{cases}
\]

This expression accounts for both the base rate and the varying cost per kilowatt-hour based on usage.

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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.

Explore how to calculate the monthly cost of electricity using a piecewise function based on usage. Understand the base rate and varying costs per kilowatt-hour. Example calculations included.

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