To determine the correct function $f(x)$ for the electric bill, let's break down the information given in the problem.

Problem Breakdown

1. Monthly Fixed Charge: $15, which applies regardless of electricity usage.
2. Cost per Kilowatt-hour (kWh):
   • For the first 300 kWh, the cost is $0.12 per kWh.
   • For usage above 300 kWh, there is an additional surcharge of $0.05 per kWh, totaling $0.17 per kWh.

Function Requirements

• $x$ represents the total electricity usage in kWh.
• If $x \leq 300$, the cost function should be based only on the fixed rate of $0.12 per kWh for the entire usage.
• If $x > 300$, the first 300 kWh should be charged at $0.12 per kWh, and any additional usage beyond 300 kWh should be charged at $0.17 per kWh.

Analysis of Each Option

1. Option 1: \begin{cases} 0.12x + 15 & \text{if } x \leq 300 \\ 0.17(x - 300) + 51 & \text{if } x > 300 \end{cases}$$ Explanation: - For $$x \leq 300$$, it correctly uses $$0.12x + 15$$. - For $$x > 300$$, it correctly charges the first 300 kWh at $$0.12$$ (which adds up to $51) and any excess at $$0.17$$ per kWh.
2. Option 2: \begin{cases} 0.12x + 15 & \text{if } x \leq 300 \\ 0.17(300 - x) + 51 & \text{if } x > 300 \end{cases}$$ - Incorrect for $$x > 300$$ because it uses $$0.17(300 - x)$$, which does not make sense mathematically.
3. Option 3: \begin{cases} 0.12x + 15 & \text{if } x \leq 300 \\ 0.17(x - 300) + 15 & \text{if } x > 300 \end{cases}$$ - Incorrect for $$x > 300$$ because it doesn’t add the initial cost of the first 300 kWh, which should be $51.
4. Option 4: \begin{cases} 0.12x + 15 & \text{if } x \leq 300 \\ 0.17(300 - x) + 15 & \text{if } x > 300 \end{cases}$$ - Also incorrect because it misuses $$0.17(300 - x)$$ for $$x > 300$$.

Conclusion

The correct answer is Option 1 because it accurately applies the $0.12 per kWh charge up to 300 kWh and adds the surcharge for any additional usage.

Would you like a deeper explanation of any part? Here are some related questions to expand your understanding:

1. What would the function $f(x)$ look like if there were a fixed fee for any usage above 300 kWh instead of a surcharge?
2. How would the cost function change if the rates for the first 300 kWh and additional usage were both lower?
3. Why is it necessary to separate the calculation into two parts for $x \leq 300$ and $x > 300$?
4. Can this type of piecewise function be represented in a graph? What would it look like?
5. How would the function change if the fixed monthly charge were proportional to total kWh usage instead of a fixed $15?

Tip: For piecewise functions, make sure each part of the function is correctly defined for its specific range!