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Think of a real-life situation that can be described by a piecewise function where not all components are linear.

Describe the scenario you are modeling
Describe the variables involved in the function
Describe how you would graph the piecewise function
Describe important aspects of the piecewise function - explain how to read it to the person you are describing it to
Describe the domain of the piecewise function pieces and what that means in context
IF YOU ARE NOT COMPLETING THE DISCUSSION WITH YOUR TEACHER, THE PERSON YOU ARE EXPLAING TO IN THE RECORDING MUST STATE THEIR NAME

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Scenario Modeling with a Piecewise Function: Utility Bill

Scenario Description: Imagine you are modeling the monthly electricity bill for a residential customer, which is affected by different rates depending on the usage of kilowatt-hours (kWh). The utility company uses a tiered pricing strategy that changes the rate per kWh based on the amount of electricity consumed.

Variables Involved:

  • x: The number of kilowatt-hours (kWh) consumed in a month.
  • y: The total electricity bill in dollars.

Piecewise Function Definition: The total bill can be defined as follows:

  • For 0 ≤ x ≤ 100: \( y = 0.10x \)
  • For 101 ≤ x ≤ 300: \( y = 10 + 0.15(x - 100) \)
  • For x > 300: \( y = 55 + 0.20(x - 300) \)

Explanation of Each Component:

  1. 0 ≤ x ≤ 100: The first segment indicates that for the first 100 kWh, the charge is $0.10 per kWh, so if you use 50 kWh, your bill is $5.
  2. 101 ≤ x ≤ 300: The second segment applies for usage between 101 and 300 kWh. Here, there's a base cost of $10 for the first 100 kWh, and then it increases to $0.15 per kWh for every kWh above 100.
  3. x > 300: For any usage above 300 kWh, the charge becomes $0.20 per kWh after the initial base cost of $55 (which accounts for the first 300 kWh).

Graphing the Piecewise Function: To graph the piecewise function:

  1. Start with the first segment, plotting the line from (0, 0) to (100, 10), which is linear.
  2. Next, plot the second segment starting from (100, 10) to (300, 55), which is also linear but with a steeper slope.
  3. Lastly, plot the third segment starting from (300, 55) onward, indicating a linear but steeper increase.

The graph will consist of three linear segments connected at the endpoints of each segment, showing clear breaks at 100 kWh and 300 kWh.

Important Aspects of the Piecewise Function:

  • When reading the function, it's essential to note the ranges for each case. The function changes at specific values (100 kWh and 300 kWh), which indicate where the pricing strategy alters.
  • Each segment represents a different pricing model, reflecting how usage influences cost.

Domain of the Piecewise Function:

  • The domain of the piecewise function includes:
    • Segment 1: 0 ≤ x ≤ 100
    • Segment 2: 101 ≤ x ≤ 300
    • Segment 3: x > 300

This means that the function describes the bill for any monthly consumption of electricity starting from 0 kWh up to any reasonable upper limit of usage, reflecting how real residential users may consume electricity.

For the Recording: If you were explaining this to someone, you would prompt them to say their name at the beginning, such as:

"Hello, [Name], today I will explain a piecewise function that describes a typical electricity bill based on different usage rates."

You would then continue with the details as described above.