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Show all work. Give complete explanations. Ask your instructor for any clarification. Answer question #1 or #2. You may do both for extra credit.

1a. Give your own example of a real life piece-wise function. State what f(x) and x represents.

Example: Consider a mobile phone plan where the cost depends on the number of minutes used. Let f(x) represent the total monthly cost, and x represent the number of minutes used in a month. The pricing structure can be described as follows:

• If x ≤ 100 minutes, then f(x) = $20 (flat fee for up to 100 minutes)
• If x > 100 minutes, then f(x) = $20 + $0.10 for each additional minute over 100

In this context, x represents the number of minutes used per month, and f(x) represents the total monthly cost charged by the mobile plan.

1b. Calculate f(x) for two values of x. Explain what these results mean.

Let's choose x = 80 and x = 150.

For x = 80 minutes: Since 80 ≤ 100, we use the first piece of the function:

• f(80) = $20

This means that using 80 minutes costs $20, which is the flat fee for up to 100 minutes.

For x = 150 minutes: Since 150 > 100, we use the second piece:

• f(150) = $20 + $0.10 × (150 - 100) = $20 + $0.10 × 50 = $20 + $5 = $25

This indicates that using 150 minutes incurs a total cost of $25, which includes the flat fee plus additional charges for the extra minutes.

These calculations demonstrate how the cost changes based on usage, reflecting the piecewise nature of the function.

1c. Explain why it is necessary for this function to be defined piece-wise.

The function is defined piece-wise because the pricing structure changes at the 100-minute mark. For the first 100 minutes, the cost is a fixed flat fee, which is simpler to represent as a constant function. Beyond 100 minutes, the cost increases at a different rate per additional minute, requiring a separate linear expression. Combining these behaviors into a single, continuous formula without segmentation would be complicated and less clear, especially considering the different billing rules. Thus, defining the function piece-wise accurately models the real-world billing system where different rules apply in different usage ranges.

Paper For Above instruction

Understanding piece-wise functions through real-world examples provides valuable insights into their practical applications and theoretical structure. A piece-wise function is a function defined by different expressions over different parts of its domain, typically used to model situations where behavior changes at certain thresholds. One clear example arises in mobile phone billing plans, where the total cost depends on usage tiers. Such functions are essential in representing real-world systems accurately, especially when distinct rules or conditions apply within different regions of the domain.

For example, consider a mobile phone plan with a fixed fee for up to 100 minutes and an additional charge for extra minutes. Let x be the number of minutes used in a month, and let f(x) be the total monthly cost. The function can be constructed as follows: for the first 100 minutes, f(x) = $20, representing the flat fee. If the usage exceeds 100 minutes, the cost increases at a rate of $0.10 per additional minute, described by the second piece: f(x) = $20 + $0.10(x - 100) for x > 100. This piece-wise definition captures the billing policy's true nature and allows users to understand how their charges are calculated based on their consumption.

Calculating specific values of the function helps illustrate its practical use. For instance, if a user uses 80 minutes, applying the first piece yields a cost of $20, which makes sense because the usage falls within the flat fee range. For 150 minutes, the second piece applies: f(150) = $20 + $0.10 × (150 - 100) = $25. This example demonstrates how additional usage incurs extra charges, reflecting the tiered pricing structure. Such calculations are vital for consumers and providers to understand billing implications and plan accordingly.

Defining this function as piece-wise is necessary due to the billing system's structure. A single formula would be inadequate to describe both the flat fee and the incremental charges effectively. A piece-wise function clearly distinguishes the two scenarios: one for 0–100 minutes and another for usage beyond 100. This segmentation simplifies the representation and provides clarity, accurately reflecting the billing policy's operational logic. It also emphasizes the importance of piece-wise functions in modeling systems with different rules applied at various thresholds, such as tax brackets, shipping fees, and insurance premiums.

Beyond utility billing, piece-wise functions are widely used in economics, engineering, and environmental science. They model situations where different conditions lead to different outcomes, such as tax rates, stress-strain relationships, or pollution levels. Recognizing the utility of piece-wise functions enhances our ability to interpret complex data and develop effective solutions for real-world problems. Mastery of their structure and application is essential for students and professionals working in quantitative fields, underpinning accurate analysis and strategic decision-making.

In conclusion, the example of a mobile plan illustrates the necessity and utility of piece-wise functions in real-world applications. They provide a flexible framework for modeling systems with segmented rules, facilitating clear understanding and precise calculations. By employing such functions, we can better analyze, predict, and optimize outcomes across diverse disciplines, demonstrating their fundamental importance in mathematical modeling and problem-solving.

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