Question

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

Answer 1

Describe the scenario you are modeling

Answer 2

Consider a mobile phone billing system that charges customers based on their usage of data, with different rates applied depending on the amount of data consumed. The billing can be modeled as a piecewise function with non-linear components.

Here's how the billing system works:

1. For the first 2 GB of data used, there is a flat rate of $20. This first segment can be represented as a constant function:
\[
f(x) = 20 \quad \text{for } 0 \leq x \leq 2
\]

2. For data usage between 2 GB and 5 GB, the user is charged $10 per additional GB (which is linear). For this range, the cost can be represented as:
\[
f(x) = 20 + 10(x - 2) \quad \text{for } 2 < x \leq 5
\]
This simplifies to:
\[
f(x) = 10x \quad \text{for } 2 < x \leq 5
\]

3. For usage above 5 GB, the user starts to incur overage charges at a rate of $15 per GB, but the first 5 GB are still charged at the previous rates. Therefore, the cost for this segment would be:
\[
f(x) = 50 + 15(x - 5) \quad \text{for } x > 5
\]
This simplifies to:
\[
f(x) = 15x - 25 \quad \text{for } x > 5
\]

Combining all these pieces, the piecewise function representing the mobile phone billing would look like this:

\[
f(x) =
\begin{cases}
20 & \text{if } 0 \leq x \leq 2 \\
10x & \text{if } 2 < x \leq 5 \\
15x - 25 & \text{if } x > 5
\end{cases}
\]

Here, the function consists of a constant (flat fee) section, a linear (increasing) section, and a nonlinear (increasing with a different slope) section, making it a piecewise function with non-linear components.