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:
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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: \[ f(x) = 20 \quad \text{for } 0 \leq x \leq 2 \]
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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: \[ f(x) = 10x \quad \text{for } 2 < x \leq 5 \]
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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.