Representing a Piece-Wise Functions

Problem #6

The post office offers flat-rate mailing of packages: 15.50 for a package weighing less than 4 oz, P22.50 for a package weighing 4 oz to less than 8 oz and P30.50 for a package weighing 8 oz to 12 oz. Express the cost of packages as a function of x ounces.

Problem #7

The base jeepney fare is P8.00 for the first four kilometers and an additional P1.50 per kilometers for the succeeding kilometers. Express the fare as a function of x kilometers.

Problem #8

Income tax of 5% will be applied on the first P50,000 of income or less, and 8% on any income in excess of P50,000. Express the tax as a function of x.

Let's break down each problem into a piecewise function. 

### Problem #6: Postal Rate Based on Weight

The post office charges different rates depending on the weight of the package:

- **\(15.50\)** for a package weighing less than 4 oz.
- **\(22.50\)** for a package weighing from 4 oz to less than 8 oz.
- **\(30.50\)** for a package weighing from 8 oz to 12 oz.

Let \(C(x)\) be the cost as a function of the package's weight \(x\) in ounces:

\[
C(x) =
\begin{cases} 
15.50 & \text{if } 0 \leq x < 4 \\
22.50 & \text{if } 4 \leq x < 8 \\
30.50 & \text{if } 8 \leq x \leq 12
\end{cases}
\]

### Problem #7: Jeepney Fare Based on Distance

The jeepney fare is calculated based on the distance traveled:

- **\(8.00\)** for the first 4 kilometers.
- **\(1.50\)** per kilometer for any additional distance beyond 4 kilometers.

Let \(F(x)\) be the fare as a function of the distance \(x\) in kilometers:

\[
F(x) =
\begin{cases} 
8.00 & \text{if } 0 \leq x \leq 4 \\
8.00 + 1.50(x - 4) & \text{if } x > 4
\end{cases}
\]

### Problem #8: Income Tax Based on Income

Income tax is calculated based on the amount of income:

- **5%** tax on the first \(50,000\) or less.
- **8%** tax on any income exceeding \(50,000\).

Let \(T(x)\) be the tax as a function of income \(x\):

\[
T(x) =
\begin{cases} 
0.05x & \text{if } 0 \leq x \leq 50,000 \\
0.05(50,000) + 0.08(x - 50,000) & \text{if } x > 50,000
\end{cases}
\]

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Would you like more details or have any questions? Here are some related questions to consider:

1. How would you graph these piecewise functions?
2. What would be the domain and range of each piecewise function?
3. How would you write the piecewise function for different tax brackets?
4. What happens if the weight or distance exceeds the given ranges?
5. How would the functions change if the base fare or tax rate was adjusted?

**Tip:** When working with piecewise functions, always carefully consider the boundary points where the function's behavior changes.

Explore how to express postal rates, jeepney fares, and income tax as piecewise functions. Understand the cost of packages based on weight, jeepney fare based on distance, and income tax based on income brackets. Suitable for high school students and learners interested in mathematical applications.

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