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**Part a) Calculate the tax on $125,000.**

The tax policy is as follows:
- The first $50,000 is taxed at 5%.
- Any income above $50,000 is taxed at 12%.

For an income of $125,000:
1. The first $50,000 is taxed at 5%, so:
   \[
   \text{Tax on first } 50,000 = 50,000 \times 0.05 = 2,500
   \]

2. The remaining income above $50,000 is:
   \[
   125,000 - 50,000 = 75,000
   \]
   This amount is taxed at 12%, so:
   \[
   \text{Tax on the remaining } 75,000 = 75,000 \times 0.12 = 9,000
   \]

3. Total tax:
   \[
   \text{Total tax} = 2,500 + 9,000 = 11,500
   \]

Thus, the total tax on $125,000 is **$11,500**.

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**Part b) Write a function that models the tax policy.**

Let $ T(x) $ represent the tax on an income $ x $, where:
- $ x \leq 50,000 $ is taxed at 5%, and
- $ x > 50,000 $ is taxed at 5% for the first $50,000 and 12% for the remainder.

The piecewise function for $ T(x) $ can be written as:

\[
T(x) = \begin{cases}
0.05x & \text{if } 0 \leq x \leq 50,000 \\
0.05(50,000) + 0.12(x - 50,000) & \text{if } x > 50,000
\end{cases}
\]

This function correctly models the tax policy described.

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Do you need more details or have any questions?

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Here are five related questions for further exploration:
1. What would be the tax on an income of $200,000 using the same tax policy?
2. How would the tax change if the rate above $50,000 was increased to 15%?
3. How does the tax compare for incomes of $50,000 and $125,000 in percentage terms?
4. What is the marginal tax rate for someone earning $75,000?
5. How would the tax function change if there was an additional tax bracket at $100,000?

**Tip**: Piecewise functions are useful when different rules apply to different sections of an input range, such as in tax calculations with different rates for different income brackets.

A certain tax policy states that the first $50,000 of income is taxed at 5% and any income above $50,000 is taxed at 12%. a) Calculate the tax on $125,000. b) Write a function that models the tax policy.

Learn how to calculate the tax on an income of $125,000 using a piecewise function that applies a 5% tax rate on the first $50,000 and a 12% tax rate on any income above $50,000. This problem covers essential concepts such as percentages and tax policies, suitable for students in Grades 9-12.

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