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To write a piecewise function $ T(x) $ that represents the tax due on an individual's income based on the provided income tax schedule, we need to break the income brackets into three parts: $ 0 \leq x \leq 15,000 $, $ 15,000 < x \leq 30,000 $, and $ x > 30,000 $.

### Tax Rates from the Table (for individuals):

1. **Income from $0 to $15,000**: The tax rate is 2.5% of the taxable income.
   \[
   T(x) = 0.025x
   \]

2. **Income from $15,000 to $30,000**: The tax is \$375 plus 5.55% of the income over \$15,000.
   \[
   T(x) = 375 + 0.0555(x - 15,000)
   \]

3. **Income over $30,000**: The tax is \$1,207.50 plus 6.75% of the income over \$30,000.
   \[
   T(x) = 1,207.50 + 0.0675(x - 30,000)
   \]

### Piecewise Definition:
\[
T(x) =
\begin{cases}
0.025x & \text{if } 0 \leq x \leq 15,000 \\
375 + 0.0555(x - 15,000) & \text{if } 15,000 < x \leq 30,000 \\
1,207.50 + 0.0675(x - 30,000) & \text{if } x > 30,000
\end{cases}
\]

Would you like a more detailed explanation, or do you have any questions about this piecewise function?

### Related Questions:
1. How does the tax rate change for incomes over $30,000?
2. What is the total tax for an income of $20,000?
3. How would the tax calculation differ for married couples filing jointly?
4. Can this piecewise function be represented graphically, and how would it look?
5. What is the marginal tax rate for someone earning exactly $30,000?

### Tip:
Piecewise functions are a great way to represent tax brackets since each income range has a different calculation. Always watch for boundaries in such functions!

Write a piecewise definition for the tax due T(x) on an income of x dollars for an individual, using the provided income tax schedule.

Learn how to create a piecewise function to calculate state income tax based on income brackets. This solution explains tax rates for incomes from $0 to over $30,000 using algebraic expressions and piecewise definitions. Suitable for students in Grades 9-12.

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Solution

Tax Rates from the Table (for individuals):

Piecewise Definition:

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