A tax rate schedule is given in the table. If x equals taxable income and y equals the tax​ due, construct a function y equals f left parenthesis x right parenthesis for the tax schedule.
If taxable income is over
But not over
The tax is this amount
Plus this​ %
Of the excess over
0
8 comma 000
0.00
plus
10 %
0
8 comma 000
30 comma 600
800.00
plus
16 %
8 comma 000
30 comma 600
74 comma 100
4 comma 416.00
plus
24 %
30 comma 600
74 comma 100
14 comma 856.00
plus
33 %
74 comma 100
Question content area bottom
Part 1
f left parenthesis x right parenthesis equals left brace Start 4 By 2 Matrix 1st Row 1st Column 0.1 x 2nd Column if 0 less than x less than or equals 8 comma 000 2nd Row 1st Column nothing plus nothing left parenthesis x minus nothing right parenthesis 2nd Column if 8 comma 000 less than x less than or equals 30 comma 600 3rd Row 1st Column nothing plus nothing left parenthesis x minus nothing right parenthesis 2nd Column if 30 comma 600 less than x less than or equals 74 comma 100 4st Row 1st Column 14 comma 856.00 plus 0.33 left parenthesis x minus 74 comma 100 right parenthesis 2nd Column if x greater than 74 comma 100 EndMatrix

To construct the tax function $ y = f(x) $ based on the tax rate schedule, we define the function piecewise according to the intervals provided in the table. Let’s break this down step-by-step.

### Tax Schedule:
1. **For $ 0 < x \leq 8,000 $**:
   - Tax is 10% of the taxable income over 0.
   \[
   f(x) = 0.10x
   \]
2. **For $ 8,000 < x \leq 30,600 $**:
   - Tax is \$800 plus 16% of the taxable income over 8,000.
   \[
   f(x) = 800 + 0.16(x - 8,000)
   \]
3. **For $ 30,600 < x \leq 74,100 $**:
   - Tax is \$4,416 plus 24% of the taxable income over 30,600.
   \[
   f(x) = 4,416 + 0.24(x - 30,600)
   \]
4. **For $ x > 74,100 $**:
   - Tax is \$14,856 plus 33% of the taxable income over 74,100.
   \[
   f(x) = 14,856 + 0.33(x - 74,100)
   \]

### Piecewise Function Representation:
We can now define the function $ y = f(x) $ as follows:
\[
f(x) = 
\begin{cases} 
0.10x & \text{if } 0 < x \leq 8,000 \\
800 + 0.16(x - 8,000) & \text{if } 8,000 < x \leq 30,600 \\
4,416 + 0.24(x - 30,600) & \text{if } 30,600 < x \leq 74,100 \\
14,856 + 0.33(x - 74,100) & \text{if } x > 74,100
\end{cases}
\]

This function provides the tax due based on the income $ x $.

---

Would you like a detailed explanation for how each piece of the function was constructed, or do you have any other questions?

### Related Questions:
1. How can you calculate the tax for an income of \$50,000 using this function?
2. What is the tax for someone earning exactly \$30,600?
3. How does the function change if a new tax bracket is added for incomes over \$100,000?
4. Can this piecewise function be represented as a single expression?
5. How do tax credits affect the computation of the tax function?

### Tip:
When working with piecewise functions, always verify the intervals to ensure you are using the correct formula for your input value.

A tax rate schedule is given in the table. If x equals taxable income and y equals the tax​ due, construct a function y = f(x) for the tax schedule. The tax schedule is as follows:
- If taxable income is over $0 but not over $8,000: The tax is 10% of taxable income.
- If taxable income is over $8,000 but not over $30,600: The tax is $800 plus 16% of the amount over $8,000.
- If taxable income is over $30,600 but not over $74,100: The tax is $4,416 plus 24% of the amount over $30,600.
- If taxable income is over $74,100: The tax is $14,856 plus 33% of the amount over $74,100.

Learn how to construct a piecewise function for a tax schedule based on different income brackets. This problem covers key algebraic concepts such as calculating tax using piecewise functions, including detailed formulas for each income range.

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