Math Problem Statement

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

Solution

To construct the tax function y=f(x)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≤8,0000 < x \leq 8,000:
    • Tax is 10% of the taxable income over 0. f(x)=0.10xf(x) = 0.10x
  2. For 8,000<x≤30,6008,000 < x \leq 30,600:
    • Tax is $800 plus 16% of the taxable income over 8,000. f(x)=800+0.16(x−8,000)f(x) = 800 + 0.16(x - 8,000)
  3. For 30,600<x≤74,10030,600 < 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)f(x) = 4,416 + 0.24(x - 30,600)
  4. For x>74,100x > 74,100:
    • Tax is $14,856 plus 33% of the taxable income over 74,100. f(x)=14,856+0.33(x−74,100)f(x) = 14,856 + 0.33(x - 74,100)

Piecewise Function Representation:

We can now define the function y=f(x)y = f(x) as follows:

\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.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Algebra

Formulas

f(x) = 0.10x if 0 < x ≤ 8,000
f(x) = 800 + 0.16(x - 8,000) if 8,000 < x ≤ 30,600
f(x) = 4,416 + 0.24(x - 30,600) if 30,600 < x ≤ 74,100
f(x) = 14,856 + 0.33(x - 74,100) if x > 74,100

Theorems

Piecewise Function Definition

Suitable Grade Level

Grades 11-12 (Pre-Calculus and Algebra)

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