Sure, Gabriella, let's work through the solution step by step.
### Part A: Calculating the Amount of Taxes Owed on a Taxable Income of \[tex]$31,000
We need to determine which tax bracket \$[/tex]31,000 falls into and use the corresponding piecewise function to calculate the taxes owed.
The tax brackets and their corresponding piecewise functions are as follows:
[tex]\[
f(x)=\left\{
\begin{array}{ll}
0.10x, & 0 \leq x \leq 10,275 \\
0.12x - 205.50, & 10,276 \leq x \leq 41,175 \\
0.22x - 4,323.00, & 41,176 \leq x \leq 89,075 \\
0.24x - 6,104.50, & 89,076 \leq x \leq 170,050 \\
0.32x - 19,708.50, & 170,051 \leq x \leq 215,950 \\
0.35x - 26,187.00, & 215,951 \leq x \leq 539,900 \\
0.37x - 36,985.00, & x \geq 539,901
\end{array}
\right.
\][/tex]
The taxable income of \[tex]$31,000 falls into the second tax bracket: \(10,276 \leq x \leq 41,175\). The piecewise function for this bracket is:
\[
f(x) = 0.12x - 205.50
\]
Plugging in \(x = 31,000\):
\[
f(31,000) = 0.12 \times 31,000 - 205.50
\]
\[
f(31,000) = 3,720 - 205.50
\]
\[
f(31,000) = 3,514.50
\]
So, the amount of taxes owed on a taxable income of \$[/tex]31,000 is \[tex]$3,514.50.
### Part B: Determining the Effective Tax Rate
The effective tax rate is calculated by dividing the total taxes owed by the taxable income and then converting it to a percentage.
\[
\text{Effective Tax Rate} = \left(\frac{\text{Taxes Owed}}{\text{Taxable Income}}\right) \times 100
\]
From Part A, we know the taxes owed are \$[/tex]3,514.50 and the taxable income is \[tex]$31,000.
\[
\text{Effective Tax Rate} = \left(\frac{3,514.50}{31,000}\right) \times 100
\]
\[
\text{Effective Tax Rate} \approx 11.3371\%
\]
So, the effective tax rate is approximately 11.34%.
### Part C: Comparison of the Piecewise Function to the Marginal Tax Rate Chart
The marginal tax rate chart is represented in the piecewise function by providing different linear functions for different intervals of taxable income. Each interval corresponds to a specific tax bracket.
For example:
- For incomes between \$[/tex]0 and \[tex]$10,275, the tax rate is 10%, represented by the function \(0.10x\).
- For incomes between \$[/tex]10,276 and \$41,175, the tax rate is 12%, but an adjustment is made for the prior bracket to ensure continuity, resulting in the function [tex]\(0.12x - 205.50\)[/tex].
Each segment of the piecewise function adjusts the tax calculation to account for the progressive rates applied to the preceding income ranges, ensuring that only the income within each bracket is taxed at the corresponding marginal rate.
