Chapter 2.4, Problem 52AYU

Federal Income Tax Refer to the 2018 tax rate schedules. If $x$ equals taxable income and y equals the tax due, construct a function $y=f(x)$ for Scheduled $Y-1$.

$\begin{array}{c}\text{2018}\text{Tax}\text{Rate}\\ \begin{array}{cccccccccccc}& \text{Schedule}& X& -& \text{Single}& & \text{Schedule}& \text{Y-1-Married}& \text{Filing}& \text{Jointly}\text{or}& \text{Qualified}& \text{Widow(er)}\\ \begin{array}{l}\text{If}\text{Taxable}\\ \text{Income}\text{is}\\ \text{Over}\end{array}& \begin{array}{l}\text{But}\text{Not}\\ \text{Over}\end{array}& \begin{array}{l}\text{The}\text{Tax}\text{is}\\ \text{This}\\ \text{Amount}\end{array}& & \begin{array}{l}\text{Plus}\\ \text{this}\text{\%}\end{array}& \begin{array}{l}Ofthe\\ Excess\\ Over\end{array}& \begin{array}{l}\text{If}\text{Taxable}\\ \text{Income}\text{is}\\ \text{Over}\end{array}& \begin{array}{l}\text{But}\text{Not}\\ \text{Over}\end{array}& \begin{array}{l}\text{The}\text{Tax}\text{is}\\ \text{This}\\ \text{Amount}\end{array}& & \begin{array}{l}\text{Plus}\\ \text{this}\text{\%}\end{array}& \begin{array}{l}Ofthe\\ Excess\\ Over\end{array}\\ \$0& \$9,525& \$0& +& 10\%& \$0& \$0& \$19,050& \$0& +& 10\%& \$0\\ 9,525& 38,700& 952.50& +& 12\%& 9,525& 19,050& 77,400& 1,905& +& 12\%& 19,050\\ 38,700& 82,500& 4,453.50& +& 22\%& 38,700& 77,400& 165,000& 8,907.00& +& 22\%& 77,400\\ 82,500& 157,500& 14,089.50& +& 24\%& 82,500& 165,000& 315,000& 28,179.00& +& 24\%& 165,000\\ 157,500& 200,000& 32,089.50& +& 32\%& 157,500& 315,000& 400,000& 64,179.00& +& 32\%& 315,000\\ 200,000& 500,000& 45,689.50& +& 35\%& 200,000& 400,000& 600,000& 91,379.00& +& 35\%& 400,000\\ 500,000& -& 150,689.50& +& 37\%& 500,000& 600,000& -& 161,379.00& +& 37\%& 600,000\end{array}\end{array}$

To determine

The function $y=f(x)$ for schedule $Y-1$, where $x$ is the taxable income and $y$ is the tax due. Federal income tax two rate schedules of $2018$ are as below:

$\begin{array}{cccccccccccc}& & & & 2018& \text{Tax Rate }& & & & & & \\ & & \text{Schedules}& X-& \text{Single}& & & \text{Schedules}& Y-1-& \text{Married}& FillingJoinly& \\ \begin{array}{c}\text{If Taxable }\\ \text{Income is}\\ \text{Over}\end{array}& \begin{array}{c}\text{But Not}\\ \text{Over}\end{array}& \begin{array}{c}\text{The Tax is}\\ \text{This}\\ \text{Amount}\end{array}& & \begin{array}{c}\text{Plus This }\\ \text{\%}\end{array}& \begin{array}{c}\text{Of the }\\ \text{Excess}\\ \text{Over }\end{array}& \begin{array}{c}\text{If Taxable }\\ \text{Income is}\\ \text{Over}\end{array}& \begin{array}{c}\text{But Not}\\ \text{Over}\end{array}& \begin{array}{c}\text{The Tax is}\\ \text{This}\\ \text{Amount}\end{array}& & \begin{array}{c}\text{Plus This }\\ \text{\%}\end{array}& \begin{array}{c}\text{Of the }\\ \text{Excess}\\ \text{Over }\end{array}\\ \$0& \$9,525& \$0& +& 10\%& \$0& \$0& \$19,050& \$0& +& 10\%& \$0\\ 9,525& 38,700& 952.5& +& 12\%& 9,525& 19,050& 77,400& 1,905& +& 12\%& 19,050\\ 38,700& 82,500& 4,453.50& +& 22\%& 38,700& 77,400& 165,000& 8,907.00& +& 22\%& 77,400\\ 82,500& 157,500& 14,089.50& +& 24\%& 82,500& 165,000& 315,000& 28,179.00& +& 24\%& 165,000\\ 157,500& 200,000& 32,089.50& +& 32\%& 157,500& 315,000& 400,000& 64,179.00& +& 32\%& 315,000\\ 200,000& 500,000& 45,689.50& +& 35\%& 200,000& 400,000& 600,000& 91,379.00& +& 35\%& 400,000\\ 500,000& -& 150,689.50& +& 37\%& 500,000& 600,000& -& 161,379.00& +& 37\%& 600,000\end{array}$

Answer to Problem 52AYU

Solution:

The function $y=f(x)$ for schedule $Y-1$ is $y=f(x)=\{\begin{array}{l}0.1x\text{if}0<x\le 19050\\ 0.12x-381\text{if}19050<x\le 77400\\ 0.22x-8121\text{if}77400<x\le 165000\\ 0.24x-11421\text{if}165000<x\le 315000\\ 0.32x-36621\text{if}315000<x\le 400000\\ 0.35x-48621\text{if}400000<x\le 600000\\ 0.37x-60621\text{if}600000<x\end{array}$

, where $x$ is the taxable income and $y$ is the tax due

Explanation of Solution

Given Information:

Federal income tax two rate schedules of $2018$ are as below:

$\begin{array}{cccccccccccc}& & & & 2018& \text{Tax Rate }& & & & & & \\ & & \text{Schedules}& X-& \text{Single}& & & \text{Schedules}& Y-1-& \text{Married}& FillingJoinly& \\ \begin{array}{c}\text{If Taxable }\\ \text{Income is}\\ \text{Over}\end{array}& \begin{array}{c}\text{But Not}\\ \text{Over}\end{array}& \begin{array}{c}\text{The Tax is}\\ \text{This}\\ \text{Amount}\end{array}& & \begin{array}{c}\text{Plus This }\\ \text{\%}\end{array}& \begin{array}{c}\text{Of the }\\ \text{Excess}\\ \text{Over }\end{array}& \begin{array}{c}\text{If Taxable }\\ \text{Income is}\\ \text{Over}\end{array}& \begin{array}{c}\text{But Not}\\ \text{Over}\end{array}& \begin{array}{c}\text{The Tax is}\\ \text{This}\\ \text{Amount}\end{array}& & \begin{array}{c}\text{Plus This }\\ \text{\%}\end{array}& \begin{array}{c}\text{Of the }\\ \text{Excess}\\ \text{Over }\end{array}\\ \$0& \$9,525& \$0& +& 10\%& \$0& \$0& \$19,050& \$0& +& 10\%& \$0\\ 9,525& 38,700& 952.5& +& 12\%& 9,525& 19,050& 77,400& 1,905& +& 12\%& 19,050\\ 38,700& 82,500& 4,453.50& +& 22\%& 38,700& 77,400& 165,000& 8,907.00& +& 22\%& 77,400\\ 82,500& 157,500& 14,089.50& +& 24\%& 82,500& 165,000& 315,000& 28,179.00& +& 24\%& 165,000\\ 157,500& 200,000& 32,089.50& +& 32\%& 157,500& 315,000& 400,000& 64,179.00& +& 32\%& 315,000\\ 200,000& 500,000& 45,689.50& +& 35\%& 200,000& 400,000& 600,000& 91,379.00& +& 35\%& 400,000\\ 500,000& -& 150,689.50& +& 37\%& 500,000& 600,000& -& 161,379.00& +& 37\%& 600,000\end{array}$

Explanation:

For schedule $Y-1$, the tax due $y=f(x)$ for the taxable income $x$ is as below:

Given that,

If the income is over $\$0$, but not greater than $\$19,050$, the tax amount is $\$0$, plus the $10\%$ of the income.

That is, the tax due $f(x)$ for income $0<x\le 19050$ is $f(x)=0+0.1x=0.1x$…(1)

Now, if the income is over $\$19,050$, but not greater than $\$77,400$, the tax amount is $\$1,905$, plus the $12\%$ of the income $(x-19050)$.

That is, the tax due $f(x)$ for income $19050<x\le 77400$ is $f(x)=1905+0.12(x-19050)=0.12x-381$…(2)

Now, if the income is over $\$77,400$, but not greater than $\$165,000$, the tax amount is $\$8,907.00$, plus the $22\%$ of the income $(x-77400)$

That is, the tax due $f(x)$ for income $77400<x\le 165000$ is $f(x)=8,907.00+0.22(x-77400)=0.22x-8121$…(3)

Now, if the income is over $\$165,000$, but not greater than $\$315,000$, the tax amount is $\$28,179.00$, plus the $24\%$ of the income $(x-165000)$.

That is, the tax due $f(x)$ for income $165000<x\le 315000$ is $f(x)=28179.00+0.24(x-165000)=0.24x-11421$…(4)

Now, if the income is over $\$315,000$, but not greater than $\$400,000$, the tax amount is $\$64,179.00$, plus the $32\%$ of the income $(x-315000)$

That is, the tax due $f(x)$ for income $315000<x\le 400000$ is $f(x)=64179+0.32(x-315000)=0.32x-36621$…(5)

Now, if the income is over $\$400,000$, but not greater than $\$600,000$, the tax amount is $\$91,379$, plus the $35\%$ of the income $(x-400000)$

That is, the tax due $f(x)$ for income $400000x\le 600000$ is

$f(x)=91,379+0.35(x-400000)=0.35x-48621$…(6)

Now, if the income is over $\$600,000$, then the tax amount is $\$161,379$, plus the $37\%$ of the income $(x-600000)$.

That is, the tax due $f(x)$ for income $600000<x$ is $f(x)=161,379+0.37(x-600000)=0.37x-60621$…(7)

From (1) to (7),

Therefore, the function $y=f(x)$ for schedule $Y-1$ is $y=f(x)=\{\begin{array}{l}0.1x\text{if}0<x\le 19050\\ 0.12x-381\text{if}19050<x\le 77400\\ 0.22x-8121\text{if}77400<x\le 165000\\ 0.24x-11421\text{if}165000<x\le 315000\\ 0.32x-36621\text{if}315000<x\le 400000\\ 0.35x-48621\text{if}400000<x\le 600000\\ 0.37x-60621\text{if}600000<x\end{array}$, where $x$ is the taxable income and $y$ is the tax due.