Chapter 2.4, Problem 51AYU

Federal Income Tax Two 2018 Tax Rate Schedules are given in the accompanying table. If $x$ equals taxable income and y equals the tax due, construct a function $y=f(x)$ for Scheduled $X$.

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

To determine

The function $y=f(x)$ for schedule $X$, 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 51AYU

Solution:

The function $y=f(x)$ for schedule $X$ is $y=f(x)=\{\begin{array}{l}0.1x\text{if}0<x\le 9525\\ 0.12x-190.5\text{if}9525<x\le 38700\\ 0.22x-4060.5\text{if}38700<x\le 82500\\ 0.24x-5710.5\text{if}82500<x\le 157500\\ 0.32x-18310.50\text{if}157500<x\le 200000\\ 0.35x-24310.50\text{if}200000<x\le 500000\\ 0.37x-34310.5\text{if}500000<x\end{array}$

, 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 $X$, 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 $\$9,525$, then the tax amount is $\$0$, plus the $10\%$ of the income

That is, the tax due $f(x)$ for income $0<x\le 9525$ is $f(x)=0+0.1x=0.1x$…(1)

Now, if the income is over $\$9,525$, but not greater than $\$38,700$, then the tax amount is $\$952.50$, plus the $12\%$ of the income $(x-9525)$.

That is, the tax due $f(x)$ for income $9525<x\le 38700$ is $f(x)=952.50+0.12(x-9525)=0.12x-190.5$…(2)

Now, if the income is over $\$38,700$, but not greater than $\$82,500$, then the tax amount is $\$4,453.50$, plus the $22\%$ of the income $(x-38700)$.

That is, the tax due $f(x)$ for income $38700<x\le 82500$ is $f(x)=4453.5+0.22(x-38700)=0.22x-4060.5$ …(3)

Now, if the income is over $\$82,500$, but not greater than $\$157,500$, then the tax amount is $\$14,089.50$, plus the $24\%$ of the income $(x-82500)$.

That is, the tax due $f(x)$ for income $82500<x\le 157500$ is $f(x)=14,089.50+0.24(x-82500)=0.24x-5710.5$…(4)

Now, if the income is over $\$157,500$, but not greater than $\$200,000$, then the tax amount is $\$32,089.50$, plus the $32\%$ of the income $(x-157500)$

That is, the tax due $f(x)$ for income $157500<x\le 200000$ is $f(x)=32,089.50+0.32(x-157500)=0.32x-18310.50$…(5)

Now, if the income is over $\$200,000$, but not greater than $\$500,000$, then the tax amount is $\$45689.50$, plus the $35\%$ of the income $(x-200000)$.

That is, the tax due $f(x)$ for income $200000<x\le 500000$ is

$f(x)=45689.50+0.35(x-200000)=0.35x-24310.50$…(6)

Now, if the income is over $\$500,000$, then the tax amount is $\$150,689.50$, plus the $37\%$ of the income $(x-500000)$

That is, the tax due $f(x)$ for income $500000<x$ is $f(x)=150,689.50+0.37(x-500000)=0.37x-34310.5$ …(7)

From (1) to (6),

Therefore, the function $y=f(x)$ for schedule $X$ is $y=f(x)=\{\begin{array}{l}0.1x\text{if}0<x\le 9525\\ 0.12x-190.5\text{if}9525<x\le 38700\\ 0.22x-4060.5\text{if}38700<x\le 82500\\ 0.24x-5710.5\text{if}82500<x\le 157500\\ 0.32x-18310.50\text{if}157500<x\le 200000\\ 0.35x-24310.50\text{if}200000<x\le 500000\\ 0.37x-34310.5\text{if}500000<x\end{array}$

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