Students will:
(1.) Discuss piecewise functions.
(2.) Discuss real-world applications of piecewise-functions.
(3.) Solve problems involving the application of piecewise functions.
(4.) Meet one of the learning objectives of the VCCS (Virginia Community College System) standards for:
MTH 161: PreCalculus I
(Presents topics in power, polynomial, rational, exponential, and logarithmic functions, and systems of equations and inequalities.).
MTH 167:
PreCalculus with Trigonometry
(Presents topics in power, polynomial, rational, exponential, and logarithmic functions, systems of equations, trigonometry, and trigonometric applications, including Law of Sines and Cosines, and an introduction to conics.).
(5.) Meet the QM (Quality Matters) and USDOE (United States Department of Education) requirements for distance
education as regards the provision of RSI (Regular and Substantive Interaction).
Federal Register: Distance Education and Innovation
St. John's University: New Federal Requirements for Distance Education: Regular and Substantive
Interaction (RSI)
Student – Content Interaction: Very high
Student – Student Interaction: Flexible
Student – Faculty Interaction: High
Another important real-world application of Piecewise Functions is in the filing of federal income taxes.
We shall illustrate this application by filing the "Single" option for the 2018 federal tax return.
We shall design a calculator for these taxes.
We shall verify our calculations with the calculators of some tax companies.
For the simple calculations (only taxes), we shall verify it using the:
calculator from TaxAct
Let us define some important terms used in Income Taxes.
The gross income is the total yearly income. This is the total income earned in a year.
It is the total pre-tax earnings for the year.
Let the gross income = $GI$
Sometimes, there are certain portions of the gross income that is not taxed.
In other words, there are certain untaxed portions of the gross income.
Those untaxed portions of the gross income are known as deductions or adjustments.
Let the adjustments = $A$
The difference between the gross income and the adjustments is known as the adjusted gross income
Let the adjusted gross income = $AGI$
The adjusted gross income is the income after allowable tax deductions.
The information used for the 2018 tax rates, standard deductions, and exemptions is found in these websites:
(1.) Bankrate
(2.) Forbes
Ask students to compare and contrast the two tables.
The information from the first website (Bankrate) is summarized here.
We shall work with this one first.
Tax Rate | Single | Head of household |
Married Filing Jointly Or Qualifying Widow |
Married Filing Separately |
---|---|---|---|---|
$10\%$ | Up to $\$9,525$ | Up to $\$13,600$ | Up to $\$19,050$ | Up to $\$9,525$ |
$12\%$ | $\$9,526 \:\:to\:\: \$38,700$ | $\$13,601 \:\:to\:\: \$51,800$ | $\$19,051 \:\:to\:\: \$77,400$ | $\$9,526 \:\:to\:\: \$38,700$ |
$22\%$ | $\$38,701 \:\:to\:\: \$82,500$ | $\$51,801 \:\:to\:\: \$82,500$ | $\$77,401 \:\:to\:\: \$165,000$ | $\$38,701 \:\:to\:\: \$82,500$ |
$24\%$ | $\$82,501 \:\:to\:\: \$157,500$ | $\$82,501 \:\:to\:\: \$157,500$ | $\$165,001 \:\:to\:\: \$315,000$ | $\$82,501 \:\:to\:\: \$157,000$ |
$32\%$ | $\$157,501 \:\:to\:\: \$200,000$ | $\$157,501 \:\:to\:\: \$200,000$ | $\$315,001 \:\:to\:\: \$400,000$ | $\$157,001 \:\:to\:\: \$200,000$ |
$35\%$ | $\$200,001 \:\:to\:\: \$500,000$ | $\$200,001 \:\:to\:\: \$500,000$ | $\$400,001 \:\:to\:\: \$600,000$ | $\$200,001 \:\:to\:\: \$300,000$ |
$37\%$ | $\$500,001 \:\:or\:\: more$ | $\$500,001 \:\:or\:\: more$ | $\$600,001 \:\:or\:\: more$ | $\$300,001 \:\:or\:\: more$ |
Calculate the taxes for single individuals whose taxable incomes are:
(1.) $\$7,000.00$
(2.) $\$12,575.00$
(3.) $\$43,750.00$
(4.) $\$120,327.00$
(5.) $\$165,428.00$
(6.) $\$234,543.00$
(7.) $\$700,712.00$
Simplify the percents.
$
10\% = \dfrac{10}{100} = 0.1 \\[5ex]
12\% = \dfrac{12}{100} = 0.12 \\[5ex]
22\% = \dfrac{22}{100} = 0.22 \\[5ex]
24\% = \dfrac{24}{100} = 0.24 \\[5ex]
32\% = \dfrac{32}{100} = 0.32 \\[5ex]
35\% = \dfrac{35}{100} = 0.35 \\[5ex]
37\% = \dfrac{37}{100} = 0.37 \\[5ex]
$
Solution: $1st$ Method: Manual/Arithmetic Method
(1.) $\$7,000.00$ falls in the first piece.
tax for $\$7000.00$ income @ $10\%$ per $\$$ earned = $7000 * 0.1 = 700$
tax for taxable income of $\$7,000.00$ = $\$700.00$
(2.) $\$12,575.00$ falls in the second piece.
Before we use the second piece, we have to go through the first piece first.
First Piece for $\$9525.00$
$9525.00$ of the $12575.00$ is taxed at $10\%$
tax for $\$9525.00$ income @ $10\%$ per $\$$ earned = $9525.00 * 0.1 = 952.50$
We are done with the maximum taxable income that can be taxed at the first piece of $10\%$
We have to move on to the second piece.
$12575.00 - 9525.00 = 3050.00$
The rest of the $\$3050.00$ is taxed at $12\%$
Second Piece for $\$3050.00$
tax for $\$3050.00$ income @ $12\%$ per $\$$ earned = $3050.00 * 0.12 = 366.00$
$952.50 + 366.00 = 1318.50$
tax for taxable income of $\$12,575.00$ = $\$1,318.50$
(3.) $\$43,750.00$ falls in the third piece.
Before we use the third piece, we have to go through the first and second pieces.
First Piece for $\$9525.00$
$9525.00$ of the $12575.00$ is taxed at $10\%$
tax for $\$9525.00$ income @ $10\%$ per $\$$ earned = $9525.00 * 0.1 = 952.50$
We are done with the maximum taxable income that can be taxed at the first piece of $10\%$
We have to move on to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate.
Student: So, do we need to subtract $9525.00$ from $43750.28$
Teacher: No, we do not.
Here is the thing.
We know already that $43750.28$ falls in the third tax bracket.
Student: That is correct.
Teacher: $9,525$ of that money is taxed at $10\%$
Student: Okay...
Teacher: Any money above $9525$ but not exceeding $38700$ is taxed at $12\%$
Not all the money (43750.00 - 9525 = 34225.00) is taxed at $12\%$
Let us look at it this way
For $43750.28$
$[0, 9525]$ is taxed at $10\%$
$(9525, 38700]$ is taxed at $12\%$
$(38700, 43750.28]$ is taxed at $22\%$
Do you realize the mistake that would have been made if $34225.00$ was taxed at $12\%$
Student: Yes, the U,S government would lose money.
Teacher: That is right...and we do not ...
Student: want them to lose money.
Teacher: Correct. That money could be used to help the poor.
Student: I hope they help poor people. There is a high rate of hunger and poverty in the world.
Teacher: I hope and pray so too.
So, as you can see:
The range of $9525 - 0 = 9525$ is taxed at $10\%$
Student: The range of $38700 - 9525 = 29175$ is taxed at $12\%$
Teacher: Correct! Go ahead...
Student: and the range of $43750.00 - 38700 = 5050.00$ is taxed at $22\%$
Teacher: Perfecto!
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$
Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We have to move on to the third piece.
$43750.00 - 38700.00 = 5050.00$
The rest of the $\$5050.00$ is taxed at $22\%$
Third Piece for $\$14575.00$
tax for $\$5050.00$ income @ $22\%$ per $\$$ earned = $5050.00 * 0.22 = 1111.00$
$952.50 + 3501.00 + 1111.00 = 5564.50$
tax for taxable income of $\$43,750.00$ = $\$5,564.50$
(4.) $\$120, 327.00$ falls in the fourth piece.
Before we use the fourth piece, we have to go through the first, second, and third pieces.
First Piece for $\$9525.00$
$9525.00$ of the $12575.00$ is taxed at $10\%$
tax for $\$9525.00$ income @ $10\%$ per $\$$ earned = $9525.00 * 0.1 = 952.50$
We are done with the maximum taxable income that can be taxed at the first piece of $10\%$
We move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate.
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$
Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$
Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the third piece of $22\%$
We move to the fourth piece.
$120327.00 - 82500.00 = 37827.00$
The rest of the $\$37827.00$ is taxed at $24\%$
Fourth Piece for $\$37827.00$
tax for $\$37827.00$ income @ $24\%$ per $\$$ earned = $37827.00 * 0.24 = 9078.48$
$952.50 + 3501.00 + 9636.00 + 9078.48 = 23167.98$
tax for taxable income of $\$120,327.00$ = $\$23,167.98$
(5.) $\$165, 428.00$ falls in the fifth piece.
Before we use the fifth piece, we have to go through the first, second, third, and fourth pieces.
First Piece for $\$9525.00$
$9525.00$ of the $12575.00$ is taxed at $10\%$
tax for $\$9525.00$ income @ $10\%$ per $\$$ earned = $9525.00 * 0.1 = 952.50$
We are done with the maximum taxable income that can be taxed at the first piece of $10\%$
We move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate.
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$
Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$
Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the third piece of $22\%$
We move to the fourth piece.
We need to calculate the range of taxable income that should be taxed at the fourth tax rate.
$157500.00 - 82500.00 = 75000.00$
$\$75000.00$ is taxed at $24\%$
Fourth Piece for $\$75000.00$
tax for $\$75000.00$ income @ $24\%$ per $\$$ earned = $75000.00 * 0.24 = 18000.00$
We are done with the maximum taxable income that can be taxed at the fourth piece of $24\%$
We move to the fifth piece.
$165428.00 - 157500.00 = 7928.00$
The rest of the $\$7928.00$ is taxed at $32\%$
Fifth Piece for $\$7928.00$
tax for $\$7928.00$ income @ $32\%$ per $\$$ earned = $7928.00 * 0.32 = 2536.96$
$952.50 + 3501.00 + 9636.00 + 18000 + 2536.96 = 34626.46$
tax for taxable income of $\$165,428.00$ = $\$34,626.46$
(6.) $\$234,543.00$ falls in the sixth piece.
Before we use the sixth piece, we have to go through the first, second, third, fourth, and fifth pieces.
First Piece for $\$9525.00$
$9525.00$ of the $12575.00$ is taxed at $10\%$
tax for $\$9525.00$ income @ $10\%$ per $\$$ earned = $9525.00 * 0.1 = 952.50$
We are done with the maximum taxable income that can be taxed at the first piece of $10\%$
We move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate.
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$
Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$
Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the third piece of $22\%$
We move to the fourth piece.
We need to calculate the range of taxable income that should be taxed at the fourth tax rate.
$157500.00 - 82500.00 = 75000.00$
$\$75000.00$ is taxed at $24\%$
Fourth Piece for $\$75000.00$
tax for $\$75000.00$ income @ $24\%$ per $\$$ earned = $75000.00 * 0.24 = 18000.00$
We are done with the maximum taxable income that can be taxed at the fourth piece of $24\%$
We move to the fifth piece.
We need to calculate the range of taxable income that should be taxed at the fifth tax rate.
$200000.00 - 157500.00 = 42500.00$
$\$42500.00$ is taxed at $32\%$
Fifth Piece for $\$42500.00$
tax for $\$42500.00$ income @ $32\%$ per $\$$ earned = $42500.00 * 0.32 = 13600.00$
We are done with the maximum taxable income that can be taxed at the fifth piece of $32\%$
We move to the sixth piece.
$234543.00 - 200000.00 = 34543.00$
The rest of the $\$34543.00$ is taxed at $35\%$
Sixth Piece for $\$34543.00$
tax for $\$34543.00$ income @ $35\%$ per $\$$ earned = $34543.00 * 0.35 = 12090.05$
$952.50 + 3501.00 + 9636.00 + 18000 + 13600 + 12090.05 = 57779.55$
tax for taxable income of $\$234,543.00$ = $\$57,779.55$
(7.) $\$700,712.00$ falls in the seventh piece.
Before we use the seventh piece (last piece), we have to go through the previous six pieces in order.
First Piece for $\$9525.00$
$9525.00$ of the $12575.00$ is taxed at $10\%$
tax for $\$9525.00$ income @ $10\%$ per $\$$ earned = $9525.00 * 0.1 = 952.50$
We are done with the maximum taxable income that can be taxed at the first piece of $10\%$
We move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate.
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$
Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$
Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the third piece of $22\%$
We move to the fourth piece.
We need to calculate the range of taxable income that should be taxed at the fourth tax rate.
$157500.00 - 82500.00 = 75000.00$
$\$75000.00$ is taxed at $24\%$
Fourth Piece for $\$75000.00$
tax for $\$75000.00$ income @ $24\%$ per $\$$ earned = $75000.00 * 0.24 = 18000.00$
We are done with the maximum taxable income that can be taxed at the fourth piece of $24\%$
We move to the fifth piece.
We need to calculate the range of taxable income that should be taxed at the fifth tax rate.
$200000.00 - 157500.00 = 42500.00$
$\$42500.00$ is taxed at $32\%$
Fifth Piece for $\$42500.00$
tax for $\$42500.00$ income @ $32\%$ per $\$$ earned = $42500.00 * 0.32 = 13600.00$
We are done with the maximum taxable income that can be taxed at the fifth piece of $32\%$
We move to the sixth piece.
We need to calculate the range of taxable income that should be taxed at the sixth tax rate.
$500000.00 - 200000.00 = 300000.00$
$\$300000.00$ is taxed at $35\%$
Sixth Piece for $\$300000.00$
tax for $\$300000.00$ income @ $35\%$ per $\$$ earned = $300000.00 * 0.35 = 105000.00$
We are done with the maximum taxable income that can be taxed at the sixth piece of $35\%$
We move to the seventh piece.
$700712.00 - 500000.00 = 200712.00$
The rest of the $\$200712.00$ is taxed at $37\%$
Seventh Piece for $\$200712.00$
tax for $\$200712.00$ income @ $37\%$ per $\$$ earned = $200712.00 * 0.37 = 74263.44$
$952.50 + 3501.00 + 9636.00 + 18000 + 13600 + 105000 + 74263.44 = 224952.94$
tax for taxable income of $\$700,712.00$ = $\$224,952.94$
Some students may ask if it is possible to have just one function that will calculate the tax for any taxable
income.
Or is it possible to find the tax for a taxable income that falls in the second piece, without having to go through
the first piece?
Those are really interesting questions!
That is one of the reasons for studying piecewise functions ☺☺☺
Please specify the importance of not rounding intermediate calculations.
Please specify the importance of rounding only the final answer to two decimal place (because it is dollars and
cents).
Solution: 2nd Method: Piecewise Function/Algebraic Method
Do we have to solve this manually all the time? That will be time consuming!
We can write it as a piecewise function and use each function to calculate the taxes that corresponds to each taxable
income.
Besides, writing it as a piecewise function helps us to write a computer program that will calculate the taxes for any
amount of taxable income.
Define the variables.
Let $p$ = taxable income (in $\$$)
Let $t$ = taxes (in $\$$)
The taxes paid is a function of the income earned.
Taxes is the dependent variable.
Taxable income is the independent variable.
$t = f(p)$
We can also write is as $t(p)$
This application has seven pieces.
For the first piece;
tax for $\$p$ income @ $10\%$ per $\$$ earned = $p * 0.1 = 0.1p$
$t(p) = 0.1p$
For the second piece;
We have to "finish" with the first piece first
First Piece for $\$9525.00$
tax for $\$9525$ income @ $10\%$ per $\$$ earned = $9525 * 0.1 = 952.5$
Then, we move to the second piece.
Second Piece for $\$p - 9525.00$
The remaining income, $(p - 9525)$ is taxed at $12\%$
So, we have to multiply the remaining income by $0.12$
$t(p) = 952.5 + 0.12(p - 9525)$
$t(p) = 952.5 + 0.12p - 1143$
$t(p) = 0.12p - 190.5$
For the third piece;
We have to "finish" with the first and second pieces
First Piece for $\$9525.00$
tax for $\$9525$ income @ $10\%$ per $\$$ earned = $9525 * 0.1 = 952.5$
Then, we move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate.
Please review the scenario in the Manual/Arithmetic Method
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$
Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We have to move on to the third piece.
Third Piece for $\$p - 38700.00$
The remaining income, $(p - 38700)$ is taxed at $22\%$
So, we have to multiply the remaining income by $0.22$
$t(p) = 952.5 + 3501 + 0.22(p - 38700)$
$t(p) = 952.5 + 3501 + 0.22p - 8514$
$t(p) = 0.22p - 4060.5$
For the fourth piece;
We have to "finish" with the first, second, and third pieces
First Piece for $\$9525.00$
tax for $\$9525$ income @ $10\%$ per $\$$ earned = $9525 * 0.1 = 952.5$
Then, we move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate.
Please review the scenario in the Manual/Arithmetic Method
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$
Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$
Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the second piece of $22\%$
We move to the fourth piece.
Fourth Piece for $\$p - 82500.00$
The remaining income, $(p - 82500)$ is taxed at $24\%$
So, we have to multiply the remaining income by $0.24$
$t(p) = 952.5 + 3501 + + 9636 + 0.24(p - 82500)$
$t(p) = 952.5 + 3501 + 9636 + 0.24p - 19800$
$t(p) = 0.24p - 5710.5$
For the fifth piece;
We have to "finish" with the first, second, third, and fourth pieces
First Piece for $\$9525.00$
tax for $\$9525$ income @ $10\%$ per $\$$ earned = $9525 * 0.1 = 952.5$
Then, we move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate.
Please review the scenario in the Manual/Arithmetic Method
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$
Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$
Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the third piece of $22\%$
We move to the fourth piece.
We need to calculate the range of taxable income that should be taxed at the fourth tax rate.
$157500.00 - 82500.00 = 75000.00$
$\$75000.00$ is taxed at $24\%$
Fourth Piece for $\$75000.00$
tax for $\$75000.00$ income @ $24\%$ per $\$$ earned = $75000.00 * 0.24 = 18000.00$
We are done with the maximum taxable income that can be taxed at the fourth piece of $24\%$
We move to the fifth piece.
Fifth Piece for $\$p - 157500.00$
The remaining income, $(p - 157500)$ is taxed at $32\%$
So, we have to multiply the remaining income by $0.32$
$t(p) = 952.5 + 3501 + + 9636 + 18000 + 0.32(p - 157500)$
$t(p) = 952.5 + 3501 + 9636 + 18000 + 0.32p - 50400$
$t(p) = 0.32p - 18310.5$
For the sixth piece;
We have to "finish" with the first, second, third, fourth, and fifth pieces
First Piece for $\$9525.00$
tax for $\$9525$ income @ $10\%$ per $\$$ earned = $9525 * 0.1 = 952.5$
Then, we move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate.
Please review the scenario in the Manual/Arithmetic Method
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$
Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$
Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the third piece of $22\%$
We move to the fourth piece.
We need to calculate the range of taxable income that should be taxed at the fourth tax rate.
$157500.00 - 82500.00 = 75000.00$
$\$75000.00$ is taxed at $24\%$
Fourth Piece for $\$75000.00$
tax for $\$75000.00$ income @ $24\%$ per $\$$ earned = $75000.00 * 0.24 = 18000.00$
We are done with the maximum taxable income that can be taxed at the fourth piece of $24\%$
We move to the fifth piece.
We need to calculate the range of taxable income that should be taxed at the fifth tax rate.
$200000.00 - 157500.00 = 42500.00$
$\$42500.00$ is taxed at $32\%$
Fifth Piece for $\$42500.00$
tax for $\$42500.00$ income @ $32\%$ per $\$$ earned = $42500.00 * 0.32 = 13600.00$
We are done with the maximum taxable income that can be taxed at the fifth piece of $32\%$
We move to the sixth piece.
Sixth Piece for $\$p - 200000.00$
The remaining income, $(p - 200000)$ is taxed at $35\%$
So, we have to multiply the remaining income by $0.35$
$t(p) = 952.5 + 3501 + + 9636 + 18000 + 13600 + 0.35(p - 200000)$
$t(p) = 952.5 + 3501 + 9636 + 18000 + 13600 + 0.35p - 70000$
$t(p) = 0.35p - 24310.5$
For the seventh piece;
We have to "finish" with the previous six pieces in order
First Piece for $\$9525.00$
tax for $\$9525$ income @ $10\%$ per $\$$ earned = $9525 * 0.1 = 952.5$
Then, we move to the second piece.
We need to calculate the range of taxable income that should be taxed at the second tax rate.
Please review the scenario in the Manual/Arithmetic Method
$38700.00 - 9525.00 = 29175.00$
$\$29175.00$ is taxed at $12\%$
Second Piece for $\$29175.00$
tax for $\$29175.00$ income @ $12\%$ per $\$$ earned = $29175.00 * 0.12 = 3501.00$
We are done with the maximum taxable income that can be taxed at the second piece of $12\%$
We move to the third piece.
We need to calculate the range of taxable income that should be taxed at the third tax rate.
$82500.00 - 38700.00 = 43800.00$
$\$43800.00$ is taxed at $22\%$
Third Piece for $\$43800.00$
tax for $\$43800.00$ income @ $22\%$ per $\$$ earned = $43800.00 * 0.22 = 9636.00$
We are done with the maximum taxable income that can be taxed at the third piece of $22\%$
We move to the fourth piece.
We need to calculate the range of taxable income that should be taxed at the fourth tax rate.
$157500.00 - 82500.00 = 75000.00$
$\$75000.00$ is taxed at $24\%$
Fourth Piece for $\$75000.00$
tax for $\$75000.00$ income @ $24\%$ per $\$$ earned = $75000.00 * 0.24 = 18000.00$
We are done with the maximum taxable income that can be taxed at the fourth piece of $24\%$
We move to the fifth piece.
We need to calculate the range of taxable income that should be taxed at the fifth tax rate.
$200000.00 - 157500.00 = 42500.00$
$\$42500.00$ is taxed at $32\%$
Fifth Piece for $\$42500.00$
tax for $\$42500.00$ income @ $32\%$ per $\$$ earned = $42500.00 * 0.32 = 13600.00$
We are done with the maximum taxable income that can be taxed at the fifth piece of $32\%$
We move to the sixth piece.
We need to calculate the range of taxable income that should be taxed at the sixth tax rate.
$500000.00 - 200000.00 = 300000.00$
$\$300000.00$ is taxed at $35\%$
Sixth Piece for $\$300000.00$
tax for $\$300000.00$ income @ $35\%$ per $\$$ earned = $300000.00 * 0.35 = 105000.00$
We are done with the maximum taxable income that can be taxed at the sixth piece of $35\%$
We move to the seventh piece.
Seventh Piece for $\$p - 500000.00$
The remaining income, $(p - 500000)$ is taxed at $37\%$
So, we have to multiply the remaining income by $0.37$
$t(p) = 952.5 + 3501 + + 9636 + 18000 + 13600 + 105000 + 0.37(p - 500000)$
$t(p) = 952.5 + 3501 + 9636 + 18000 + 13600 + 105000 + 0.37p - 185000$
$t(p) = 0.37p - 34310.5$
We can now write the piecewise function as:
$$
t(p) =
\begin{cases}
\\[3ex]
0.1p; & \quad 0 \leq p \leq 9525 \\[3ex]
0.12p - 190.5; & \quad 9525 \lt p \leq 38700 \\[3ex]
0.22p - 4060.5; & \quad 38700 \lt p \leq 82500 \\[3ex]
0.24p - 5710.5; & \quad 82500 \lt p \leq 157500 \\[3ex]
0.32p - 18310.5; & \quad 157500 \lt p \leq 200000 \\[3ex]
0.35p - 24310.5; & \quad 200000 \lt p \leq 500000 \\[3ex]
0.37p - 34310.5; & \quad p \gt 500000
\end{cases}
$$
Let us recalculate all the questions using the Piecewise Function method.
(1.) $\$7,000.00$ falls in the first piece.
$
t(p) = 0.1p \\[3ex]
t(7000) = 0.1(7000) \\[3ex]
= 700 \\[3ex]
$
tax for taxable income of $\$7,000.00$ = $\$700.00$
(2.) $\$12,575.00$ falls in the second piece.
$
t(p) = 0.12p - 190.5 \\[3ex]
t(12575) = 0.12(12575) - 190.5 \\[3ex]
= 1509 - 190.5 \\[3ex]
= 1318.5 \\[3ex]
$
tax for taxable income of $\$12,575.00$ = $\$1,318.50$
(3.) $\$43,750.00$ falls in the third piece.
$
t(p) = 0.22p - 4060.5 \\[3ex]
t(43750) = 0.22(43750) - 4060.5 \\[3ex]
= 9625 - 4060.5 \\[3ex]
= 5564.5 \\[3ex]
$
tax for taxable income of $\$43,750.00$ = $\$5,564.50$
(4.) $\$120,327.00$ falls in the fourth piece.
$
t(p) = 0.24p - 5710.5 \\[3ex]
t(120327) = 0.24(120327) - 5710.5 \\[3ex]
= 28878.48 - 5710.5 \\[3ex]
= 23167.98 \\[3ex]
$
tax for taxable income of $\$120,327.00$ = $\$23,167.98$
(5.) $\$165,428.00$ falls in the fifth piece.
$
t(p) = 0.32p - 18310.5 \\[3ex]
t(165428) = 0.32(165428) - 18310.5 \\[3ex]
= 52936.96 - 18310.5 \\[3ex]
= 34626.46 \\[3ex]
$
tax for taxable income of $\$165,428.00$ = $\$34,626.46$
(6.) $\$234,543.00$ falls in the sixth piece.
$
t(p) = 0.35p - 24310.5 \\[3ex]
t(234543) = 0.35(234543) - 24310.5 \\[3ex]
= 82090.05 - 24310.5 \\[3ex]
= 57779.55 \\[3ex]
$
tax for taxable income of $\$234,543.00$ = $\$57,779.55$
(7.) $\$700,712.00$ falls in the seventh piece.
$
t(p) = 0.37p - 34310.5 \\[3ex]
t(700712) = 0.37(700712) - 34310.5 \\[3ex]
= 259263.44 - 34310.5 \\[3ex]
= 224952.94 \\[3ex]
$
tax for taxable income of $\$700,712.00$ = $\$224,952.94$
Which of the two methods do you prefer?
What are your reasons?
What are the pros and cons that you see for each method?
Do you have any other method for solving Piecewise Function applications?
The Piecewise Function for the Head of Household is:
$$
t(p) =
\begin{cases}
\\[3ex]
0.1p; & \quad 0 \leq p \leq 13600 \\[3ex]
0.12p - 272; & \quad 13600 \lt p \leq 51800 \\[3ex]
0.22p - 5452; & \quad 51800 \lt p \leq 82500 \\[3ex]
0.24p - 7102; & \quad 82500 \lt p \leq 157500 \\[3ex]
0.32p - 19702; & \quad 157500 \lt p \leq 200000 \\[3ex]
0.35p - 25702; & \quad 200000 \lt p \leq 500000 \\[3ex]
0.37p - 35702; & \quad p \gt 500000
\end{cases}
$$
The Piecewise Function for the Married Filing Jointly or Qualifying Widow is:
$$
t(p) =
\begin{cases}
\\[3ex]
0.1p; & \quad 0 \leq p \leq 19050 \\[3ex]
0.12p - 381; & \quad 19050 \lt p \leq 77400 \\[3ex]
0.22p - 8121; & \quad 77400 \lt p \leq 165000 \\[3ex]
0.24p - 11421; & \quad 165000 \lt p \leq 315000 \\[3ex]
0.32p - 36621; & \quad 315000 \lt p \leq 400000 \\[3ex]
0.35p - 48621; & \quad 400000 \lt p \leq 600000 \\[3ex]
0.37p - 60621; & \quad p \gt 600000
\end{cases}
$$
The Piecewise Function for the Married Filing Separately is:
$$
t(p) =
\begin{cases}
\\[3ex]
0.1p; & \quad 0 \leq p \leq 9525 \\[3ex]
0.12p - 190.5; & \quad 9525 \lt p \leq 38700 \\[3ex]
0.22p - 4060.5; & \quad 38700 \lt p \leq 82500 \\[3ex]
0.24p - 5710.5; & \quad 82500 \lt p \leq 157000 \\[3ex]
0.32p - 18270.5; & \quad 157000 \lt p \leq 200000 \\[3ex]
0.35p - 24270.5; & \quad 200000 \lt p \leq 300000 \\[3ex]
0.37p - 30270.5; & \quad p \gt 300000
\end{cases}
$$
(1.) You may work as a group (peer tutor to teach/correct one another).
However, this is an individual project.
In other words, you will submit your own project.
(2.) No two projects should be the same.
(a.) You may not do any of my examples even if the rates of the company have changed.
Use my examples as guides to completing your own project.
(b.) You may not do any of the projects of my previous students.
The Student Project samples are provided below. Use them as guides to completing your own project.
(3.) Research any real-world application of piecewise function.
(a.) All information used for this project should be verifiable on the direct website of the company/organization.
(b.) Textbook examples are NOT allowed.
(c.) Find any application that has at least 2 pieces, and has at least a function in each piece that includes the independent variable.
(d.) You may only do the income taxes of any of the filing options for the current year or future year(s).
If you do the income taxes of the filing options for the current year, it should be before the first IRS deadline
for filing taxes.
If the IRS deadline is past, then you have to do for future years.
You may NOT do the income taxes of any of the filing options of any previous year(s).
(e.) Please write the direct website of the company/organization where you got the tax information.
(f.) Please note that it is your responsibity to ensure that the tax information is correct.
In that regard, please compare your information with the information across several reputable websites.
(g.) Please work only on Federal Taxes. Do not work on State, City, or County taxes.
(h.) For any filing option you do, please cover all the pieces.
You may file:
(A.) Single OR
(B.) Head of Household OR
(C.) Married Filing Jointly or Qualifying Widow OR
(D.) Married Filing Separately
(4.) As a student, you have free access to Microsoft Office suite of apps.
(a.) Please download the desktop apps of Microsoft Office on your desktop/laptop (Windows and/or Mac only).
Do not use a chromebook.
Do not use a tablet/iPad.
Do not use a smartphone.
Do not use the web app/sharepoint access of Microsoft Office.
(Please contact the IT/Tech Support for assistance if you do not know how to download the desktop app.)
In that regard, the project is to be typed using the desktop version/app of Microsoft Office Word only.
(b.) The file name for the Microsoft Office Word project should be saved as:
firstName–lastName–project
Use only hyphens between your first name and your last name; and between your last name and the word, project.
No spaces.
(c.) For all English terms/work (entire project): use Times New Roman; font size of 14; line spacing of 1.5.
Further, please make sure you have appropriate spacing between each heading and/or section as applicable.
Your work should be well-formatted and visually appealing.
(d.) For all Math terms/work: symbols, variables, numbers, formulas, expressions, equations and fractions among others,
the Math Equation Editor is required.
(i.) The font is set to Cambria Math by default (set it to that font if it is not); font size of 14, and
align accordingly (preferably left-aligned).
(ii.) To ensure appropriate spacing between your Math work, use a line spacing of 2.0.
Alternatively, you may use line spacing of 1.5 but insert a space after each equation as applicable.
Your work should be well-formatted, organized, well-spaced (not compact), and visually appealing.
(iii.) Align the functions in each piece of the piecewise function accordingly.
(e.) Include page numbers. You may include at the top of the pages or at the bottom of the pages but not both.
(5.) Research Skills: Cite your source properly. Use APA, MLA, or Chicago Manual of Style.
Indicate the style you used.
(6.) Writing Skills: Write the complete address of the direct page of the website where you found the application.
Write the main application completely.
Write all the information accoridngly as noted in the Example Guide and the Checklist.
(7.) (a.) Mathematical Skills: Show all work.
Arithmetic: This is the Arithmetic method.
Use random numbers to test the real-world application manually for each piece.
Show all work including intermediate calculations/values.
Write down your results.
(b.) Algebra: This is the Algebraic Method (Piecewise Function Method).
Define your variables accordingly.
Write a piecewise function for that application.
Test the piecewise function with the same random numbers that you used for the Arithmetic method.
Please NOTE:
(i.) The intermediate results should be the same for both methods.
(ii.) The final results should be the same for both methods.
If either of the results are not the same, there are issues. Please fix them.
These are the required information.
You may or may not not use a table format. Use whatever format that is visually appealing.
Please be creative.
(8.) Mr. C (SamDom For Peace) wants you to do this real-world project very well.
Hence, he highly recommends that you submit a draft so he can give you feedback.
(a.) First: (Required): Please submit the required information in the
Projects: Company Names and Websites page in the Discussions forum on the Canvas course.
An example was given in the forum.
The name of the company that has the application should be written.
The complete web address of the web page that has the application should be written.
The objectives should be written.
Please be very specific. In other words, I do not have to click on anything to see the application.
Once I visit the webpage that you provided, I must see the application directly on that page.
I shall review and respond.
(b.) Second: (Highly Recommended): When the company name and website is approved, please submit your draft.
Draft projects are not graded because they are drafts. They are only for feedback.
If your professor gives you an opportunity to submit a draft, please use that opportunity.
Submitting drafts is highly recommended. Submitting drafts is not required.
It is highly recommended because I want to give you the opportunity to do your project very well and make an excellent grade in it.
Please turn in your draft in the Discussions page → Projects: Drafts forum in the Canvas course
(if you would like your colleagues to read my comments and avoid any mistakes that you made).
You may also send it to me via email (if you do not want your colleagues to see my comments and learn from the comments).
I shall review and provide feedback.
Then, review my feedback and make changes as necessary.
Keep working with me until I give you the green light to turn in your actual project. This must be done before
the final due date to turn in the actual project.
When everything is fine (after you make changes as applicable based on my feedback), please submit your
work in the appropriate area: Assignments page → Piecewise Functions Project in the Canvas
course.
Only the projects submitted in the appropriate place in the Canvas course are graded.
(9.) All work must be turned in by the final due date to receive credit.
Please note the due dates listed in the course syllabus for the submission of the draft and the actual project.
In the course syllabus, we have the:
(a.) Initial due date for the Project Draft: Please turn in your draft.
(b.) Initial due date for the Project: If your draft is not ready for submission, keep working with me. Make changes
based on my feedback and keep working with me.
If you prefer not to turn in a draft, please review all the resources provided for you and do your project well and
submit.
(c.) Final due date for the Project Draft: This is necessary if you want a written feedback for your draft.
After this date, written feedback would not be provided for your draft. However, verbal feedback would still be provided
during Office Hours/Student Engagement Hours/Live Sessions.
(d.) Final due date for the Project: All work must be turned in by this date to receive credit.
After this date, no work may be accepted.
Name: | Your name |
Date: | The date |
Instructor: | Samuel Chukwuemeka |
Project: | Federal Tax Rates, Brackets, and Standard Deductions |
Company: |
Bankrate.com or IRS.com or or any other website you used (https://www.bankrate.com/taxes/tax-brackets/) (Name of the company and the direct Website of company that has the piecewise function) |
Objectives: |
(1.) Calculate the federal income taxes of any United States resident/citizen within each taxable income
bracket manually using Arithmetic method for year 2018. (2.) Write a piecewise function of the federal income tax rates for year 2018. (3.) Recalculate the same income tax of the United States resident/citizen for those income brackets algebraically using the Piecewise Function method for year 2018. |
Information: | Write the direct information from the direct website |
Arithmetic Method: | Test each piece manually |
Piecewise Function: | Write the piecewise function of the information |
Piecewise Function Method: | Test the same piece algebraically |
Citation: |
Indicate the type of citation format. Cite your source(s) accordingly. |
Checklist for Piecewise Function Project: Federal Tax
(1.) Project is a piecewise function that has at least 2 pieces, and has at least a function in each piece that includes the independent variable.
(2.) Title of project.
(3.) Name of Student.
(4.) Name of Instructor.
(5.) Objectives of project.
(6.) Name of company.
(7.) Direct website of company.
(8.) Information on website written as is
(9.) Arithmetic Method:
(a.) Numbers in each piece, including 0 are tested.
(b.) All work including conversions are shown.
(c.) Intermediate calculations and values (before rounding) are shown.
(d.) Final results (rounded) are written and unit (currency) is included.
(10.) Piecewise Function:
(a.) Variables are defined.
(b.) All work is shown to determine the algebraic function for each piece.
(c.) Piecewise function is developed and written well using beginning brace.
(d.) The function in each piece and the corresponding domain is written well using any of the approved formats including the use of semicolon or comma.
(e.) The functions in the piecewise function are aligned properly.
(11.) Algebraic Method (Piecewise Function Method):
(a.) The same numbers including 0 tested using the Arithmetic Method, are also tested using the Algebraic Method.
Testing each number implies that each number within the domain of each piece in the piecewise function is tested in the function for that piece.
(b.) Intermediate calculations and values (before rounding) are shown. Intermediate values must be the same as the Arithmetic Method.
(c.) Final results (rounded) are the same as the Arithmetic Method and the unit is included.
(12.) References:
(a.) The reference style is specified.
(b.) Sources are cited according to the specified reference style.
(13.) Page numbers are included in every page.
(14.) Times New Roman font, font size of 14, and line spacing of 1.5 is used for the project.
(15.) Cambria Math font, font size of 14, and the Math Equation Editor is used for all Math terms including numbers, arithmetic operations, symbols, variables,
fractions, formulas, expressions, and equations among others.
(16.) Correct file name.
Chukwuemeka, S.D (2016, April 30). Samuel Chukwuemeka Tutorials - Math, Science, and Technology.
Retrieved from https://precalculus.appspot.com/
2020 Federal Tax Rates, Brackets, & Standard Deductions. (2020, January 25). IRS.Com.
https://www.irs.com/articles/2020-federal-tax-rates-brackets-standard-deductions/
Bankrate.com. (2018, November 28). 2018-2019 Tax Brackets | Bankrate.com. Retrieved from
https://www.bankrate.com/finance/taxes/tax-brackets.aspx