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Piecewise Functions

For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka

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Samdom For Peace

I greet you this day,

You may use the calculators to check your answers. You are encouraged to solve the questions first, before you check your answers. These topics are covered in my Videos on Piecewise Functions.

I wrote the codes for these calculators using JavaScript, a client-side scripting language. Please use the latest Internet browsers. I used the AJAX Javascript library for the rest of the codes.

Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. Should you need to contact me, please use the form at the bottom of the page. Thank you for visiting!!!

Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

Story

The event will hold in three days.
Well-cooked and delicious: rice and beans
Fresh avocados
Options of fish, chicken, cabeza
Son: Dad, what about vegan?
Dad: Let's not worry about vegan. Those folks would really be hungry.
They need to eat well.
Son: Does it mean that those who are vegan do not eat well?
Dad: I did not say that. Do not force words into my mouth.
Son: I just asked a question, Dad.
What about al pastor?
Dad: No, we do not eat pork.
Why should we give people what we do not eat?
Okay, let's go and order these burritos.


Dad: Buenas tardes, señora
How are you doing today?
Son: Good afternoon, Mama Esther.
Esther: Good afternoon. I am doing well.
How are you doing too?
Dad: We are doing well.
Thank you.
How much are your burritos?
Esther: It's $\$7.00$ each.
Guacamole costs extra ¢$50$ each.
Dad: We want to place an order for $1000$ burittos.
Son: Well-cooked: rice and beans; fresh avocados...everything
No guacamole.
Esther: Okay...
Dad: $300$ pescado; $300$ cabeza; $400$ chicken
How much is the cost?
Esther: How soon do you need them?
Dad: In three days.
Esther: Would I deliver them, or would you pick them up?
Dad: We shall pick them up.
Esther: It's going to be $\$7$ each for the first $300$ burittos
$\$6$ each for the next $300$, and $\$5.00$ each for the remaining $400$.
Fair enough?
So, the price is ...
$\$2100$ for the first $300$ + $\$1800$ for the next $300$ + $\$2000$ for the remaining $400$
Total of $5900$
Add $10\%$ tax of $5900$
That gives $\$590$
Total cost of $\$6490.00$


Son: Dad, I could calculate the cost another way.
Dad: What way?
Is it better than the way Esther did?
Son: For many orders, yes!
And for writing a computer program to calculate it.
Dad: Interesting. What is it?
Son: We learned it in school.
Mama Esther did it as a simple Arithmetic.
Mr. C called it the Manual method.
Then, he taught us to do it algebraically ...
using the Piecewise Function method.
I think it is much better.
Dad: Okay, what's the method?
Son: We begin by defining the variables involved.
Let $b$ = buritto
and $c$ = cost
Which variable depends on the other variable?
Son: Does the buritto depend on the cost, or
does the cost depend on the number of burittos?
Dad: The cost depends on the number of burittos.
Son: That is correct.
We write it this way: $c = f(b)$
It can also be written as: $c(b)$
Dad: Okay...
Son: So, $c(b)$ =


$$ c(b) = \begin{cases} 7b; & \quad 1 \leq b \leq 300 \\[3ex] 6b + 300; & \quad 300 \lt b \leq 600 \\[3ex] 5b + 900; & \quad 600 \lt b \leq 1000 \end{cases} $$


Dad: How is this better than the way Esther did it?
Son: Say you wanted to order $900$ burittos
How much would it cost excluding the taxes?
Dad: $\$7$ for the first $300$ gives $\$2100$
$\$6$ for the next $300$ gives $\$1800$
$\$5$ for the remaining $300$ gives $\$1500$
That gives a total of $\$5400$
Son: How many steps did you do before you got the answer?
Dad: Three steps...
Son: I shall do only two steps before I get that answer.
$c(900) = 5(900) + 900 = 4500 + 900 = 5400$
Dad: You used the $3^{rd}$ equation because $900$ falls in ...
Son: that piece...yes...in that domain.
Dad: That is interesting! How did you get those equations?
Son: Welcome to Piecewise Functions!, Dad.

Overview of Piecewise Functions

Objectives

Students will:

(1.) Define piecewise functions.

(2.) Discuss piecewise functions.

(3.) Discuss real-world applications of piecewise-functions.

(4.) Give examples of mathematical functions that are piecewise functions.

(5.) Graph piecewise functions.

(6.) Analyze the graphs of piecewise functions.

(7.) Solve problems involving the application of piecewise functions.


Skills Measured/Acquired

(1.) Use of prior knowledge

(2.) Critical Thinking

(3.) Interdisciplinary connections/applications

(4.) Technology

(5.) Active participation through direct questioning

(6.) Research


Homework Assignment

Students should research real-world applications of piecewise functions.

(1.) Write the word problem (real-world application).

(2.) Solve solve questions (numbers) "manually" for each piece/interval.

(3.) Write the piecewise function for the word problem.

(4.) Solve those questions again using the piecewise function.

(5.) Check to ensure that you get the same answers (verify your answers) using both methods (manually and piecewise function).

(6.) Write a computer program that solves that word problem.

(7.) Verify your answers again with the computer program.


Check for prior knowledge. Ask students about these terms.

Vocabulary Words

Bring it to English: piece, step, hybrid, exception, except, condition

Bring it to Math: piecewise, function, piecewise-defined, hybrid, domain, range, intercepts, x-intercept, y-intercept, graph, continuity, discontinuity

Bring it to Computer Science: conditional statements, if, if-else, if-elif


Why Study Piecewise Functions?

Ask students to give examples of scenarios where exceptions have been made, or where certain conditions have been used.

Let us begin with a simple example.

At some Golden Corral restaurants in the U.S, children aged 3 years and under (at most 3 years) eat free.
Let us analyze the Kid's Buffet. The information is found here.

                    It is also written here for you: 
Ages 3 and under: Free
Ages 4 - 8: $5.99
Ages 9 - 12: $6.99

As you see, there are different prices (costs) for children in a certain age group.

So, you only eat free if you are at most $3$ years old. That is a condition that must be satisfied before you eat free.

When writing a computer program for this function, you have to use "conditional statements".


Let us write this function as a piecewise function.

First: Define the variables.
Two things are involved in this function: age and cost.
Let $a$ be the age of the child.
Let $c$ be the cost of the meal.
As you can see, the cost of the meal depends on the age of the child.
This means that the cost is the dependent variable.
And the age of the child is the independent variable.
So, the cost of the meal is a function of the age of the child.
$c = f(a)$. We can write this as $c(a)$ pronounced as $c \: of \: a$
$$ c(a) = \left \{\begin{aligned} & $0; && 0 < a ≤ 3 \\[2ex] & $5.99; && 3 < a ≤ 8 \\[2ex] & $6.99; && 8 < a ≤ 12 \end{aligned} \right.$$

Exercises
Use the piecewise function above and answer these questions.

(1.) Peter is 3 years old.
How much (excluding taxes) is the cost of his meal?

Peter eats free because he is $3$ years old. His age is included in the first "piece".
The cost of his meal is $$0.00$

(2.) Mary is 7 years old.
How much (including taxes) is the cost of her meal?
Assume Golden Corral charges a tax rate of 7.5%.

Mary is $7$ years old. So, her age is included in the second "piece".
The cost of her meal is $$5.99$
$7.5\% = \dfrac{7.5}{100} = 0.075$

$7.5\%$ tax on the cost = $0.075 * 5.99 = 0.44925$
Total cost = 5.99 + 0.44925 = 6.43925 = $6.44

Students should give more examples as seen in national parks, car rentals, and retail stores among others.

They should cite their sources "properly". Teachers should emphasize the importance of citing sources accurately, and citing them in the correct format.


What is a Piecewise Function?

A piecewise function is a function consisting of sub-functions (pieces) defined on a sequence of intervals (domains).

It is also known as a piecewise-defined function or hybrid function.

The domain of each sub-function must not overlap.

Ask students why the domains must not overlap. What would happen if the domains of any two sub-functions overlap?

Some students may ask you that question first. If they do, ask them for their opinions.

Residential Service Rates by Georgia Power; State of Georgia, USA

As of the 30th day of December, 2018; the power rates for the Residential Service during the Summer Period (June - September) by Georgia Power is found here.

We shall focus on the Residential Service Rates. These costs exclude taxes and other costs.

                    It is also written here for you: 
Basic Service: $10.00
First 650KWh: ¢5.7 per KWh
650 - 1000KWh: ¢9.4 per KWh
Over 1000KWh: ¢9.7 per KWh

KWh means kilowatt hour.
$1$ KWh = $3600000$ J
$1$ kilowatt hour is equivalent to $3600000$ Joules.
The S.I (System International) unit of power is the Joule, $J$
The basic service fee is in dollars.
The rate of consumption is in cents.
We have to convert to the same unit.
We shall use the dollars (not the cents).
$ \$1 = ¢100 \\[2ex] ¢100 = \$1 \\[2ex] ¢5.7 = \$0.057 \\[2ex] ¢9.4 = \$0.094 \\[2ex] ¢9.7 = \$0.097 $

But, please wait a minute!!!

That information is not "really" correct.
Why would Georgia Power display incorrect information?
The "Interactive Sample Bill" (hmmmm....it is not interactive though ☺☺☺) found here is also incorrect.
Why?

The correct information is found in the Georgia Power Bill Calculator of the Georgia Public Service Commission.
The correct information is here
It is great to know that the Georgia Public Service Commission developed a Power Bill calculator.
We shall check our work with their calculator.
And of course...we shall also check our work with my calculator.

                    The correct information is written here for you: 
Basic Service: $10.00
Tier Usage Cost per KWh
1st tier up to 650KWh $0.056582
2nd tier next 350KWh $0.093983
3rd tier over 1000KWh $0.097273

Ask students to explain the differences between the information found in the Georgia Power website (the first one) and that found in the Georgia Public Service Commission (the second one).
Students should also explain the differences based on the domains of each piece in both websites.
What are the errors in the first one?
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).

Calculate the power costs for the following consumption of power.

(1.) $0$ KWh
(2.) $300$ KWh
(3.) $700$ KWh
(4.) $1000$ KWh
(5.) $1200$ KWh
Solution - $1^{st}$ Method: Manual/Arithmetic Method
This application has three pieces.
The basic service fee is not a piece. It must be paid whether one consumed any power or not.
Let $p$ = power (in KWh)
Let $r$ = cost (in dollars per KWh)
(1.) $0$ KWh falls in the first piece.
Basic service fee = $$10.00$
cost for $0$ KWh @ $$0.056582$ per KWh = $0 * 0.056582 = 0$
$10.00 + 0.00 = 10.00$
cost for $0$ KWh = $$10.00$

(2.) $300$ KWh falls in the first piece.
Basic service fee = $$10.00$
cost for $300$ KWh @ $$0.056582$ per KWh = $300 * 0.056582 = 16.9746$
$10.00 + 16.9746 = 26.9746$
cost for $300$ KWh = $$26.97$

(3.) $700$ KWh falls in the second piece.
Basic service fee = $$10.00$
Before we use the second piece, we have to go through the first piece first.
First Piece for $650$ KWh
cost for $650$ KWh @ $$0.056582$ per KWh = $650 * 0.056582 = 36.7783$
$700 - 650 = 50$
We need to find the cost for the remaining $50$ KWh
That takes us to the second piece.
Second Piece for $50$ KWh
cost for $50$ KWh @ $$0.093983$ per KWh = $50 * 0.093983 = 4.69915$
$10.00 + 36.7783 + 4.69915 = 51.47745$
cost for $700$ KWh = $$51.48$

(4.) $1000$ KWh falls in the second piece.
Basic service fee = $$10.00$
Before we use the second piece, we have to go through the first piece first.
First Piece for $650$ KWh
cost for $650$ KWh @ $$0.056582$ per KWh = $650 * 0.056582 = 36.7783$
$1000 - 650 = 350$
We need to find the cost for the remaining $350$ KWh
That takes us to the second piece.
Second Piece for $350$ KWh
cost for $350$ KWh @ $$0.093983$ per KWh = $350 * 0.093983 = 32.89405$
$10.00 + 36.7783 + 32.89405 = 79.67235$
cost for $1000$ KWh = $$79.67$

(5.) $1200$ KWh falls in the third piece.
Basic service fee = $$10.00$
Before we use the third piece, we have to go through the first piece and also through the second piece.
First Piece for $650$ KWh
cost for $650$ KWh @ $$0.056582$ per KWh = $650 * 0.056582 = 36.7783$
$1000 - 650 = 350$
We need to find the cost for $350$ KWh
That takes us to the second piece.
Second Piece for $350$ KWh
cost for $350$ KWh @ $$0.093983$ per KWh = $350 * 0.093983 = 32.89405$
$1200 - 1000 = 200$
We need to find the cost for the remaining $200$ KWh
That takes us to the third piece.
Third Piece for $200$ KWh
cost for $200$ KWh @ $$0.097273$ per KWh = $200 * 0.097273 = 19.4546$
$10.00 + 36.7783 + 32.89405 + 19.4546 = 99.12695$
cost for $1200$ KWh = $$99.13$

Some students may ask if it is possible to have just one function that will find the cost for any consumption of power.
Or is it possible to find the cost for the consumption of power 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 - $2^{nd}$ Method: Piecewise Function/Algebraic Method
What if we have to calculate the power rates for "several" consumption of power?
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 for the consumption of power that correspond to that piece.
Besides, writing it as a piecewise function helps us to write a computer program that will find the rate for any consumption of power.
This application has three pieces.
Let $p$ = power consumed(in KWh)
Let $c$ = cost per KWh or power consumed (in dollars)
$c = f(p)$
This can be written as: $c(p)$

For the first piece;
Basic service fee = $$10.00$
cost for $p$ KWh @ $$0.056582$ per KWh = $p * 0.056582 = 0.056582p$
$10 + 0.056582p = 0.056582p + 10$
$c(p) = 0.056582p + 10$

For the second piece;
Basic service fee = $$10.00$
We have to "finish" with the first piece first
cost for $650$ KWh @ $$0.056582$ per KWh = $650 * 0.056582 = 36.7783$
$10 + 36.7783 = 46.7783$
Then, we can multiply the remaining consumption of power by $0.093983$
$c(p) = 46.7783 + 0.093983(p - 650)$
$c(p) = 46.7783 + 0.093983p - 61.08895$
$c(p) = 0.093983p - 14.31065$

For the third piece;
Basic service fee = $$10.00$
We have to "finish" with the first piece first
cost for $650$ KWh @ $$0.056582$ per KWh = $650 * 0.056582 = 36.7783$
$10 + 36.7783 = 46.7783$
Then we have to finish with the second piece next
$1000 - 650 = 350$
cost for $350$ KWh @ $$0.093983$ per KWh = $350 * 0.093983 = 32.89405$
$46.7783 + 32.89405 = 79.67235$
Then, we can multiply the remaining consumption of power by $0.097273$
$c(p) = 79.67235 + 0.097273(p - 1000)$
$c(p) = 79.67235 + 0.097273p - 97.273$
$c(p) = 0.097273p - 17.60065$

We can now write the piecewise function as:
$$ c(p) = \begin{cases} 0.056582p + 10; & \quad 0 \leq p \leq 650 \\[3ex] 0.093983p - 14.31065; & \quad 650 \lt p \leq 1000 \\[3ex] 0.097273p - 17.60065; & \quad p \gt 1000 \end{cases} $$ Let us recalculate all the questions using the Piecewise Function method.

(1.) $0$ KWh falls in the first piece.

$ c(p) = 0.056582p + 10 \\[3ex] c(0) = 0.056582(0) + 10 \\[3ex] = 0 + 10 \\[3ex] = 10 \\[3ex] $ cost for $0$ KWh = $$10.00$

(2.) $300$ KWh falls in the first piece.

$ c(p) = 0.056582p + 10 \\[3ex] c(300) = 0.056582(300) + 10 \\[3ex] = 16.9746 + 10 \\[3ex] = 26.9746 \\[3ex] = 26.97 \\[3ex] $ cost for $300$ KWh = $$26.97$

(3.) $700$ KWh falls in the second piece.

$ c(p) = 0.093983p - 14.31065 \\[3ex] c(700) = 0.093983(700) - 14.31065 \\[3ex] = 65.7881 - 14.31065 \\[3ex] = 51.47745 \\[3ex] = 51.48 \\[3ex] $ cost for $700$ KWh = $$51.48$

(4.) $1000$ KWh falls in the second piece.

$ c(p) = 0.093983p - 14.31065 \\[3ex] c(1000) = 0.093983(1000) - 14.31065 \\[3ex] = 93.983 - 14.31065 \\[3ex] = 79.67235 \\[3ex] = 79.67 \\[3ex] $ cost for $1000$ KWh = $$79.67$

(5.) $1200$ KWh falls in the third piece.

$ c(p) = 0.097273p - 17.60065 \\[3ex] c(1200) = 0.097273(1200) - 17.60065 \\[3ex] = 116.7276 - 17.60065 \\[3ex] = 99.12695 \\[3ex] = 99.13 \\[3ex] $ cost for $1200$ KWh = $$99.13$

Ask students their preferred method - solving it manually or solving it by piecewise function.
They should give reasons for their answers.
Ask students if any of them can come up with another method besides the two methods already discussed.



Calculator for the Summer Residential Power Rates by Georgia Power, State of Georgia, USA

Power Consumption and Monthly Bill Statement

You are welcome to test the calculator and see the output.
Please complete all fields. Then, click "Bill Statement".

  • Power Consumption

in

  • Monthly Bill Statement

Calculator for the Winter Residential Power Rates by Georgia Power, State of Georgia, USA

Power Consumption and Monthly Bill Statement

You are welcome to test the calculator and see the output.
Please complete all fields. Then, click "Bill Statement".

  • Power Consumption

in

  • Monthly Bill Statement

Student Project

All information should be verifiable.

Each student will work independently.
No two projects should be the same.
Research any real-world case where piecewise function is used. It should be on the "direct website" of the company or organization. No textbook examples.
Find any application that uses "at least 3" pieces, and has "at least" a function in each piece that includes a variable (the independent variable).
The application should be on a verifiable website.
Write the "complete" address of the "direct page" of the website where you found the application.
*If the "direct" web address is too long, please shorten it by pasting the "complete web address" into www.tinyurl.com
* *This is only for traditional students (onsite) students*
*Generate a short address and write that address "as is".*
For online students, please "copy and paste" the link "as is". Please set the link to open in a new window.
Research Skills: Cite your source properly. Use APA, MLA, or Chicago Manual of Style. Indicate the style you used.
Writing Skills: Write or type the "main" application entirely.
Mathematical Skills - Arithmetic: Use random numbers to test the real-world application manually for each "piece". Write down your results.
Mathematical Skills - Algebra: Write a piecewise function for that application. Test the piecewise function with the same random numbers. The results should be the same. If the results are not the same, fix it.

Technology Skills: Use any programming language to write the piecewise function.

The requirements for the Technology Skills are:
Program asks user to enter the:
First and last names.
Initial and final meter readings.
Tax (if applicable).

The deliverables for the Technology Skills are:
Title of Application
Greeting. Greet the customer appropriately. Include the customer's first and last names.
Specify the dates for the monthly power bill.
Display the initial and final meter readings entered by the user.
Calculate the KWh of power consumed.
Calculate the bill for the KWh of power consumed.
Calculate the tax (as applicable).
Calculate the final bill.
Test your program with the same random numbers. The results should be the same. If the results are not the same, fix it.

Checks:
Initial power reading cannot be negative.
Final power reading cannot be negative.
The tax cannot be negative.
The initial power reading cannot be greater than the final power reading.
If any of the conditions are not met; the program should display an error message of the condition that was violated, then the program should quit. Do not allow the program to run.

The teacher should guide each student to the successful completion of the project.
Let students know you are willing to help.

Further Work
Writing Skills: Submit a Reflection Journal. Include your challenges, and how you overcame those challenges.
Technology Skills:
Include penalties for late payments in your project.
Write a mobile application for the project.
Develop a mobile application that sends reminders to customers so they do not miss payments.
Develop a rewards program that rewards customers who make payments on time.

Students should come up with additional functionalities and advancements.
Please note their responses and advise.

Water Rates by Calhoun County Water Authority; State of Alabama, USA

As of the 7th day of July, 2018; the water rates by the Calhoun County Water Authority is found here.

We shall focus on the Residential Rates. These rates exclude taxes.

                    It is also written here for you: 
First 3,000 Gallons: $17.35 Minimum Per Month
Next 2,000 Gallons: $5.24 per 1,000 Gallons
Next 5,000 Gallons: $3.96 per 1,000 Gallons
All over 10,000 Gallons: $3.10 per 1,000 Gallons

Calculate the water rates for the following consumption of water.

(1.) $2500$ gallons
(2.) $3700$ gallons
(3.) $5000$ gallons
(4.) $7500$ gallons
(5.) $12000$ gallons
(6.) $4692$ gallons
(7.) $6456$ gallons
Solution - $1^{st}$ Method: Manual/Arithmetic Method
This application has four pieces.
Let $g$ = gallons of water (in gallons)
Let $r$ = rate (in dollars per gallons)
(1.) $2500$ gallons falls in the first piece.
rate for $2500$ gallons @ $$17.35$ = $$17.35$

(2.) $3700$ gallons falls in the second piece.
Before we use the second piece, we have to go through the first piece first.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$3700 - 3000 = 700$
We need to find the rate for the remaining $700$ gallons
The first piece (for $3000$ gallons) is a constant rate. It is a flat fee.
However, the second piece is $$5.24$ per $1000$ gallons
The second piece is $$5.24$ per $1000$ (not per $700$) gallons

But, we have $700$ (not $1000$ gallons remaining). What do we do?
We have to use Proportional Reasoning Method to calculate the rate per gallons (rather than using the rate per thousand gallons)
Remember: per gallon means for $1$ gallon; and per $1000$ gallons means for a thousand gallons

Let us set up the Proportional Reasoning
Let $x$ be the cost per gallon
                                        dollars                  gallons
                                        5.24                      1000
                                          x                         1
                    
This means that:
$$ \dfrac{5.24}{x} = \dfrac{1000}{1} \\[5ex] 1000 * x = 5.24 * 1 \\[3ex] 1000 * x = 5.24 \\[3ex] x = \dfrac{5.24}{1000} \\[5ex] x = 0.00524 $$ Similarly;
$$ \dfrac{3.96}{x} = \dfrac{1000}{1} \\[5ex] 1000 * x = 3.96 * 1 \\[3ex] 1000 * x = 3.96 \\[3ex] x = \dfrac{3.96}{1000} \\[5ex] x = 0.00396 $$ And;
$$ \dfrac{3.10}{x} = \dfrac{1000}{1} \\[5ex] 1000 * x = 3.1 * 1 \\[3ex] 1000 * x = 3.1 \\[3ex] x = \dfrac{3.1}{1000} \\[5ex] x = 0.0031 $$
Can we re-write the application?

                    First 3,000 Gallons:                $17.35 Minimum Per Month 
Next 2,000 Gallons: $0.00524 per gallon
Next 5,000 Gallons: $0.00396 per gallon
All over 10,000 Gallons: $0.0031 per gallon
Ask students if they understood how the rates per gallon were calculated.

So, back to completing the second question:
rate for $700$ gallons @ $0.00524 per gallon = $0.00524 * 700$ = $$3.668$
Please do not approximate intermediate calculations especially if it deals with money!
rate for $3700$ gallons @ = 17.35 + 3.668 = 21.018
Now, you can round your final answer to the nearest cent.
NOTE: If your professor does not want you to round, do not round.
rate for $3700$ gallons = $$21.02$

(3.) $5000$ gallons falls in the second piece.
Before we use the second piece, we have to go through the first piece first.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$5000 - 3000 = 2000$
rate for $2000$ gallons @ $0.00524 per gallon = $0.00524 * 2000$ = $$10.48$
rate for $5000$ gallons @ = 17.35 + 10.48 = 27.83
rate for $5000$ gallons = $$27.83$

(4.) $7500$ gallons falls in the third piece.
Before we use the third piece, we have to go through the first and second pieces.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$7500 - 3000 = 4500$
rate for $2000$ gallons @ $0.00524 per gallon = $0.00524 * 2000$ = $$10.48$
$4500 - 2000 = 2500$
rate for $2500$ gallons @ $0.00396 per gallon = $0.00396 * 2500$ = $$9.90$
rate for $7500$ gallons @ = 17.35 + 10.48 + 9.90 = 37.73
rate for $7500$ gallons = $$37.73$

(5.) $12000$ gallons falls in the fourth piece.
Before we use the third piece, we have to go through the first, second, and third pieces.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$12000 - 3000 = 9000$
rate for $2000$ gallons @ $0.00524 per gallon = $0.00524 * 2000$ = $$10.48$
$9000 - 2000 = 7000$
rate for $5000$ gallons @ $0.00396 per gallon = $0.00396 * 5000$ = $$19.80$
$7000 - 5000 = 2000$
rate for $2000$ gallons @ $ per gallon = $0.0031 * 2000$ = $$6.20$
rate for $12000$ gallons @ = 17.35 + 10.48 + 19.80 + 6.20 = 53.83
rate for $12000$ gallons = $$53.83$

(6.) $4692$ gallons falls in the second piece.
Before we use the second piece, we have to go through the first piece first.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$4692 - 3000 = 1692$
rate for $1692$ gallons @ $0.00524 per gallon = $0.00524 * 1692$ = $$8.86608$
rate for $4692$ gallons @ = 17.35 + 8.86608 = 26.21608
rate for $4692$ gallons = $$26.22$

(7.) $6456$ gallons falls in the third piece.
Before we use the third piece, we have to go through the first and second pieces.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$6456 - 3000 = 3456$
rate for $2000$ gallons @ $0.00524 per gallon = $0.00524 * 2000$ = $$10.48$
$3456 - 2000 = 1456$
rate for $1456$ gallons @ $0.00396 per gallon = $0.00396 * 1456$ = $$5.76576$
rate for $6456$ gallons @ = 17.35 + 10.48 + 5.76576 = 33.59576
rate for $6456$ gallons = $$33.60$

Some students may ask if it is possible to have just one function that will find the rate for any gallon(s) of water?.
Or is it possible to find the rate for gallons of water that is contained in the second piece, without having to go through the first piece?
Solution - $2^{nd}$ Method: Piecewise Function/Algebraic Method
What if we have to calculate the water rates for "several" gallons of water?
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 for the number of gallons that correspond to that piece.
Besides, writing it as a piecewise function helps us to write a computer program that will find the rate for any gallon of water.
This application has four pieces.
Let $g$ = gallons of water (in gallons)
Let $r$ = rate (in dollars per gallons)
$r = f(g)$
This can be written as: $r(g)$

For the first piece;
$r(g)$ = $$17.35$

For the second piece;
We have to "finish" with the first piece first
Then, we can multiply the remaining gallons of water by $0.00524$
$r(g) = 17.35 + 0.00524(g - 3000)$
$r(g) = 17.35 + 0.00524g - 15.72$
$r(g) = 0.00524g + 1.63$

For the third piece;
We have to "finish" the second piece first
Then, we can multiply the remaining gallons of water by $0.00396$
What is the complete gallons for the second piece? - It is $5000$
In other words, $5000$ is the end point for the second piece. $5000$ is included.
Let us find the rate for that end point, $5000$
$r(g) = 0.00524(5000) + 1.63$
$r(g) = 26.2 + 1.63 = 27.83$
So, $$27.83$ is the most that can be charged for the second piece.
Any remaining gallons over $5000$ would be multiplied by $0.00396$
$r(g) = 27.83 + 0.00396(g - 5000)$
$r(g) = 27.83 + 0.00396g - 19.8$
$r(g) = 0.00396g + 8.03$

For the fourth piece;
We have to "finish" the third piece first
Then, we can multiply the remaining gallons of water by $0.0031$
What is the complete gallons for the second piece? - It is $10000$
In other words, $10000$ is the end point for the third piece. $10000$ is included.
Let us find the rate for that end point, $10000$
$r(g) = 0.00396(10000) + 8.03$ $r(g) = 39.6 + 8.03 = 47.63$
So, $$47.63$ is the most that can be charged for the third piece.
Any remaining gallons over $10000$ would be multiplied by $0.0031$
$r(g) = 47.63 + 0.0031(g - 10000)$
$r(g) = 47.63 + 0.0031g - 31$
$r(g) = 0.0031g + 16.63$

We can now write the piecewise function as:
$$ r(g) = \begin{cases} $17.35; & \quad 0 \leq g \leq 3000 \\[2ex] 0.00524g + 1.63; & \quad 3000 \lt g \leq 5000 \\[2ex] 0.00396g + 8.03; & \quad 5000 \lt g \leq 10000 \\[2ex] 0.0031g + 16.63; & \quad g \gt 10000 \end{cases} $$ Let us recalculate all the questions using the Piecewise Function method.

(1.) $2500$ gallons falls in the first piece.
The rate for the first piece is $$17.35$
$\therefore$ the rate for $2500$ gallons of water = $$17.35$

(2.) $3700$ gallons falls in the second piece.
The rate for the second piece is $0.00524g + 1.63$
For $g = 3700$, $r = 0.00524(3700) + 1.63$
$r = 19.388 + 1.63$
$r = 21.018$
$\therefore$ the rate for $3700$ gallons of water = $$21.02$

(3.) $5000$ gallons falls in the second piece.
The rate for the second piece is $0.00524g + 1.63$
For $g = 5000$, $r = 0.00524(5000) + 1.63$
$r = 26.2 + 1.63$
$r = 27.83$
$\therefore$ the rate for $5000$ gallons of water = $$27.83$

(4.) $7500$ gallons falls in the third piece.
The rate for the third piece is $0.00396g + 8.03$
For $g = 7500$, $r = 0.00396(7500) + 8.03$
$r = 29.7 + 8.03$
$r = 37.73$
$\therefore$ the rate for $7500$ gallons of water = $$37.73$

(5.) $12000$ gallons falls in the fourth piece.
The rate for the fourth piece is $0.0031g + 16.63$
For $g = 12000$, $r = 0.0031(12000) + 16.63$
$r = 37.2 + 16.63$
$r = 53.83$
$\therefore$ the rate for $12000$ gallons of water = $$53.83$

(6.) $4692$ gallons falls in the second piece.
The rate for the second piece is $0.00524g + 1.63$
For $g = 4692$, $r = 0.00524(4692) + 1.63$
$r = 24.58608 + 1.63$
$r = 26.21608$
$\therefore$ the rate for $4692$ gallons of water = $$26.22$

(7.) $6456$ gallons falls in the third piece.
The rate for the third piece is $0.00396g + 8.03$
For $g = 6456$, $r = 0.00396(6456) + 8.03$
$r = 25.56576 + 8.03$
$r = 33.59576$
$\therefore$ the rate for $6456$ gallons of water = $$33.60$

Ask students their preferred method - solving it manually or solving it by piecewise function.
They should give reasons for their answers.
Ask students if any of them can come up with another method besides the two methods already discussed.

Calculator for the Water Rates by Calhoun County Water Authority, State of Alabama, USA

Water Usage and Monthly Bill Statement

You are welcome to test the calculator and see the output.
Please complete all fields. Then, click "Bill Statement".

  • Water Usage

in

  • Monthly Bill Statement
Student Project
All information should be verifiable.

Each student will work independently.
No two projects should be the same.
Research any real-world case where piecewise function is used. It should be on the "direct website" of the company or organization. No textbook examples.
Find any application that uses "at least 3" pieces, and has "at least" a function in each piece that includes a variable (the independent variable).
The application should be on a verifiable website.
Write the "complete" address of the "direct page" of the website where you found the application.
*If the "direct" web address is too long, please shorten it by pasting the "complete web address" into www.tinyurl.com
* *This is only for traditional students (onsite) students*
*Generate a short address and write that address "as is".*
For online students, please "copy and paste" the link "as is". Please set the link to open in a new window.
Research Skills: Cite your source properly. Use APA, MLA, or Chicago Manual of Style. Indicate the style you used.
Writing Skills: Write or type the "main" application entirely.
Mathematical Skills - Arithmetic: Use random numbers to test the real-world application manually for each "piece". Write down your results.
Mathematical Skills - Algebra: Write a piecewise function for that application. Test the piecewise function with the same random numbers. The results should be the same. If the results are not the same, fix it.

Technology Skills: Use any programming language to write the piecewise function.

The requirements for the Technology Skills are:
Program asks user to enter the:
First and last names.
Initial and final water meter readings.
Tax (if applicable).

The deliverables for the Technology Skills are:
Title of Application
Greeting. Greet the customer appropriately. Include the customer's first and last names.
Specify the dates for the monthly water bill.
Display the initial and final water meter readings entered by the user.
Calculate the gallons of water used.
Calculate the bill for the gallons of water used.
Calculate the tax (as applicable).
Calculate the final bill.
Test your program with the same random numbers. The results should be the same. If the results are not the same, fix it.

Checks:
Initial water reading cannot be negative.
Final water reading cannot be negative.
The tax cannot be negative.
The initial water reading cannot be greater than the final water reading.
If any of the conditions are not met; the program should display an error message of the condition that was violated, then the program should quit. Do not allow the program to run.

The teacher should guide each student to the successful completion of the project.
Let students know you are willing to help.

Further Work
Writing Skills: Submit a Reflection Journal. Include your challenges, and how you overcame those challenges.
Technology Skills:
Include penalties for late payments in your project.
Write a mobile application for the project.
Develop a mobile application that sends reminders to customers so they do not miss payments.
Develop a rewards program that rewards customers who make payments on time.

Federal Income Tax Rates - 2018

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 $\$9,525$
$12\%$ $\$9,526 \:\:to\:\: \$38,700$ $\$13,601 \:\:to\:\: \$51,800$ $\$19,051 \:\:to\:\: \$77,400$ $\$9,526 \:\:to\:\: \$38,700$
$22\%$ $\$38,701 \:\:to\:\: \$82,500$ $\$51,801 \:\:to\:\: \$82,500$ $\$77,401 \:\:to\:\: \$165,000$ $\$38,701 \:\:to\:\: \$82,500$
$24\%$ $\$82,501 \:\:to\:\: \$157,500$ $\$82,501 \:\:to\:\: \$157,500$ $\$165,001 \:\:to\:\: \$315,000$ $\$82,501 \:\:to\:\: \$157,000$
$32\%$ $\$157,501 \:\:to\:\: \$200,000$ $\$157,501 \:\:to\:\: \$200,000$ $\$315,001 \:\:to\:\: \$400,000$ $\$157,001 \:\:to\:\: \$200,000$
$35\%$ $\$200,001 \:\:to\:\: \$500,000$ $\$200,001 \:\:to\:\: \$500,000$ $\$400,001 \:\:to\:\: \$600,000$ $\$200,001 \:\:to\:\: \$300,000$
$37\%$ $\$500,001 \:\:or\:\: more$ $\$500,001 \:\:or\:\: more$ $\$600,001 \:\:or\:\: more$ $\$300,001 \:\:or\:\: more$

Let us calculate the taxes for the filing "Single" option.
We shall test each piece.


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 $
Solution - $1^{st}$ 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 - $2^{nd}$ Method: Piecewise Function/Algebraic Method
What if we have to calculate the taxes for "several" taxable incomes?
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 $\$700,712.00$ = $\$224,952.94$

Ask students their preferred method - solving it manually or solving it by piecewise function.
They should give reasons for their answers.
Ask students if any of them can come up with another method besides the two methods already discussed.



Student Project

All information should be verifiable.

Each student will work independently.
No two projects should be the same.
Research any real-world case where piecewise function is used. It should be on the "direct website" of the company or organization. No textbook examples.
Find any application that uses "at least 3" pieces, and has "at least" a function in each piece that includes a variable (the independent variable).
The application should be on a verifiable website.
Write the "complete" address of the "direct page" of the website where you found the application.
*If the "direct" web address is too long, please shorten it by pasting the "complete web address" into www.tinyurl.com
* *This is only for traditional students (onsite) students*
*Generate a short address and write that address "as is".*
For online students, please "copy and paste" the link "as is". Please set the link to open in a new window.
Research Skills: Cite your source properly. Use APA, MLA, or Chicago Manual of Style. Indicate the style you used.
Writing Skills: Write or type the "main" application entirely.
Mathematical Skills - Arithmetic: Use random numbers to test the real-world application manually for each "piece". Write down your results.
Mathematical Skills - Algebra: Write a piecewise function for that application. Test the piecewise function with the same random numbers. The results should be the same. If the results are not the same, fix it.

Technology Skills: Use any programming language to write the piecewise function.

The requirements for the Technology Skills are:
Program asks user to enter the:
First and last names.
Initial and final meter readings.
Tax (if applicable).

The deliverables for the Technology Skills are:
Title of Application
Greeting. Greet the customer appropriately. Include the customer's first and last names.
Specify the dates for the monthly power bill.
Display the initial and final meter readings entered by the user.
Calculate the KWh of power consumed.
Calculate the bill for the KWh of power consumed.
Calculate the tax (as applicable).
Calculate the final bill.
Test your program with the same random numbers. The results should be the same. If the results are not the same, fix it.

Checks:
Initial power reading cannot be negative.
Final power reading cannot be negative.
The tax cannot be negative.
The initial power reading cannot be greater than the final power reading.
If any of the conditions are not met; the program should display an error message of the condition that was violated, then the program should quit. Do not allow the program to run.

The teacher should guide each student to the successful completion of the project.
Let students know you are willing to help.

Further Work
Writing Skills: Submit a Reflection Journal. Include your challenges, and how you overcame those challenges.
Technology Skills:
Include penalties for late payments in your project.
Write a mobile application for the project.
Develop a mobile application that sends reminders to customers so they do not miss payments.
Develop a rewards program that rewards customers who make payments on time.

Students should come up with additional functionalities and advancements.
Please note their responses and advise.

Greatest Integer Function

The greatest integer function is also called the floor function.

The greatest integer of a number is defined as the greatest integer less than or equal to that number.

The greatest integer of a number, say $x$ is denoted by ${\lfloor x \rfloor}$ or $[\![ x ]\!]$

$\mathbb{Z}$ is the set of all integers.

Say we have a decimal: $\large x.y$;
$x$ is the integer part.

$$ {\lfloor x \rfloor} = \left \{\begin{aligned} & x; && x \: \epsilon \: \mathbb{Z} \\[2ex] & x; && x.y > 0 \\[2ex] & x-1; && x.y < 0 \end{aligned} \right.$$

Greatest Integer Function Calculator

The greatest integer of $\:$ =



Least Integer Function

The least integer function is also called the ceiling function.

The least integer of a number is defined as the least integer greater than or equal to that number.

The least integer of a number, say $x$ is denoted by ${\lceil x \rceil}$ or $]\!]x[\![$

$\mathbb{Z}$ is the set of all integers.

Say we have a decimal: $\large x.y$;
$x$ is the integer part.

$$ {\lceil x \rceil} = \begin{cases} x, & \quad \text{if }\: x \: \epsilon \: \mathbb{Z} \\[2ex] x + 1, & \quad \text{if }\: x.y > 0 \\[2ex] x, & \quad \text{if }\: x.y < 0 \end{cases} $$

Did you notice we wrote this piecewise function in another way? It is still acceptable this way.


Least Integer Function Calculator

The least integer of $\:$ =



Absolute Value Function

The absolute value function is sometimes called the modulus function. (not modulo function)

The absolute value of a number is defined as the magnitude of the number regardless of sign.

This implies that the absolute value of a number must either be zero, or positive.

The number can be negative, zero, or positive.

However, the absolute value cannot be negative.

It can either be zero (if the number is zero), or positive (if the number is negative or positive).

The absolute value of a number, say $x$ is denoted by $|x|$

$$ |x| = \begin{cases} -x; & \quad x< 0 \\[2ex] x; & \quad x≥ 0 \end{cases} $$

Absolute Value Function Calculator

The absolute value of $\:$ =



References

Chukwuemeka, S.D (2016, April 30). Samuel Chukwuemeka Tutorials - Math, Science, and Technology. Retrieved from https://www.samuelchukwuemeka.com

Bankrate.com. (2018, November 28). 2018-2019 Tax Brackets | Bankrate.com. Retrieved from https://www.bankrate.com/finance/taxes/tax-brackets.aspx

Calhoun County Water Association. (n.d.). Calhoun County Water Association. Retrieved from http://www.calhouncwa.com/rates.htm

Georgia Public Service Commission. (n.d.). Georgia Public Service Commission. Retrieved from http://www.psc.state.ga.us/calc/electric/GPcalc.asp

Kelly Phillips Erb. (2019, January 30). New: IRS Announces 2018 Tax Rates, Standard Deductions, Exemption Amounts And More. Retrieved from https://www.forbes.com/sites/kellyphillipserb/2018/03/07/new-irs-announces-2018-tax-rates-standard-deductions-exemption-amounts-and-more/#3c6ec333133d

TaxAct. (n.d.). TaxAct. Retrieved from https://www.taxact.com/tools/tax-bracket-calculator

CrackACT. (n.d.). Retrieved from http://www.crackact.com/act-downloads/