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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

RELATIONS AND FUNCTIONS

Relations and Functions 7
Relations and Functions 1
Relations and Functions 2
Relations and Functions 3
Relations and Functions 4
Relations and Functions 5
Relations and Functions 6
Begin from First Example$\rightarrow$

Welcome to Our Site

I greet you this day,
First: read the notes. Second: view the videos. Third: solve the questions/solved examples. Fourth: check your solutions with my thoroughly-explained solutions. Fifth: check your answers with the calculators as applicable.
I wrote the codes for some of the calculators using JavaScript, a client-side scripting language.
The Wolfram Alpha widgets (many thanks to the developers) was used for some calculators.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me.
If you are my student, please do not contact me here. Contact me via the school's system.
Thank you for visiting!!!
Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

Objectives

For Algebra and Discrete Mathematics Students

Students will:
(1.) Define relations.
(2.) Define functions.
(3.) Differentiate between relations and functions.
(4.) Discuss the types of functions.
(5.) List the different ways to represent relations.
(6.) Represent relations using those different ways.
(7.) Define the domain of a relation.
(8.) Define the codomain of a relation.
(9.) Define the range of a relation.
(10.) Represent the domain of a relation in set notation.
(11.) Represent the domain of a relation in interval notation.
(12.) Represent the range of a relation in set notation.
(13.) Represent the range of a relation in interval notation.
(14.) Evaluate functions.
(15.) Discuss the symmetry of functions.
(16.) Determine whether a function is even, odd, or neither even nor odd.
(17.) Perform arithmetic operations on functions.
(18.) Discuss linear functions.
(19.) Discuss the slope concept.
(20.) Determine the slope of a linear function.
(21.) Express linear functions in different forms.
(22.) Discuss the concepts of parallel and perpendicular lines based on the slope concept.
(23.) Interpret the slopes of linear functions.
(24.) Solve applied problems on linear functions.
(25.) Discuss quadratic functions.
(26.) Determine the vertex of a parabola.
(27.) Express quadratic functions in different forms.
(28.) Interpret the vertices of quadratic functions.
(29.) Solve applied problems on quadratic functions.
(30.) Calculate the difference quotient of a function.
(31.) Compose functions.
(32.) Decompose functions.
(33.) Determine the inverse of a function.
(34.) Test if a function is an inverse of another function.
(35.) Solve applied problems on relations.
(36.) Solve applied problems on the inverse of functions.


For Discrete Mathematics Students

(37.) Discuss the properties of relations.
(38.) Discuss order relations.
(39.) Determine the minimal and the maximal element of a poset.
(40.) Determine the lower bounds and the upper bounds of a poset.
(41.) Determine the least upper bound and the greatest lower bound of a poset.
(42.) Draw Hasse diagrams of posets.
(43.) Discuss lattices.
(44.) Discuss compatibility and topological sorting.
(45.) Solve applied problems on order relations.


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


These topics will be covered "holistically". Students can focus on the sections relevant to their curriculum.

Vocabulary Words

Ask students to suggest possible vocabulary words for this topic based on the objectives.

Connect with English Language: relation, relative ("blood relative", "relative to ..."), relationship, compatible, compatibility, rule, least, greatest, minimum, maximum, sort

Connect with Physics: reflection, reflexive, reflexivity

Mathematics: relation, function, one-to-one function, injective function, onto function, surjective function, bijective function, set, table, mapping, rule, correspondence, equation, graph, domain, codomain, image, range, set notation, interval notation, evaluate, symmetry, arithmetic operations, difference quotient, slope, intercept, x-intercept, y-intercept, slope-intercept form, point-slope form, two-points form, difference quotient, derivative, function composition, function decomposition, linear functions, quadratic functions, parabolas, vertical parabola, vertex, axis, line of symmetry, function transformations, even functions, odd functions, order relations, total order, partial order, pseudo order, quasi order, reflexivity, symmetric, antisymmetric, transitivity, poset, maximal element, minimal element, Hasse diagrams, upper bound, lower bound, least upper bound, greatest lower bound, lattices, lexicographic order, topological sorting

Symbols and Meanings

  • $\mathbb{N}$ = set of natural numbers
  • $\mathbb{W}$ = set of whole numbers
  • $\mathbb{Z}$ = set of integers
  • $\mathbb{Z}_+$ = set of positive integers
  • $\mathbb{Z}_-$ = set of negative integers
  • $\mathbb{Q}$ = set of rational numbers
  • $\mathbb{I}$ = set of irrational numbers
  • $\mathbb{R}$ = set of real numbers
  • $\mathbb{C}$ = set of complex numbers
  • $\mathbb{P}$ = set of prime numbers
  • { } (braces) = used in set notation
  • [ ] (brackets) and ( ) (parenthesis) = used in interval notation
  • [ ] = closed interval (closed at both ends)
  • ( ) = open interval (open at both ends)
  • [ ) = half-closed half-open interval (closed at $1^{st}$ end, open at $2^{nd}$ end)
  • ( ] = half-open half-closed interval (open at $1^{st}$ end, closed at $2^{nd}$ end)
  • [c, d] = closed interval - includes $c$ and $d$
  • (c, d) = open interval - excludes $c$ and $d$
  • [c, d) = half-closed half-open interval - includes $c$, excludes $d$
  • (c, d] = half-open half-closed interval - excludes $c$, includes $d$
  • $D$ = domain
  • $C$ = codomain
  • $R$ = range
  • $x | $ or $x: $ = $x$ such that
  • $f(x)$ read as $f \:of\: x$
  • $g(x)$ read as $g \:of\: x$
  • $h(x)$ read as $h \:of\: x$
  • $(f + g)(x)$ read as $f \:plus\: g \:of\: x$
  • $(f - g)(x)$ read as $f \:minus\: g \:of\: x$
  • $(f * g)(x)$ read as $f \:times\: g \:of\: x$
  • $(f \div g)(x)$ read as $f \:divided\: by\: g \:of\: x$
  • $f(x) + g(x)$ read as $f \:of\: x \:plus\: g \:of\: x$
  • $f(x) - g(x)$ read as $f \:of\: x \:minus\: g \:of\: x$
  • $f(x) * g(x)$ read as $f \:of\: x \:times\: g \:of\: x$
  • $f(x) \div g(x)$ read as $f \:of\: x \:divided\: by\: g \:of\: x$
  • $(f \circ g)(x)$ read as $f$ composed with $g$ of $x$
  • $(f \circ g)(x)$ read as The composition of $f$ and $g$ of $x$
  • $f(g(x))$ read as $f \:of\: g \:of\: x$
  • $(g \circ f)(x)$ read as $g$ composed with $f$ of $x$
  • $(g \circ f)(x)$ read as The composition of $g$ and $f$ of $x$
  • $g(f(x))$ read as $g \:of\: f \:of\: x$
  • $DNE$ read as $Does \:Not\: Exist$

Definitions

A Relation is a set of ordered pairs, $(x, y)$.
$x = input$
$y = output$
For a relation;
An input value has at least an output value.
At least one means one or more
This implies that for a relation;
An input value can have only one output value.
An input value can also have several output values.


A Function is a relation in which each input has a unique output.
It is first an foremost, a relation.
Each input value has a unique output value.
"Unique Output" means that any input cannot have two or more output values
For a function;
An input value can have only one output value.
Two input values can also have the same output value.
However, an input value cannot have more than one output value.

Ask students if they can tell the difference between a function and relation.

Let us look at this way:
Two students can make the same grade on the same mathematics test.
But, no student can make more than one grade on the same mathematics test.


A One-to-One Function is a function in which each input value has a corresponding different output value.
It is also known as an Injective Function.
It is first an foremost, a function.
If the relation is not a function, it cannot be a one-to-one function.
Each input value has a corresponding different output value.
For a one-to-one function;
One input value must have only one output value.

Ask students if they see the reason why it is called one-to-one.

No two input values can have the same output value.
No input value can have more than one output value.
Let us look at this way:
Each student at Arizona Western College (AWC) has only one Student Identification Number (Student ID).
That ID corresponds to each student. It is an "output value", the result of an "input value" - that is being a student.
That ID is also different for each student. No two students have the same ID.


An Onto Function is a function in which each output value is an image of at least an input value.
It is also known as a Surjective Function.
It is first an foremost, a function.
If the relation is not a function, it cannot be an onto function.
Each output value is a product of ("came from", "is as a result of") an input value.
That product is called an image.
For an onto function;
Each output value is an image of at least one input value.
An output value can be an image of only one input value.
An output value can also be an image of more than one input value.
But, remember: if the relation is not a function, it cannot be an onto function.

Ask students to give an example of a case where it is possible to see a relation as an onto function, when the relation is not a function.
Ask students to give an example of a case where it is possible to see a relation as an onto function, when the relation is a function.

Let us look at this way:
Each person in this world was created by GOD.
Evolution is a lie. Do not believe any of it.
GOD created everyone.
Also, each person in this world was born by a woman.


A Bijective Function is a function that is both injective and surjective.
It is first an foremost, a function.
For any relation/function to be bijective;
It must be one-to-one and it must be onto.


The Domain of a function is defined as the set of all input values that will give an output.

The Codomain of a function is defined as all the output values of the function.

The Range of a function is defined as all the output values of the function which are images of the input values.

The image is the result or output value based on the input value.

Teacher: What is the difference between the "Range" and the "Codomain"?
Did you notice the difference?

A Linear Function is a function in which the highest exponent of the variable is $1$

Introduction

Please observe the two pictures. Do you notice any difference(s)?

Which picture did I take in Nigeria?
Which picture did I take in the United States?

No burgers Yes burgers
pic in Nigeria pic in the United States

Do you see what "eating cheeseburgers" contributed to?... in terms of my weight?

So, we have two variables here:
the independent variable = number of cheeseburgers and
the dependent variable = weight

Why is it called the independent variable?
Why is it called the dependent variable?
Which variable "depends" on the other variable?

Say: weight = $w$ and number of burgers = $b$
$w$ depends on $b$
The weight I gained in the United States depended on the number of burgers I ate.

Student: But, you didn't eat only burgers. Did you?
Teacher: No, I did not. But, let's just focus on burgers at the moment.
Yes, there are other variables that can lead to weight gain such as:

Students: too much screen time(television, games), inadequate number of hours of sleep among others
Teacher: Correct. But, let's just focus on the number of burgers.

So, the weight I gained is a "function" of the number of cheeseburgers I ate.
$w$ is a function of $b$
$w = f(b)$
Just as in Algebra and Calculus; Make that connection right away
$y$ is a function of $x$
$y = f(x)$
So, $y$ is the dependent variable and
$x$ is the independent variable.

Bring it to Statistics
$y$ is the response variable.
$x$ is the predictor or explanatory variable.

Bring it to Philosophy
$y$ is the effect.
$x$ is the cause.
Talk about the existence of GOD based on cause-effect relationship
GOD exists!!!

Bring it to Economics/Business
$y$ is the output.
$x$ is the input.

Bring it to Psychology/Human Behavior/Sociology
$y$ is the consequence.
$x$ is the action.


Say I really wanted to know how many pounds I gain due to the number of burgers I ate.
That means I have to do some statistics which involves research.
Assume that I:
gained $3$ pounds when I ate a burger
gained $6$ pounds when I ate $2$ burgers
gained $9$ pounds when I ate $3$ burgers
gained $12$ pounds when I ate $4$ burgers
and so on and so forth.

Besides writing the information in English language (like we just did) what other ways can we represent this information?

We can represent the information as a Table

Number of burgers, $b$ Weight, $w$
$1$ $3$
$2$ $6$
$3$ $9$
$4$ $12$

We can represent the information as a Set of ordered pairs
Do you remember how we write the set notation?
We have to use braces.
Burger-Weight = {$(1, 3), (2, 6), (3, 9), (4, 12)$}


We can represent the information as a Graph
graph


We can write the information as an Equation or Rule
$w = 3 * b$
$w = 3b$


We can represent this information as a Mapping or Correspondence
mapping

Teacher: So, how many ways can we represent a relation?
Name them.

We can represent a relation in five ways namely:
(1.) Equation or Rule
(2.) Table
(3.) Mapping or Correspondence
(4.) Set of ordered pairs.
(5.) Graph

Domain and Range Calculator

This calculator will:
(1.) Determine the domain of a function.
(2.) Determine the range of a function.
(3.) Write the domain of the function in set notation.
(4.) Write the range of the function in set notation.
(5.) Graph the domain on a number line.
(6.) Graph the range on a number line.

To use the calculator, please:
(1.) Type your function (equation) or expression in the textbox (the bigger textbox).
(2.) Type it according to the examples I listed.
(3.) Copy and paste the function (equation) you typed, into the small textbox of the calculator.
(4.) Click the "Submit" button.
(5.) Check to make sure it is the correct function or expression you typed.
(6.) Review the answer.

Function:

Evaluating Functions

Evaluating Functions means: determining the value of the output for a certain value of the input.
So, you are given the function and a certain input and
you are asked to determine the output corresponding to that input.

To evaluate a function:
(1.) Substitute the value of the "given input" into the function.
(2.) Simplify as applicable.

Evaluating Functions Calculator

This calculator will:
(1.) Evaluate a function for a specified value.
(2.) Return the answer in the simplest form.
(3.) Graph the function and indicate the specified value.

To use the calculator, please:
(1.) Assume the function is $f(x)$.
(2.) Type your expression in the first textbox - bigger Textbox 1.
(3.) Type your specified value in the second textbox - bigger Textbox 2.
(4.) Type them according to the examples I listed.
(5.) Delete the default expression in the first textbox of the calculator.
(6.) Delete the default value in the second textbox of the calculator.
(7.) Copy and paste the expression you typed, into the first textbox of the calculator.
(8.) Copy and paste the specified value you typed, into the second textbox of the calculator.
(9.) Click the "Submit" button.
(10.) Check to make sure that the expression and specified value are your questions.
(11.) Review the answer(s). At least one of the answers is what you need.

Function:

Specified Value:

Linear Functions

A Linear Function is a function in which the highest exponent of the independent variable is $1$.
A Linear Equation is an equation in which the highest exponent of the independent variable is $1$.
For a linear function, only two variables are considered in this context.
The variables are the: input/independent variable, and the output/dependent variable.
The graph of a linear function is a straight line.
The graph of a linear equation is a straight line.

Student: Why do you write both "function" and "equation"? What is the difference?
Teacher: What do you think? - (Typical Nigerian - asking questions with questions ☺☺☺)
Student: I do not know. That's why I'm asking
Teacher: What is the main thing/sign that characterizes an equation?
Student: It's an equal sign, $=$
Teacher: Good answer. So, a function is also...
Student: an equation ...
Teacher: because a function has the ...
Student: equal sign.
Teacher: An equation can also be expressed as a...
Student: function.

Back to functions
Let us consider an input-output (two-variable) function.
How does the output behave when the input is changed?
If you walk more miles, would you burn more calories?
If you study for more hours, would your score in mathematics exam increase?
Remember the passage in the Bible about the laborers that worked for different number of hours but received the same pay? Student: What passage is it?
Teacher: Matthew $20: 9 - 10$

An important concept in Algebra has been to determine how the output changes due to changes in the input.
Student: But, there may be several input factors/variables affecting an output.
Teacher: That is correct. However, let us study two variables as a start.

So, we gave three examples so far.
In the first example; as the input increases (the number of miles increases), the output (the calories stored in your body decreases).
You begin to lose weight (calories, then pounds).
In this case, our slope is negative.
If you draw a graph of the function where $x$ is the number of miles walked and $y$ is the number of calories lost.

Student: What is slope?
Teacher: It is the ratio of the changes in the output with respect to changes in the input.
It describes the steepness or grade of a line.
It is also known as gradient.
Make up some Table of Values to demonstrate it.
So, when we look at the graph from left to right, we see that there is a fall from left to right.
So, for a negative slope; there is a fall from left to right.
Remember, it has to be from left to right.

Let's say that for each mile you walk, you burn $3$ calories
This implies that the output decreases by $3$ units for each unit increase in the input.
This means that our slope is $-3$

Let the:
output, number of calories = $y$ (calories)
input, number of miles = $x$ (miles)
$ y = f(x) \\[2ex] y = -3x $
negative slope

Assume we select the points: $(-2, 6)$ and $(2, -6)$
Point $1$ is $(-2, 6)$
$x_1 = -2, y_1 = 6$

Point $2$ is $(2, -6)$
$x_2 = 2, y_2 = -6$

Slope = $m$
$ m = \dfrac{y_2 - y_1}{x_2 - x_1} \\[5ex] m = \dfrac{-6 - 6)}{2 - (-2)} \\[5ex] m = \dfrac{-12}{2 + 2} \\[5ex] m = \dfrac{-12}{4} \\[5ex] m = -3 $
Negative slope


In the second example; as the input increases (the number of study hours increases), the output (the mathematics exam score increases).
In this case, our slope is positive.
If you draw a graph of the function where $x$ is the number of study hours and $y$ is the mathematics scores.

When we look at the graph from left to right, we see that there is a rise from left to right.
So, for a positive slope; there is a rise from left to right.
Remember, it has to be from left to right.

Let's say that for each hour you study, your mathematics exam score increases by 3 points
This implies that the output increases by $3$ units for each unit increase in the input.
This means that our slope is $3$

Let the:
output, mathematics exam scores = $y$ ($\%$)
input, number of hours of study = $x$ (hours)
$ y = f(x) \\[2ex] y = 3x $
positive slope

Assume we select the points: $(-2, -6)$ and $(2, 6)$
Point $1$ is $(-2, -6)$
$x_1 = -2, y_1 = -6$

Point $2$ is $(2, 6)$
$x_2 = 2, y_2 = 6$

Slope = $m$
$ m = \dfrac{y_2 - y_1}{x_2 - x_1} \\[5ex] m = \dfrac{6 - (-6)}{2 - (-2)} \\[5ex] m = \dfrac{6 + 6}{2 + 2} \\[5ex] m = \dfrac{12}{4} \\[5ex] m = 3 $
Positive slope


In the third example; as the input increases (the number of work hours increases), the output (the wage stays the same).
In this case, our slope is zero.
There is no steepness.
If you draw a graph of the function where $x$ is the number of work hours for someone who was hired in the morning, and $y$ is the wage.

When we look at the graph from left to right, we see that it is a straight horizontal line.
So, for a zero slope; there is a horizontal line from left to right.
In this case, the function is a constant.
Remember, it has to be from left to right.

Let's say that for each hour the person works, the wage remains the same.

Student: I do not think it's fair.
Teacher: In reality, it is not fair.
But, this is a parable.
Student: What is a parable?
Teacher: A parable is an earthly story with a heavenly meaning.
Student: What is the meaning?
Teacher: Romans 8:1


This implies that the output is $$3$ regardless of the number of hours worked.
This means that our slope is $0$

Let the:
output, wage = $y$ ($)
input, number of work hours = $x$ (hours)
$ y = f(x) \\[2ex] y = 3 $
zero slope

Assume we select the points: $(-2, 3)$ and $(2, 3)$
Point $1$ is $(-2, 3)$
$x_1 = -2, y_1 = 3$

Point $2$ is $(2, 3)$
$x_2 = 2, y_2 = 3$

Slope = $m$
$ m = \dfrac{y_2 - y_1}{x_2 - x_1} \\[5ex] m = \dfrac{3 - 3}{2 - (-2)} \\[5ex] m = \dfrac{0}{2 + 2} \\[5ex] m = \dfrac{0}{4} \\[5ex] m = 0 $
Zero slope
The slope of a horizontal line is $0$


Teacher: Are we missing any case?
Student: A vertical line?
Teacher: That is right.
What do you think is the slope of a vertical line?
Student: The slope is undefined.
Teacher: Correct. Can you show it?
Also, would you consider a vertical line to be a function? Student: A vertical line is not a function because it does not pass the Vertical Line test
Teacher: Okay. But, give the reason based on the definition of a function.
Student: A vertical line is not a function because for a function, an input value cannot have more than one output value. Teacher: Correct!


Assume we select the points: $(2, -3)$ and $(2, 3)$
Point $1$ is $(2, -3)$
$x_1 = 2, y_1 = -3$

Point $2$ is $(2, 3)$
$x_2 = 2, y_2 = 3$

Slope = $m$
$ m = \dfrac{y_2 - y_1}{x_2 - x_1} \\[5ex] m = \dfrac{3 - (-3)}{2 - 2} \\[5ex] m = \dfrac{3 + 3}{0} \\[5ex] m = \dfrac{6}{0} \\[5ex] m = undefined $
Undefined slope
The slope of a vertical line is undefined


Intercepts
Intercept is the point where the graph touches/crosses the axis.
Be reminded that a point is defined by two co-ordinates: the $x-coordinate$ and the $y-coordinate$

$x-intercept$ is the point where the graph touches/crosses the $x-axis$
To determine the $x-intercept$:
Set $y = 0$ and
Solve for $x$

$y-intercept$ is the point where the graph touches/crosses the $y-axis$
To determine the $y-intercept$:
Set $x = 0$ and
Solve for $y$

NOTE: Intercept is a point.
A point has two coordinates - the $x-coordinate$ and the $y-coordinate$
The $x-intercept$ has two coordinates - the $x-coordinate$ and the $y-coordinate$ (which is $0$)
The $y-intercept$ has two coordinates - the $x-coordinate$ (which is $0$), and the $y-coordinate$

What do Intercepts Really Mean?

Summary


The Slope of a Linear Function
This is defined as the ratio of the changes in the output values of the function with respect to ($wrt$) to the changes in the input values of the function.
It is also known as the average rate of change.

For any two points, say:
Point $1$ with coordinates $(x_1, y_1)$ and
Point $2$ with coordinates $(x_2, y_2)$;

Slope Formula
$m = \dfrac{y_2 - y_1}{x_2 - x_1}$

The slope of any horizontal line is $0$

The slope of any vertical line is undefined.

The Equations/Forms of a Linear Function are:

(1.) Slope-Intercept Form: $y = mx + b$
where
$m$ is the slope
$b$ is the y-intercept
$x$ is the independent variable
$y$ is the dependent variable

(2.) Standard Form: $Ax + By = C$
where
$A$, $B$, and $C$ are constants and $A \gt 0$
$x$ is the independent variable
$y$ is the dependent variable

(3.) Point-Slope Form: $y - y_1 = m(x - x_1)$
where
$m$ is the slope
$x_1$ is the x-coordinate of Point $1$
$y_1$ is the y-coordinate of Point $1$
$x$ is the independent variable
$y$ is the dependent variable

(4.) Two-Points Form: $\dfrac{y - y_1}{x - x_1} = \dfrac{y_2 - y_1}{x_2 - x_1}$
where
$x_1$ is the x-coordinate of Point $1$
$y_1$ is the y-coordinate of Point $1$
$x_2$ is the x-coordinate of Point $2$
$y_2$ is the y-coordinate of Point $2$
$x$ is the independent variable
$y$ is the dependent variable

What do you notice about this equation?
Did you notice it combines the Slope Formula and the Point-Slope Form?

Based on the Slope Formula;
$m = \dfrac{y_2 - y_1}{x_2 - x_1}$

Based on the Point-Slope Form;
$y - y_1 = m(x - x_1)$
$m(x - x_1) = y - y_1$
This means that: $m = \dfrac{y - y_1}{x - x_1}$

$m = m$
$\therefore \dfrac{y - y_1}{x - x_1} = \dfrac{y_2 - y_1}{x_2 - x_1}$

For any two parallel lines; Let $m_1$ = slope of Line $1$
$m_2$ = slope of Line $2$
$m_1 = m_2$

For any two perpendicular lines; Let $m_1$ = slope of Line $1$
$m_2$ = slope of Line $2$
$m_1 * m_2 = -1$
OR
$m_1 = \dfrac{-1}{m_2}$
OR
$m_2 = \dfrac{-1}{m_1}$

Intercepts

$x-intercept$
To determine the $x-intercept$,
Set $y = 0$ and
Calculate $x$

$y-intercept$
To determine the $y-intercept$,
Set $x = 0$ and
Calculate $y$

Quadratic Functions

Pre-requisite Topic: Quadratic Equations

A Quadratic Function is a function in which the highest exponent of the independent variable is $2$.
A Quadratic Function is a polynomial function of degree $2$.
A Quadratic Equation is an equation in which the highest exponent of the independent variable is $2$.

For a linear function, only two variables are considered in this context.
The variables are the: input/independent variable, and the output/dependent variable.
The graph of a linear function is a straight line.
The graph of a linear equation is a straight line.

Student: Why do you write both "function" and "equation"? What is the difference?
Teacher: What do you think? - (Typical Nigerian - asking questions with questions ☺☺☺)
Student: I do not know. That's why I'm asking
Teacher: What is the main thing/sign that characterizes an equation?
Student: It's an equal sign, $=$
Teacher: Good answer. So, a function is also...
Student: an equation ...
Teacher: because a function has the ...
Student: equal sign.
Teacher: An equation can also be expressed as a...
Student: function.


A Parabola is the graph of a quadratic function.

A Vertical Parabola is the graph of a quadratic function of the form: $y = ax^2 + bx + c$ where $a \ne 0$

Ask students what happens if $a = 0$
Must we put that condition in the definition?
Also, relate it with transformation of functions. What transformation of function is it?

A Horizontal Parabola is of the form: $x = ay^2 + by + c$ where $a \ne 0$

Ask students what happens if $a = 0$
Must we put that condition in the definition?
Also, relate it with transformation of functions. What transformation of function is it?

The Vertex of a vertical parabola is the lowest point (minima) or the highest point (maxima) on the parabola.
NOTE: Vertex is a point.
A point has two coordinates - the $x-coordinate$ and the $y-coordinate$
Vertex has two coordinates - the $x-coordinate$ and the $y-coordinate$
Vertex: Look for these words: maximum, minimum, greatest, least

The Axis of a vertical parabola is the vertical line through the vertex of the parabola.
The axis is the $x-coordinate$ of the vertex.

The Line of Symmetry of a vertical parabola is the axis that shows two "same halves" of the parabola when the parabola is folded across that axis.



Why Do We Study Parabolas?

Ask students to state the two main goals of every business.

Extrema (Maxima and Minima)
(1.) Maximize profits - Vertex of a parabola
Sarah Technologies produces desktop computers.
The mathematician/economist that works at the firm determined that the desktop sales during the summer season is a quadratic function business model.
How many laptops should be sold during the summer season to generate the maximum revenue, and hence, the maximum profit.

(2.) Minimize losses / Minimize costs - Vertex of a parabola
Micah Farms typically orders citrus fertilizers during the winter season.
The mathematician/economist determined that the production costs of citrus trees during the winter season resembles a quadratic business model.
How many fertilizers should be ordered each month during the winter season to minimize the total cost of producing the citrus trees?

vertex

(3.) Have you ever wondered why the light beam from the projectors, headlights of cars, and torches is strong?
Parabolas have a special reflecting property. Hence, they are used in the design of automobile headlights, torch lights, telescopes, television and radio antenna among others.
headlight

(4.) Why do the newest and most popular type of skis have parabolic cuts on both sides?
Under a load, parabola designs on skis deforms to a perfect arc. This shortens the turning area, hence making it easier to turn the skis.
ski

(5.) What is the shape of a water fountain?
water fountain

(6.) What happens when you throw football or soccer or basketball?
It bounces to the ground and bounces up, creating the shape of a parabola.
ball



Given: the general form/standard form of a Quadratic Function: y = $ax^2 + bx + c$;

First term = $ax^2$
Second term = $bx$
Third term = $c$
Coefficient of $x^2 = a$
Coefficient of $x = b$

The Forms of a Quadratic Function are:

(1.) Standard Form/General Form
Note that the standard form is written in descending order of $x$

$ y = ax^2 + bx + c \\[3ex] OR \\[3ex] f(x) = ax^2 + bx + c \\[3ex] Vertex = \left(-\dfrac{b}{2a}, f\left(-\dfrac{b}{2a}\right)\right) \\[5ex] $

(2.) Vertex Form

$ y = a(x - h)^2 + k \\[3ex] OR \\[3ex] f(x) = a(x - h)^2 + k \\[3ex] Vertex = (h, k) $

Sometimes, you will need to convert from one form to another form.
You may be given a Quadratic Equation in Standard Form and asked to write it in Vertex Form
Similarly, you may be given a Quadratic Equation in Vertex Form and asked to write it in Standard Form

Converting a Quadratic Equation Between Standard Form and Vertex Form
There are two ways to do this.

First Method: Vertex Formula
Vertex in Vertex Form = Vertex in Standard Form

$ (h, k) = \left(-\dfrac{b}{2a}, f\left(-\dfrac{b}{2a}\right)\right) \\[5ex] This\:\: means\:\: that \\[3ex] h = -\dfrac{b}{2a} \\[5ex] k = f\left(-\dfrac{b}{2a}\right) \\[5ex] $ Second Method: Completing the Square Method

Symmetry of Functions

Ask students if they have observed: a human being (yes a human being)
Imagine you cut a human being in two equal halves. Don't do it!!!
What do you notice?
What about a butterfly?
What about a starfish?
What about a square table? rectangular desk? Which ones are bilaterally symmetrical? radially symmetrical?
Name several other things that are symmetrical
Avoid defecting to Physics. Avoid deflecting to Geometry.
Alright, back to Algebra. Back to Symmetry of Functions.

Consider a function graph in two axis - the $x-axis$ and the $y-axis$
Imagine you cut through the graph of that function - cut in two equal halves
Does one part look identical to the other part?
If no, find a graph that you has identical halves when you cut through it.
If yes, go to the next line.
Wait a minute. How did you cut the graph?
Did you cut it vertically - about the $y-axis$
Or did you cut it horizontally - about the $x-axis$
Show students the graph of $y = x^2$; $y = -x^2$; $x = y^2$; $y = x^3$; and $y = -x^3$
Ask them what they notice

So;
If you cut the graph of a function vertically, and
you notice identical halves; left and right
this means that you have the same y-value for corresponding opposite x-values
From the graph of $y = x^2$ that you showed them;
$y = f(x)$
$y = x^2$
$f(x) = y$
$f(x) = x^2$
$f(2) = 2^2 = 4$
$f(-2) = (-2)^2 = 4$
Same result (same y-value) for corresponding opposite x-values
Same $4$ for $2$ and $-2$
$(2, 4) = (-2, 4)$ on the same graph
So, $(x, y) = (-x, y)$
$f(x) = f(-x)$
$f(-x) = f(x)$
this implies that the graph is symmetrical about the y-axis
therefore; the function is even.
It is an even function.

So;
If you cut the graph of a function horizontally, and
you notice identical halves; above (up) and below (down)
this means that you have the same x-value for corresponding opposite y-values
From the graph of $x = y^2$ that you showed them;
$y^2 = x$
$y = \pm \sqrt{x}$
Ask students "what type of function" this is
Hmmmm...is this even a function?
Does it pass the Vertical Line test?
Take them back to the definition of a function - can the same input value have two different output values?
Can a student make two different grades on the same test?
$x = y^2$
$y = \pm \sqrt{x}$
For $x = 4$;
$y = \pm \sqrt{4}$
$y = \pm 2$
$(4, 2)$ and $(4, -2)$ on the same graph
Same x-value for corresponding opposite y-values
So, $(x, y) = (x, -y)$
this implies that the graph is symmetrical about the x-axis
It is not a function.

Ask students if it is possible to have both ways - symmetrical about the y-axis and symmetrical about the x-axis?
What happens if you cut it both vertically and horizontally; and there are identical halves?
Some students may ask about cutting it diagonally. Please respond.
From the graph of $y = x^3$ that you showed them;
$y = f(x)$
$y = x^3$
$f(x) = y$
$f(x) = x^3$
$f(2) = 2^3 = 8$
$f(-2) = (-2)^3 = -8$
We can also have $x-values$ and $y-values$ for corresponding opposite $x-values$ and $y-values$
$(2, 8) = (-2, -8)$ on the same graph
We can have: $(x, y) = (-x, -y)$
$f(-x) = -f(x)$
this implies that the graph is symmetrical about the origin
therefore; the function is odd.
It is an odd function.

Summary

(1.) For each $(x, y)$ on the graph,
there is also $(-x, y)$ on the same graph
(1.) For each $(x, y)$ on the graph,
there is also $(-x, -y)$ on the same graph
(1.) For each $(x, y)$ on the graph,
there is also $(x, -y)$ on the same graph
(1.) For each $(x, y)$ on the graph,
there is neither $(-x, y)$ nor $(-x, -y)$ on the same graph
(2.) Same $y-value$ for corresponding opposite $x-values$ (2.) Opposite $y-values$ for corresponding opposite $x-values$ (2.) Same $x-value$ for corresponding opposite $y-values$ (2.) No same $y-value$ for corresponding opposite $x-values$
No opposite $y-values$ for corresponding opposite $x-values$
(3.) $f(-x) = f(x)$
Easier to test with $x = 1$
(3.) $f(-x) = -f(x)$
Easier to test with $x = 1$
(3.) $f(x) = -f(x)$
Easier to test with $x = 1$
(3.) $f(-x) \ne f(x)$ and $f(-x) \ne -f(x)$
Easier to test with $x = 1$
(4.) Identical halves when graph is cut vertically (4.) Identical halves when graph is cut vertically and horizontally
Center is the origin $(0, 0)$
(4.) Identical halves when graph is cut horizontally (4.) No identical halves
Graph is: symmetric about the $y-axis$ symmetric about the origin symmetric about the $x-axis$ not symmetric
Function is: even odd not a function
neither even nor odd
neither even nor odd

Operations on Functions

Operations on Functions are the basic arithmetic operations performed on functions.
These include the arithmetic operations of addition, subtraction, multiplication, and division.

How do we apply this in real-life?
Say you are involved in the production of an item, say laptop computers
Ask students to list the two main goals of every business - even the non-profit / 501(c) businesses.
They can include those goals in their resumes.

So, we want to minimize the production costs of each laptop computer, and
We want to maximize the sales, thereby maximizing the revenue from the sales, and thereby maximizing the profits
But, how do we calculate the profit?
$Profit = Revenue - Cost$
That is an arithmetic operation of subtraction.

Let:
$x$ be the number of items or service. In this case, the item is the laptop computer
$C(x)$ be the Cost function
$R(x)$ be the Revenue function
$P(x)$ be the Profit function
$P(x) = R(x) - C(x)$
$C(x), R(x), \:and\: P(x)$ are all functions of $x$
As you can see, we have performed the arithmetic operation of subtraction on those two functions, the cost function and the revenue function to calculate the profit function.

Ask students to give real-world examples where we add functions, multiply functions, and divide functions.

Given: any two functions of $x$ say $f(x)$ and $g(x)$ such that $f(x)$ and $g(x)$ are defined for all values of $x$; then:
(1.) Sum Function: $(f + g)(x) = f(x) + g(x)$
Similarly, $f(x) + g(x) = (f + g)(x)$

(2.) Difference Function: $(f - g)(x) = f(x) - g(x)$
Similarly, $f(x) - g(x) = (f - g)(x)$

(3.) Product Function: $(f * g)(x) = f(x) * g(x)$
Similarly, $f(x) * g(x) = (f * g)(x)$

(4.) Quotient Function: $(f \div g)(x) = f(x) \div g(x)$ where $g(x) \ne 0$
Similarly, $f(x) \div g(x) = (f \div g)(x)$ where $g(x) \ne 0$
OR
Quotient Function: $\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)}$ where $g(x) \ne 0$

Similarly, $\dfrac{f(x)}{g(x} = \left(\dfrac{f}{g}\right)(x)$ where $g(x) \ne 0$

Let us do an example.
Example
Add, subtract, multiply and divide the functions.

(1.) $f(x) = -4x + 3$
$g(x) = 7x + 5$

$(f + g)(x) \\[2ex] = f(x) + g(x) \\[2ex] = (-4x + 3) + (7x + 5) \\[2ex] = -4x + 3 + 7x + 5 \\[2ex] = 3x + 8 $
$(f - g)(x) \\[2ex] = f(x) - g(x) \\[2ex] = (-4x + 3) - (7x + 5) \\[2ex] = -4x + 3 - 7x - 5 \\[2ex] = -11x - 2 $
$(f * g)(x) \\[2ex] = f(x) * g(x) \\[2ex] = (-4x + 3)(7x + 5) \\[2ex] = -28x^2 - 20x + 21x + 15 \\[2ex] = -28x^2 + x + 15 $
$\left(\dfrac{f}{g}\right)(x) \\[3ex] = \dfrac{f(x)}{g(x} \\[3ex] = \dfrac{-4x + 3}{7x + 5} $

Arithmetic Operations on Functions Calculator

This calculator will:
(1.) Add two functions.
(2.) Subtract two functions.
(3.) Multiply two functions.
(4.) Divide two functions.
(5.) Calculate the result of a function raised to an exponent value.
(6.) Perform arithmetic operations on functions according to the order of operations.
(7.) Graph the result of the arithmetic operation.

To use the calculator, please:
(1.) Assume the first function is $f(x)$.
(2.) Assume the second function is $g(x)$.
(3.) Type the first function in the first textbox - bigger Textbox 1.
(4.) Type the second function in the first textbox - bigger Textbox 2.
(5.) Type the arithmetic operation(s) in the third textbox - bigger Textbox 3.
(6.) Type them according to the examples I listed.
(7.) Copy and paste the first expression you typed, into the first textbox of the calculator.
(8.) Copy and paste the second expression you typed, into the second textbox of the calculator.
(9.) Copy and paste the arithmetic operation(s) you typed, into the third textbox of the calculator.
(10.) Click the "Submit" button.
(11.) Check to make sure that the expressions you typed are the actual expressions of your question.
(12.) Review all the answer(s). At least one of the answers is what you need.

$f(x) = $

$g(x) = $

$Arithmetic\: Operation:$

Difference Quotient

Difference Quotient

This topic prepares us for Calculus.
It is actually the first principle used in determining the derivative of a function.
The limit of the difference quotient of a function as the increment approaches zero is the derivative of the function.

Teacher: Let's rewind to geometry
What is the difference between a secant line and a tangent line

A secant line is a line that cuts across (intersects) two points on a curve.

A tangent line is a line that touches only one point on a curve.

Compare and Contrast the Difference Quotient of a Function and the Derivative of a Function

Difference Quotient of a Function Derivative of a Function
is denoted by $\dfrac{\Delta y}{\Delta x}$ is denoted by $\dfrac{dy}{dx}$
is the slope of the secant line. is the slope of the tangent line.

Difference Quotient Calculator

This calculator will:
(1.) Calculate the difference quotient of a function.
(2.) Return the answer in the simplest form.

To use the calculator, please:
(1.) Assume the function is $f(x)$.
(2.) Assume the function value is $f(x + h)$.
(3.) Calculate the function value first. Use the "Evaluating Functions Calculator" to calculate it.
Find the value of $f(x + h)$ first using the "Evaluating Functions Calculator".
In other words, evaluate the function, $f(x)$ for $x = x + h$ using the "Evaluating Functions Calculator".
(4.) Type your expression in the first textbox - bigger Textbox 1.
(5.) Type the result (the calculated value you got from the Evaluating Functions Calculator) in the second textbox - bigger Textbox 2.
(6.) Type them according to the examples I listed.
(7.) Copy and paste the expression you typed, into the first textbox of the calculator.
(8.) Copy and paste the calculated value you typed, into the second textbox of the calculator.
(9.) Click the "Difference Quotient" button.
(10.) Check to make sure that those values were correctly substituted in the formula for Difference Quotient.
(11.) Review the answer(s). At least one of the answers is what you need.

$f(x)$:

$f(x + h)$:

Composition of Functions

Say you visit your favorite retail store.

Ask students to list their favorite retail stores.
Say you visit a Walmart store
You want to buy a bag
Say the cost of each bag = $$20.00$
You want to buy $10$ bags
$10$ bags @ $$20.00$ / bag = $10 * 20$ = $$200.00$
Say the sales tax = $10\%$
$10\%$ sales tax = ${10 \over 100} * 200$ = $$20.00$
Total Cost = $200.00 + 20.00$ = $$220.00$
Do you see how we write all these steps just to calculate the total cost of a single item?

Ask students if they can find another way (preferably an easier way) to calculate the total cost

Can we just write functions to make it easier for us to do?

Let:
$x$ = number of bags
Cost function = $C(x)$
This means that:
$C(x) = 20x$
This is one function.
Let us form another function.
Some students may ask why.
Ask students the meaning of "composite".
Emphasize that "composite" involves "more than one". Then, explain the "Composition of Functions".

Let:
$d$ = dollar
Total cost function = $T(d)$
$T(d) = d + 10\%d$
But, the cost is a function of $x$
The total cost is also a function of $x$
Yes, it is a function of the dollar; but the dollar is also a function of the number of bags, $x$
$T(d) = d + 0.1d$
$T(d) = 1.1d$

Let us now find the composition of $T$ and $C$
This will be written as:
$T(C(x))$
But why???
Before we find the cost function, we have to find the total cost function.
You have to work from "inside" to "outside"
You have to be good "inwards" before you are good "outwards"
Do you remember the saying: "Charity begins at home".
Here is the thing: I want to one function that will just calculate the total cost.
Give me one function of one variable (one independent variable) that will just give me the total cost without having to go through all those initial steps.
Make life easier!
So, $T(d) = 1.1 d$
$\therefore T(C(x) = 1.1 * C(x)$
$T(C(x)) = 1.1 * 20x$
$T(C(x)) = 22x$
This is what I need!
So, for $x = 10$ bags,
$T(C(20)) = 22 * 10$ = $$220.00$

Does this make sense?
Do you see the benefit of the composition of functions?

With the composition of functions, you can get two or more different functions of the same variable, and "compose" them into just one function of that variable.
Say you have two functions of a variable, $x$;
$f(x)$ and $g(x)$
$f(x)$ is read as $f \:of\: x$
$g(x)$ is read as $g \:of\: x$
Then;
$f$ composed with $g$ of $x$ = $(f \circ g)(x)$
Alternatively, you can say:
The composition of $f$ and $g$ of $x$ = $(f \circ g)(x)$
Similarly;
$(g \circ f)(x)$ is the composition of $g$ and $f$ of $x$ OR
$(g \circ f)(x)$ is $g$ composed with $f$ of $x$
Welcome to $f \circ g$ and $g \circ f$

$(f \circ g)(x) = f(g(x))$
$(g \circ f)(x) = g(f(x))$
Do the inner function first.
Do the outer function of the result of the inner function next.

For $f(g(x))$;
Do $g(x)$ first
Then, do $f(...what \:you\: have)$ next

For $g(f(x))$;
Do $f(x)$ first
Then, do $g(...what \:you\: have)$ next

Can you have more than two functions composed as a single function?

Say you have three functions of $x$:
$x$ = independent variable
$f(x)$
$g(x)$
$h(x)$

$f(g(h(x)))$ means that you should do:
$h(x)$ first
then, do $g(...what \:you\: have)$ next
then, do $f(...what \:you\: have)$ next

Do you notice any relationship between "Evaluating Functions", "Operations on Functions", and the "Composition of Functions"?
Do you see why we learn the topics in that order?

Domain of Composite Functions
Given: two functions of $x$, say $f(x)$ and $g(x)$;
$(f \circ g)(x) = f(g(x))$
The domain is all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$
This means that:
(1.) We have to find the domain of $g(x)$ first. If there is any restriction in that domain, we have to restrict those value(s) in the composite function as well.
(2.) Then, we have to find the domain of the composite function.
In other words, we have to look at the intersection of both domains - the domain of the inner function, $g(x)$ and the domain of the composite function, $f(g(x))$

Similarly,
$(g \circ f)(x) = g(f(x))$
The domain is all $x$ in the domain of $f$ such that $f(x)$ is in the domain of $g$
(1.) We have to find the domain of $f(x)$ first. If there is any restriction in that domain, we have to restrict those value(s) in the composite function as well.
(2.) Then, we have to find the domain of the composite function.
In other words, we have to look at the intersection of both domains - the domain of the inner function, $f(x)$ and the domain of the composite function, $g(f(x))$

Let us do some examples.
Example 1
Calculate:
$(f \circ g)(x) \\[3ex] (g \circ f)(x) \\[3ex] $ the domain for each composite function

(1.) $f(x) = 5x - 3$
$g(x) = 2x + 1$

$ (f \circ g)(x) \\[3ex] = f(g(x)) \\[3ex] = f(2x + 1) \\[3ex] But, f(x) = 5x - 3 \\[3ex] f(2x + 1) = 5(2x + 1) - 3 \\[3ex] = 10x + 5 - 3 \\[3ex] = 10x + 2 \\[3ex] (f \circ g)(x) = 10x + 2 \\[3ex] $ The domain of $(f \circ g)(x)$ is all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$

Let us look at $g(x)$
$ g(x) = 2x + 1 \\[3ex] D = (-\infty, \infty) \\[3ex] $ Next, we look at $(f \circ g)(x)$
$ (f \circ g)(x) = 10x + 2 \\[3ex] D = (-\infty, \infty) \\[3ex] $ Let us look at the intersection of both domains.
$\therefore D = (-\infty, \infty)$
$D$ = {$x | x \in \mathbb{R}$}



$ (g \circ f)(x) \\[3ex] = g(f(x)) \\[3ex] = g(5x - 3) \\[3ex] But, g(x) = 2x + 1 \\[3ex] g(5x - 3) = 2(5x - 3) + 1 \\[3ex] = 10x - 6 + 1 \\[3ex] = 10x - 5 \\[3ex] (g \circ f)(x) = 10x - 5 \\[3ex] $ The domain of $(g \circ f)(x)$ is all $x$ in the domain of $f$ such that $f(x)$ is in the domain of $g$

Let us look at $f(x)$
$ f(x) = 5x - 3 \\[3ex] D = (-\infty, \infty) \\[3ex] $ Next, we look at $(g \circ f)(x)$
$ (g \circ f)(x) = 10x - 5 \\[3ex] D = (-\infty, \infty) \\[3ex] $ Let us look at the intersection of both domains.
$\therefore D = (-\infty, \infty)$
$D$ = {$x | x \in \mathbb{R}$}

Example 2
Calculate:
$(f \circ g)(x)$
$(g \circ f)(x)$
the domain for each composite function

(1.) $f(x) = 4x + 1$
$g(x) = \sqrt{x}$

$ (f \circ g)(x) \\[3ex] = f(g(x)) \\[3ex] = f(\sqrt{x}) \\[3ex] But, f(x) = 4x + 1 \\[3ex] f(\sqrt{x}) = 4 * \sqrt{x} + 1 \\[3ex] = 4\sqrt{x} + 1 \\[3ex] (f \circ g)(x) = 4\sqrt{x} + 1 \\[3ex] $ The domain of $(f \circ g)(x)$ is all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$

Let us look at $g(x)$
$ g(x) = \sqrt{x} \\[3ex] D = [0, \infty) \\[3ex] $ Next, we look at $(f \circ g)(x)$
$ (f \circ g)(x) = 4\sqrt{x} + 1 \\[3ex] D = [0, \infty) \\[3ex] $ Let us look at the intersection of both domains.
$\therefore D = [0, \infty)$
$D$ = {$x | x \ge 0$}



$ (g \circ f)(x) \\[3ex] = g(f(x)) \\[3ex] = g(4x + 1) \\[3ex] But, g(x) = \sqrt{x} \\[3ex] g(4x + 1) = \sqrt{4x + 1} \\[3ex] (g \circ f)(x) = \sqrt{4x + 1} \\[3ex] $ The domain of $(g \circ f)(x)$ is all $x$ in the domain of $f$ such that $f(x)$ is in the domain of $g$

Let us look at $g(x)$
$ f(x) = 4x + 1 \\[3ex] D = (-\infty, \infty) \\[3ex] $ Next, we look at $(g \circ f)(x)$
$ (g \circ f)(x) = \sqrt{4x + 1} \\[3ex] $ To find the domain of \sqrt{4x + 1}
$ 4x + 1 \ge 0 \\[3ex] 4x \ge 0 - 1 \\[3ex] 4x \ge -1 \\[3ex] x \ge -\dfrac{1}{4} \\[5ex] D = \left[-\dfrac{1}{4}, \infty\right) \\[5ex] $ Let us look at the intersection of both domains.
$\therefore D = \left[-\dfrac{1}{4}, \infty\right)$

$D$ = {$x | x \ge -\dfrac{1}{4}$}

Inverse of Functions

Test for the Inverses of Functions
Two functions say $f(x)$ and $g(x)$ are inverses of each other if:
$(f \circ g)(x) = (g \circ f)(x) = x$

Or you may simply write it this way:
Given: a function of $x$, say $f(x)$ and it's inverse, $f^{-1}(x)$
$(f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x$

References

Chukwuemeka, S.D (2016, April 30). Samuel Chukwuemeka Tutorials - Math, Science, and Technology. Retrieved from https://www.samuelchukwuemeka.com
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Coburn, J., & Coffelt, J. (2014). College Algebra Essentials ($3^{rd}$ ed.). New York: McGraw-Hill
Kaufmann, J., & Schwitters, K. (2011). Algebra for College Students (Revised/Expanded ed.). Belmont, CA: Brooks/Cole, Cengage Learning.
Lial, M., & Hornsby, J. (2012). Beginning and Intermediate Algebra (Revised/Expanded ed.). Boston: Pearson Addison-Wesley.
Sullivan, M., & Sullivan, M. (2017). Algebra & Trigonometry ($7^{th}$ ed.). Boston: Pearson.
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Desmos Graphing Calculator (n.d.). Retrieved from https://www.desmos.com/calculator
RentCalculators Graphing Calculator (n.d.). Retrieved from https://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html