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COURSE101

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Vivamus suscipit tortor eget felis porttitor volutpat. Vivamus suscipit tortor eget felis porttitor volutpat. Vestibulum ante ipsum primis in faucibus orci luctus et ultrices posuere cubilia Curae; Donec velit neque, auctor sit amet aliquam vel, ullamcorper sit amet ligula. Donec sollicitudin molestie malesuada.

Vivamus suscipit tortor eget felis porttitor volutpat. Pellentesque in ipsum id orci porta dapibus. Donec sollicitudin molestie malesuada. Donec rutrum congue leo eget malesuada.

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Sed porttitor lectus nibh. Proin eget tortor risus. Nulla quis lorem ut libero malesuada feugiat. Proin eget tortor risus.

Vivamus suscipit tortor eget felis porttitor volutpat. Vivamus suscipit tortor eget felis porttitor volutpat. Vestibulum ante ipsum primis in faucibus orci luctus et ultrices posuere cubilia Curae; Donec velit neque, auctor sit amet aliquam vel, ullamcorper sit amet ligula. Donec sollicitudin molestie malesuada.

Vivamus suscipit tortor eget felis porttitor volutpat. Pellentesque in ipsum id orci porta dapibus. Donec sollicitudin molestie malesuada. Donec rutrum congue leo eget malesuada.

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MAT1L Lesson Template

Content Blocks

Learning Objectives

  1. Add learning objectives here
  2. Examples of learning objectives can include:
  3. Understand how to solve for 'x'
  4. Read and create functions
  5. And so on
Photo of a pen on top of paper with math expressions

Swap out this image with an image of your own.

Introduction to the Lesson should go here. You should introduce the topic in one or two paragraphs. 

Praesent nonummy mi in odio. Phasellus magna. In dui magna, posuere eget, vestibulum et, tempor auctor, justo. Nulla neque dolor, sagittis eget, iaculis quis, molestie non, velit.

You can also add mathematical expressions and functions using the LaTex notation. For example the following syntax:

$$\{(1,\text{ }2),\text{ }(2,\text{ }4),\text{ }(3,\text{ }6),\text{ }(4,\text{ }8),\text{ }(5,\text{ }10)\}$$

Produces the following expression:

$$\{(1,\text{ }2),\text{ }(2,\text{ }4),\text{ }(3,\text{ }6),\text{ }(4,\text{ }8),\text{ }(5,\text{ }10)\}$$

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This is a template for creating Lessons.

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Remix of Functions and Function Notation

Content Blocks

Learning Objectives

In this section, you will:

  • Determine whether a relation represents a function.
  • Find the value of a function.
  • Determine whether a function is one-to-one.
  • Use the vertical line test to identify functions.
  • Graph the functions listed in the library of functions.

A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

Determining Whether a Relation Represents a Function

A relation is a set of ordered pairs. The set consisting of the first components of each ordered pair is called the domain and the set consisting of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.

$$\{(1,\text{ }2),\text{ }(2,\text{ }4),\text{ }(3,\text{ }6),\text{ }(4,\text{ }8),\text{ }(5,\text{ }10)\}$$

The domain is \(\{1,\text{ }2,\text{ }3,\text{ }4,\text{ }5\}\). The range is \(\{2,\text{ }4,\text{ }6,\text{ }8,\text{ }10\}\).

Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter \(x\). Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter \(y\).

A function \(f\) is a relation that assigns a single value in the range to each value in the domain. In other words, no x-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, \(\{1, 2, 3, 4, 5\}\), is paired with exactly one element in the range, \(\{2, 4, 6, 8, 10\}\).

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

$$\{{(\text{odd},\text{ }1),\text{ }(\text{even},\text{ }2),\text{ }(\text{odd},\text{ }3),\text{ }(\text{even},\text{ }4),\text{ }(\text{odd},\text{ }5)}\}$$

Notice that each element in the domain, \(\{even, odd\}\) is not paired with exactly one element in the range, \(\{1, 2, 3, 4, 5\}\). For example, the term “odd” corresponds to three values from the range, \(\{1, 3, 5\}\) and the term “even” corresponds to two values from the range, \(\{2, 4\}\). This violates the definition of a function, so this relation is not a function.

Figure 1 compares relations that are functions and not functions.

Three relations that demonstrate what constitute a function.

Figure 1 (a) This relationship is a function because each input is associated with a single output. Note that input  q  and  r  both give output  n.  (b) This relationship is also a function. In this case, each input is associated with a single output.

Functions

function is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.”

The input values make up the domain, and the output values make up the range.

Examples

Given a relationship between two quantities, determine whether the relationship is a function.

Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.

To represent “height is a function of age,” we start by identifying the descriptive variables \(h\) for height and \(a\) for age. The letters \(f\), \(g\), and \(h\) are often used to represent functions just as we use \(x, y\), and \(z\) to represent numbers and \(A, B\), and \(C\) to represent sets.

$$ \begin{array}{lcccc}h\text{ is }f\text{ of }a&&&&\text{We name the function }f;\text{ height is a function of age}.\\h=f(a)&&&&\text{We use parentheses to indicate the function input}\text{. }\\f(a)&&&&\text{We name the function }f;\text{ the expression is read as “}f\text{ of }a\text{.”}\end{array} $$

Remember, we can use any letter to name the function; the notation \(h(a)\) shows us that \(h\) depends on \(a\). The value a  a must be put into the function \(h\) to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.

We can also give an algebraic expression as the input to a function. For example \(f(a+b)\) means "first add a and b, and the result is the input for the function f." The operations must be performed in this order to obtain the correct result.

Function Notation

The notation \(y=f(x)\) defines a function named \(f\). This is read as "\(y is a function of x\)". The letter \(x\) represents the input value, or independent variable. The letter \(y\), or \(f(x)\), represents the output value, or dependent variable.

Practice Quiz

Content for this page has been sourced from OpenStax - Access for free at https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction-to-prerequisites

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A demonstration of the MathJax authoring capabilities. Content for this page has been sourced from OpenStax - Access for free at https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction-to-prerequisites

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Daring Dragon - Remix of Indigenous Vocabulary: Algonquin Words

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Statue of Keewakwa Abenaki Keenahbeh in Opechee Park in Laconia, New Hampshire (standing at 36 ft.)

Statue of Keewakwa Abenaki Keenahbeh in Opechee Park in Laconia, New Hampshire (standing at 36 ft.). Sources: Sculptor: Peter Wolf Toth / Photo by: Niranjan Arminius - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=51375010

The Abenaki (Abnaki, Abinaki, Alnôbak) are a Native American tribe and First Nation. They are one of the Algonquian-speaking peoples of northeastern North America. The Abenaki originated in a region called Wabanahkik in the Eastern Algonquian languages (meaning "Dawn Land"), a territory now including parts of Quebec and the Maritimes of Canada and northern sections of the New England region of the United States. The Abenaki are one of the five members of the Wabanaki Confederacy.

The Abenaki language is closely related to the Panawahpskek (Penobscot) language. Other neighboring Wabanaki tribes, the Pestomuhkati (Passamaquoddy), Wolastoqiyik (Maliseet), and Miꞌkmaq, and other Eastern Algonquian languages share many linguistic similarities. It has come close to extinction as a spoken language. Tribal members are working to revive the Abenaki language at Odanak (means "in the village"), a First Nations Abenaki reserve near Pierreville, Quebec, and throughout New Hampshire, Vermont and New York state.

Twenty Basic Words in Algonquin

Abenaki is an Algonquian language, related to other languages like Lenape and Ojibwe. We have included twenty basic Algonquin words here.

Algonquin Word Set

English (Français) Algonquin Words
One (Un) Pejig
Two (Deux) Nìj
Three (Trois) Niswi
Four (Quatre) New
Five (Cinq) Nànan
Man (Homme) Ininì
Woman (Femme) Ikwe
Dog (Chien) Animosh
Sun (Soleil) Kìzis
Moon (Lune) Tibik-kìzis
Water (Eau) Nibì
White (Blanc) Wàbà
Yellow (Jaune) Ozàwà
Red (Rouge) Miskwà
Black (Noir) Makadewà
Eat (Manger) Mìdjin
See (Voir) Wàbi
Hear (Entendre) Nòndam
Sing (Chanter) Nigamo
Leave (Partir) Màdjà or Nagadàn

Test your memory

Please note: the content on this page, sourced from http://www.native-languages.org/algonquin_words.htm, is meant for demonstration purposes only.

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Abenaki is an Algonquian language, related to other languages like Lenape and Ojibwe. We have included twenty basic Algonquin words here.

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Remix (Test) of Indigenous Vocabulary: Algonquin Words

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Statue of Keewakwa Abenaki Keenahbeh in Opechee Park in Laconia, New Hampshire (standing at 36 ft.)

Statue of Keewakwa Abenaki Keenahbeh in Opechee Park in Laconia, New Hampshire (standing at 36 ft.). Sources: Sculptor: Peter Wolf Toth / Photo by: Niranjan Arminius - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=51375010

The Abenaki (Abnaki, Abinaki, Alnôbak) are a Native American tribe and First Nation. They are one of the Algonquian-speaking peoples of northeastern North America. The Abenaki originated in a region called Wabanahkik in the Eastern Algonquian languages (meaning "Dawn Land"), a territory now including parts of Quebec and the Maritimes of Canada and northern sections of the New England region of the United States. The Abenaki are one of the five members of the Wabanaki Confederacy.

The Abenaki language is closely related to the Panawahpskek (Penobscot) language. Other neighboring Wabanaki tribes, the Pestomuhkati (Passamaquoddy), Wolastoqiyik (Maliseet), and Miꞌkmaq, and other Eastern Algonquian languages share many linguistic similarities. It has come close to extinction as a spoken language. Tribal members are working to revive the Abenaki language at Odanak (means "in the village"), a First Nations Abenaki reserve near Pierreville, Quebec, and throughout New Hampshire, Vermont and New York state.

Twenty Basic Words in Algonquin

Abenaki is an Algonquian language, related to other languages like Lenape and Ojibwe. We have included twenty basic Algonquin words here.

Algonquin Word Set

English (Français) Algonquin Words
One (Un) Pejig
Two (Deux) Nìj
Three (Trois) Niswi
Four (Quatre) New
Five (Cinq) Nànan
Man (Homme) Ininì
Woman (Femme) Ikwe
Dog (Chien) Animosh
Sun (Soleil) Kìzis
Moon (Lune) Tibik-kìzis
Water (Eau) Nibì
White (Blanc) Wàbà
Yellow (Jaune) Ozàwà
Red (Rouge) Miskwà
Black (Noir) Makadewà
Eat (Manger) Mìdjin
See (Voir) Wàbi
Hear (Entendre) Nòndam
Sing (Chanter) Nigamo
Leave (Partir) Màdjà or Nagadàn

Please note: the content on this page, sourced from http://www.native-languages.org/algonquin_words.htm, is meant for demonstration purposes only.

Test your memory

Assessment

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Abenaki is an Algonquian language, related to other languages like Lenape and Ojibwe. We have included twenty basic Algonquin words here.

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My Remix of Functions and Function Notation

Content Blocks

Learning Objectives

In this section, you will:

  • Determine whether a relation represents a function.
  • Find the value of a function.
  • Determine whether a function is one-to-one.
  • Use the vertical line test to identify functions.
  • Graph the functions listed in the library of functions.

A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

Determining Whether a Relation Represents a Function

A relation is a set of ordered pairs. The set consisting of the first components of each ordered pair is called the domain and the set consisting of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.

$$\{(1,\text{ }2),\text{ }(2,\text{ }4),\text{ }(3,\text{ }6),\text{ }(4,\text{ }8),\text{ }(5,\text{ }10)\}$$

The domain is \(\{1,\text{ }2,\text{ }3,\text{ }4,\text{ }5\}\). The range is \(\{2,\text{ }4,\text{ }6,\text{ }8,\text{ }10\}\).

Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter \(x\). Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter \(y\).

A function \(f\) is a relation that assigns a single value in the range to each value in the domain. In other words, no x-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, \(\{1, 2, 3, 4, 5\}\), is paired with exactly one element in the range, \(\{2, 4, 6, 8, 10\}\).

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

$$\{{(\text{odd},\text{ }1),\text{ }(\text{even},\text{ }2),\text{ }(\text{odd},\text{ }3),\text{ }(\text{even},\text{ }4),\text{ }(\text{odd},\text{ }5)}\}$$

Notice that each element in the domain, \(\{even, odd\}\) is not paired with exactly one element in the range, \(\{1, 2, 3, 4, 5\}\). For example, the term “odd” corresponds to three values from the range, \(\{1, 3, 5\}\) and the term “even” corresponds to two values from the range, \(\{2, 4\}\). This violates the definition of a function, so this relation is not a function.

Figure 1 compares relations that are functions and not functions.

Three relations that demonstrate what constitute a function.

Figure 1 (a) This relationship is a function because each input is associated with a single output. Note that input  q  and  r  both give output  n.  (b) This relationship is also a function. In this case, each input is associated with a single output.

Functions

function is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.”

The input values make up the domain, and the output values make up the range.

Examples

Given a relationship between two quantities, determine whether the relationship is a function.

Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.

To represent “height is a function of age,” we start by identifying the descriptive variables \(h\) for height and \(a\) for age. The letters \(f\), \(g\), and \(h\) are often used to represent functions just as we use \(x, y\), and \(z\) to represent numbers and \(A, B\), and \(C\) to represent sets.

$$ \begin{array}{lcccc}h\text{ is }f\text{ of }a&&&&\text{We name the function }f;\text{ height is a function of age}.\\h=f(a)&&&&\text{We use parentheses to indicate the function input}\text{. }\\f(a)&&&&\text{We name the function }f;\text{ the expression is read as “}f\text{ of }a\text{.”}\end{array} $$

Remember, we can use any letter to name the function; the notation \(h(a)\) shows us that \(h\) depends on \(a\). The value a  a must be put into the function \(h\) to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.

We can also give an algebraic expression as the input to a function. For example \(f(a+b)\) means "first add a and b, and the result is the input for the function f." The operations must be performed in this order to obtain the correct result.

Function Notation

The notation \(y=f(x)\) defines a function named \(f\). This is read as "\(y is a function of x\)". The letter \(x\) represents the input value, or independent variable. The letter \(y\), or \(f(x)\), represents the output value, or dependent variable.

Practice Quiz

Content for this page has been sourced from OpenStax - Access for free at https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction-to-prerequisites

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A demonstration of the MathJax authoring capabilities. Content for this page has been sourced from OpenStax - Access for free at https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction-to-prerequisites

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Anatomy & Physiology

Instructional Summary
This is an example of a Module resource. Modules are comprised of lessons and h5p resources. Resources are added to the Module's Table of Contents to create a bundled resource.

The content on this page/module is for demonstration purposes only. You can view the original work here: https://training.seer.cancer.gov/anatomy/
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<span>Photo by <a href="https://unsplash.com/@pierreacobas?utm_source=unsplash&amp;utm_medium=referral&amp;utm_content=creditCopyText">Pierre Acobas</a> on <a href="https://unsplash.com/s/photos/anatomy?utm_source=unsplash&amp;utm_medium=referral&amp;utm_content=creditCopyText">Unsplash</a></span>
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Introduction

The Anatomy and Physiology module introduces the structure and function of the human body. You will read about the cells, tissues and membranes that make up our bodies and how our major systems function to help us develop and stay healthy.

In this module you will learn to:

  • Describe basic human body functions and life process.
  • Name the major human body systems and relate their functions.
  • Describe the anatomical locations, structures and physiological functions of the main components of each major system of the human body.
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Remix of Functions and Function Notation

Content Blocks

Learning Objectives

In this section, you will:

  • Determine whether a relation represents a function.
  • Find the value of a function.
  • Determine whether a function is one-to-one.
  • Use the vertical line test to identify functions.
  • Graph the functions listed in the library of functions.

A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

Determining Whether a Relation Represents a Function

A relation is a set of ordered pairs. The set consisting of the first components of each ordered pair is called the domain and the set consisting of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.

$$\{(1,\text{ }2),\text{ }(2,\text{ }4),\text{ }(3,\text{ }6),\text{ }(4,\text{ }8),\text{ }(5,\text{ }10)\}$$

The domain is \(\{1,\text{ }2,\text{ }3,\text{ }4,\text{ }5\}\). The range is \(\{2,\text{ }4,\text{ }6,\text{ }8,\text{ }10\}\).

Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter \(x\). Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter \(y\).

A function \(f\) is a relation that assigns a single value in the range to each value in the domain. In other words, no x-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, \(\{1, 2, 3, 4, 5\}\), is paired with exactly one element in the range, \(\{2, 4, 6, 8, 10\}\).

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

$$\{{(\text{odd},\text{ }1),\text{ }(\text{even},\text{ }2),\text{ }(\text{odd},\text{ }3),\text{ }(\text{even},\text{ }4),\text{ }(\text{odd},\text{ }5)}\}$$

Notice that each element in the domain, \(\{even, odd\}\) is not paired with exactly one element in the range, \(\{1, 2, 3, 4, 5\}\). For example, the term “odd” corresponds to three values from the range, \(\{1, 3, 5\}\) and the term “even” corresponds to two values from the range, \(\{2, 4\}\). This violates the definition of a function, so this relation is not a function.

Figure 1 compares relations that are functions and not functions.

Three relations that demonstrate what constitute a function.

Figure 1 (a) This relationship is a function because each input is associated with a single output. Note that input  q  and  r  both give output  n.  (b) This relationship is also a function. In this case, each input is associated with a single output.

Functions

function is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.”

The input values make up the domain, and the output values make up the range.

Examples

Given a relationship between two quantities, determine whether the relationship is a function.

Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.

To represent “height is a function of age,” we start by identifying the descriptive variables \(h\) for height and \(a\) for age. The letters \(f\), \(g\), and \(h\) are often used to represent functions just as we use \(x, y\), and \(z\) to represent numbers and \(A, B\), and \(C\) to represent sets.

$$ \begin{array}{lcccc}h\text{ is }f\text{ of }a&&&&\text{We name the function }f;\text{ height is a function of age}.\\h=f(a)&&&&\text{We use parentheses to indicate the function input}\text{. }\\f(a)&&&&\text{We name the function }f;\text{ the expression is read as “}f\text{ of }a\text{.”}\end{array} $$

Remember, we can use any letter to name the function; the notation \(h(a)\) shows us that \(h\) depends on \(a\). The value a  a must be put into the function \(h\) to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.

We can also give an algebraic expression as the input to a function. For example \(f(a+b)\) means "first add a and b, and the result is the input for the function f." The operations must be performed in this order to obtain the correct result.

Function Notation

The notation \(y=f(x)\) defines a function named \(f\). This is read as "\(y is a function of x\)". The letter \(x\) represents the input value, or independent variable. The letter \(y\), or \(f(x)\), represents the output value, or dependent variable.

Practice Quiz

Content for this page has been sourced from OpenStax - Access for free at https://openstax.org/books/algebra-and-trigonometry/pages/1-introductio…

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Indigenous Vocabulary: Algonquin Words

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Statue of Keewakwa Abenaki Keenahbeh in Opechee Park in Laconia, New Hampshire (standing at 36 ft.)

Statue of Keewakwa Abenaki Keenahbeh in Opechee Park in Laconia, New Hampshire (standing at 36 ft.). Sources: Sculptor: Peter Wolf Toth / Photo by: Niranjan Arminius - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=51375010

The Abenaki (Abnaki, Abinaki, Alnôbak) are a Native American tribe and First Nation. They are one of the Algonquian-speaking peoples of northeastern North America. The Abenaki originated in a region called Wabanahkik in the Eastern Algonquian languages (meaning "Dawn Land"), a territory now including parts of Quebec and the Maritimes of Canada and northern sections of the New England region of the United States. The Abenaki are one of the five members of the Wabanaki Confederacy.

The Abenaki language is closely related to the Panawahpskek (Penobscot) language. Other neighboring Wabanaki tribes, the Pestomuhkati (Passamaquoddy), Wolastoqiyik (Maliseet), and Miꞌkmaq, and other Eastern Algonquian languages share many linguistic similarities. It has come close to extinction as a spoken language. Tribal members are working to revive the Abenaki language at Odanak (means "in the village"), a First Nations Abenaki reserve near Pierreville, Quebec, and throughout New Hampshire, Vermont and New York state.

Twenty Basic Words in Algonquin

Abenaki is an Algonquian language, related to other languages like Lenape and Ojibwe. We have included twenty basic Algonquin words here.

Algonquin Word Set

English (Français) Algonquin Words
One (Un) Pejig
Two (Deux) Nìj
Three (Trois) Niswi
Four (Quatre) New
Five (Cinq) Nànan
Man (Homme) Ininì
Woman (Femme) Ikwe
Dog (Chien) Animosh
Sun (Soleil) Kìzis
Moon (Lune) Tibik-kìzis
Water (Eau) Nibì
White (Blanc) Wàbà
Yellow (Jaune) Ozàwà
Red (Rouge) Miskwà
Black (Noir) Makadewà
Eat (Manger) Mìdjin
See (Voir) Wàbi
Hear (Entendre) Nòndam
Sing (Chanter) Nigamo
Leave (Partir) Màdjà or Nagadàn

Test your memory

Please note: the content on this page, sourced from http://www.native-languages.org/algonquin_words.htm, is meant for demonstration purposes only.

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kamillacastro, 9 months 3 weeks ago

Thank you, this was really helpful.

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Functions and Function Notation

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Learning Objectives

In this section, you will:

  • Determine whether a relation represents a function.
  • Find the value of a function.
  • Determine whether a function is one-to-one.
  • Use the vertical line test to identify functions.
  • Graph the functions listed in the library of functions.

A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

Determining Whether a Relation Represents a Function

A relation is a set of ordered pairs. The set consisting of the first components of each ordered pair is called the domain and the set consisting of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.

$$\{(1,\text{ }2),\text{ }(2,\text{ }4),\text{ }(3,\text{ }6),\text{ }(4,\text{ }8),\text{ }(5,\text{ }10)\}$$

The domain is \(\{1,\text{ }2,\text{ }3,\text{ }4,\text{ }5\}\). The range is \(\{2,\text{ }4,\text{ }6,\text{ }8,\text{ }10\}\).

Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter \(x\). Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter \(y\).

A function \(f\) is a relation that assigns a single value in the range to each value in the domain. In other words, no x-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, \(\{1, 2, 3, 4, 5\}\), is paired with exactly one element in the range, \(\{2, 4, 6, 8, 10\}\).

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

$$\{{(\text{odd},\text{ }1),\text{ }(\text{even},\text{ }2),\text{ }(\text{odd},\text{ }3),\text{ }(\text{even},\text{ }4),\text{ }(\text{odd},\text{ }5)}\}$$

Notice that each element in the domain, \(\{even, odd\}\) is not paired with exactly one element in the range, \(\{1, 2, 3, 4, 5\}\). For example, the term “odd” corresponds to three values from the range, \(\{1, 3, 5\}\) and the term “even” corresponds to two values from the range, \(\{2, 4\}\). This violates the definition of a function, so this relation is not a function.

Figure 1 compares relations that are functions and not functions.

Three relations that demonstrate what constitute a function.

Figure 1 (a) This relationship is a function because each input is associated with a single output. Note that input  q  and  r  both give output  n.  (b) This relationship is also a function. In this case, each input is associated with a single output.

Functions

function is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.”

The input values make up the domain, and the output values make up the range.

Examples

Given a relationship between two quantities, determine whether the relationship is a function.

Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.

To represent “height is a function of age,” we start by identifying the descriptive variables \(h\) for height and \(a\) for age. The letters \(f\), \(g\), and \(h\) are often used to represent functions just as we use \(x, y\), and \(z\) to represent numbers and \(A, B\), and \(C\) to represent sets.

$$ \begin{array}{lcccc}h\text{ is }f\text{ of }a&&&&\text{We name the function }f;\text{ height is a function of age}.\\h=f(a)&&&&\text{We use parentheses to indicate the function input}\text{. }\\f(a)&&&&\text{We name the function }f;\text{ the expression is read as “}f\text{ of }a\text{.”}\end{array} $$

Remember, we can use any letter to name the function; the notation \(h(a)\) shows us that \(h\) depends on \(a\). The value a  a must be put into the function \(h\) to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.

We can also give an algebraic expression as the input to a function. For example \(f(a+b)\) means "first add a and b, and the result is the input for the function f." The operations must be performed in this order to obtain the correct result.

Function Notation

The notation \(y=f(x)\) defines a function named \(f\). This is read as "\(y is a function of x\)". The letter \(x\) represents the input value, or independent variable. The letter \(y\), or \(f(x)\), represents the output value, or dependent variable.

Practice Quiz

Content for this page has been sourced from OpenStax - Access for free at https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction-to-prerequisites

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