Relations and Functions | Algebra

Summary

This video script explores the concepts of relations and functions in mathematics. It explains that a relation is a set of pairs, with 'x' values representing the domain and 'y' values the range. The script teaches how to identify the domain and range of a relation and how to determine if a relation is a function, emphasizing that each input must have a unique output. It also introduces the vertical line test for graph-based relations, demonstrating how to use it to ascertain if a graph represents a function. Additionally, the script promotes the creator's website, video-dash-tutor.net, for specialized educational content.

Takeaways

• 📚 A relation is a set of ordered pairs of input and output values, with the x-values representing the domain and the y-values representing the range.
• 📝 To determine the domain and range of a relation, list all unique x-values in ascending order for the domain and all unique y-values in ascending order for the range.
• 🔍 A relation is a function if every input value has exactly one output value; if an input value corresponds to more than one output value, the relation is not a function.
• 🔑 To identify if a relation is a function, look for repeating x-values with different y-values, which indicates the relation is not a function.
• 🌐 The speaker promotes their website, video-dash-tutor.net, for specialized content and encourages joining an email list for updates on new materials.
• 📈 A mapping diagram can be used to visualize the relation between domain and range values, helping to determine if the relation is a function by checking for one-to-one correspondence.
• 📊 A function table lists input values (domain) next to output values (range), and if there are identical x-values with different y-values, the relation is not a function.
• 📐 The vertical line test is a method to determine if a graph represents a function; if any vertical line intersects the graph at more than one point, the graph does not represent a function.
• ⭕ For a circle, the vertical line test will fail because a vertical line will intersect a circle at two points, indicating it does not represent a function.
• 📉 The vertical line test confirms a relation as a function only if every vertical line touches the graph at exactly one point, ensuring a one-to-one correspondence between domain and range.
• 📚 The video script provides a comprehensive guide on understanding relations and functions, including how to list domain and range, identify functions, and apply the vertical line test.

Q & A

What is a relation in the context of the video?

A relation is a set of pairs of input and output values, where each input value (x) is associated with a domain and each output value (y) is associated with a range of the relation.

How do you determine the domain of a relation?

The domain of a relation is determined by making a list of all the x values, which are the input values, and writing them in ascending order.

What is the range of a relation and how is it found?

The range of a relation is the set of all possible output values (y values). It is found by listing the y values in ascending order.

What is the difference between a relation and a function?

A relation becomes a function only if every input value has exactly one output value. If an input value corresponds to two or more output values, the relation is not a function.

How can you quickly determine if a relation is not a function?

You can quickly determine if a relation is not a function by looking for repeating x values that correspond to different y values.

What is a mapping diagram and how is it used in the context of the video?

A mapping diagram is a visual representation of a relation where the domain (x values) is arranged on one side and the range (y values) on the other, showing the correspondence between each input and output value. It helps to determine if a relation is a function by checking for one-to-one correspondence.

What is a function table and how does it help in determining if a relation is a function?

A function table is a tabular representation of a relation that lists input values (x values) alongside corresponding output values (y values). It helps in determining if a relation is a function by showing if there are any identical x values with different y values, which would indicate it is not a function.

What is the vertical line test and how is it used to determine if a graph represents a function?

The vertical line test is a method used to determine if a graph represents a function by drawing vertical lines across the graph. If the line touches the graph at more than one point, the graph does not represent a function. If it only touches at one point, it does represent a function.

Why does the circle in the video's example not represent a function according to the vertical line test?

The circle does not represent a function because, when applying the vertical line test, a vertical line touches the circle at more than one point, indicating that there are multiple output values for at least one input value.

What is the significance of the website video-tutor.net mentioned in the video?

The website video-tutor.net is mentioned as a resource for those who want to be notified about specialized content such as videos, ebooks, articles, digital courses, or podcasts released by the video creator. It also provides access to a page with all of the creator's playlists, including final exam and test prep videos.

How can one join the email list on video-tutor.net to get access to additional content?

To join the email list on video-tutor.net, one needs to sign up on the website. After confirming their email, they will gain access to a page listing all of the creator's playlists, including specialized content for exams and test preparation.

Outlines

00:00

📚 Understanding Relations and Functions

This paragraph introduces the concept of relations and functions in mathematics. A relation is defined as a set of pairs consisting of input (x values) and output (y values). The domain of the relation is the set of all unique x values, while the range is the set of all unique y values. The paragraph explains how to list the domain and range for two given relations, emphasizing the importance of listing them in ascending order. It also discusses the criteria for a relation to be considered a function: each input value must correspond to exactly one output value. The paragraph provides examples to illustrate this, noting that if an input value corresponds to multiple output values, the relation is not a function. The speaker also promotes their website, video-tutor.net, for specialized content and resources.

05:01

📈 Mapping Diagrams and Function Tables

The second paragraph delves into visual representations of relations through mapping diagrams and function tables. It explains how to create a mapping diagram by arranging the domain (x values) and range (y values) and connecting corresponding pairs. The paragraph clarifies that if each input value has a unique output value, the relation is a function. Conversely, if there are repeating x values with different y values, the relation is not a function. The speaker also demonstrates how to construct a function table, listing input and output values side by side, and points out the importance of matching values in the table. The paragraph concludes with an example of a relation that is not a function due to the presence of identical x values with different y values.

10:02

📊 Vertical Line Test for Function Determination

The final paragraph discusses the use of the vertical line test to determine whether a graph represents a function. The vertical line test involves drawing a vertical line across the graph; if the line intersects the graph at more than one point, the relation is not a function. The paragraph provides examples of graphs that pass and fail the vertical line test, including a circle, which does not represent a function because it intersects a vertical line at multiple points. The speaker emphasizes that for a graph to represent a function, it must touch a vertical line at only one point, ensuring that each input value corresponds to a single output value.

Keywords

💡Relation

A relation is a mathematical concept that refers to a set of ordered pairs, where each pair consists of an input value (x) and an output value (y). In the context of the video, the domain of the relation consists of all possible x values, while the range consists of all possible y values. The script uses the term to explain the basic setup for understanding functions and their properties, with examples provided to illustrate the concept.

💡Domain

The domain in mathematics is the set of all possible input values for a function or relation. In the video, the domain is identified by listing all unique x values from a given set of ordered pairs. It is crucial for determining the set of values that a function can accept as inputs, as seen in the script when the domain for each relation is listed in ascending order.

💡Range

The range of a function or relation is the set of all possible output values, or y values, that result from the input values in the domain. The script explains how to determine the range by listing the y values associated with each relation, which are already in ascending order, providing a clear example of how the range is derived from a given set of data.

💡Function

A function is a special type of relation where each input value is associated with exactly one output value. The script emphasizes this one-to-one correspondence as a defining characteristic of functions. It also explains that if a relation does not have this property, meaning if an input value corresponds to more than one output value, then it is not a function.

💡Ordered Pairs

Ordered pairs in mathematics are pairs of numbers where the first number is the input (x) and the second is the output (y). The video script uses ordered pairs to illustrate the concept of relations and functions, showing how each pair represents a specific input-output relationship.

💡Vertical Line Test

The vertical line test is a graphical method used to determine whether a curve on a graph represents a function. If any vertical line drawn through the graph intersects it at more than one point, the curve does not represent a function. The script explains this concept and applies it to determine the nature of various graphs presented in the video.

💡Mapping Diagram

A mapping diagram is a visual representation used to depict the relationship between the domain and range of a function or relation. The video script describes how to create a mapping diagram by arranging the domain and range values and connecting each input value to its corresponding output value, which helps in understanding and determining if a relation is a function.

💡Function Table

A function table is a tabular representation of a relation or function, showing input values in one column and their corresponding output values in another. The script explains how to construct a function table and uses it to analyze whether a given relation is a function by checking for any repeating x values that correspond to different y values.

💡Input Value

In the context of functions and relations, an input value is the value that is put into a function to receive an output. The script discusses input values in relation to the domain and how each unique input value in a function should map to a single output value.

💡Output Value

An output value is the result produced by a function when an input value is applied. The script explains how output values are determined from the input values and are part of the range of a relation or function, emphasizing that in a function, each input value should result in a unique output value.

💡Video-dash-Tutor.net

Video-dash-Tutor.net is mentioned in the script as the website of the video's creator, where specialized content such as videos, ebooks, articles, digital courses, and podcasts are released. The script encourages viewers to join the email list to be notified of new content and to access playlists, including final exam and test prep videos.

Highlights

A relation is defined as a set of pairs of input and output values.

The domain of a relation consists of all unique x-values, while the range consists of all unique y-values.

To determine the domain and range, list all x-values in ascending order for the domain and y-values for the range.

A relation is a function if every input value has exactly one output value.

The first relation is a function because each input value corresponds to a unique output value.

The second relation is not a function due to the same input value (-2) corresponding to two different output values (4 and 7).

A quick way to identify a non-function is by looking for repeating x-values with different y-values.

The presenter introduces their website, video-dash-tutor.net, for specialized educational content.

Joining the email list on the website provides access to playlists including final exam and test prep videos.

A mapping diagram can be used to visualize the domain and range of a relation.

If a mapping diagram shows an x-value corresponding to one y-value, the relation is a function.

If an x-value corresponds to multiple y-values in a mapping diagram, the relation is not a function.

A function table lists input and output values, representing the domain and range of a relation.

Identical x-values with different y-values in a function table indicate the relation is not a function.

The vertical line test is a method to determine if a graph represents a function.

A graph passes the vertical line test if no vertical line intersects the graph at more than one point.

A circle does not represent a function as it fails the vertical line test by intersecting a vertical line at multiple points.

A graph that passes the vertical line test at every point represents a function.

Transcripts

in this video we're going to focus on

relations and functions

so what is a relation

a relation is a set of pairs of input

and output values

here we have three ordered pairs

in the first relation on the left

the x value

is the input value

the y value is the output value

the x values is associated with the

domain of the relation the y values is

associated with

the range of the relation

so now let's focus on part a list the

domain and range of each relation

so let's start with the domain

so what we're going to do is we're going

to make a list of all of the x values

and i'm going to write it in ascendant

order

so first we have negative three

and then zero

and two

now let's write the range

of that relation

so we're going to focus on

the y values and it's already listed in

ascendant order

so 1 4

and 5.

now let's do the same thing for the

other relation

so let's write out the domain

so the lowest

x value is negative two

next is one and then three

now let's write out the range

of that relation

the lowest y value

is negative two

and then it's

three four and seven

so that's how you can write out the

domain and range of each relation

now how can we determine if the relation

is a function

in order for the relation to be a

function

every input value

must have

only one output value

if an input value corresponds to two or

more output values

that relation is not a function

now

let's focus on the first relation

so we have the ordered pair 2 1 the

input value is 2 the output value is 1.

and then negative 3 4. so negative 3

corresponds to 4

and then 0 corresponds to 5.

so for the first relation we can see

that

for every input value there's only one

output value

now let's focus on the second relation

so we have the ordered pair one comma

three

next is negative two four

and then it's three negative two

and then finally negative two seven

so for the second relation notice that

negative two corresponds

to two different output values

now that's a problem

if you put in an input value of negative

two should the output be four or seven

so whenever you have that situation you

know that relation

is not

a function

the first one is a function

every input value corresponds to an

output value just one output value

so a quick way to

look at a relation to see if it's

if it's not going to be a function

look for repeating x values

if you see

two x values that are the same

but correspond to two different y values

then you know the relation is not a

function

i want to take a minute to talk about

my website

video dash tutor.net

it's a very simple website not too

complicated

but for those of you who want to be

notified

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an ebook

an article

it could be a digital course or podcast

if you want to be notified

feel free to join the email list

and once you confirm your email

you're going to get access to a page

that has all of my playlists

listed on it

and this includes my final exam videos

and also

my test prep videos so feel free to join

the email list when you get a chance

and let's get back to the video

now let's move on to the next example

draw a mapping diagram of each relation

shown below

so let's start with the relation on the

left

we're going to map out the domain and

arrange

so for the domain we have the values

negative 2

1 and 3.

for the range we have the y values

negative 6

0

and 4.

now negative two

corresponds to zero

one corresponds to four

three

corresponds to negative six

so for every input value on the domain

side there's one corresponding

output value on a range side

so this

is a relation

i mean

this relation is a function so the

answer is yes for

the first relation

now let's move on to the second

relation

so let's create a mapping diagram as

well

so let's start with the domain

the lowest x value is negative two

next we have

zero

and then the last one is three

now looking at the y values

the lowest one is negative one

and then it's going to be one

two and five

so negative two corresponds

to positive one

zero corresponds to five

three corresponds to two

and zero

corresponds to negative one

so just by seeing

the repeat x values that we see here we

could tell that

this is not going to be

a function

the two x values have two different y

values

you can see zero points to negative one

and five

so the second relation

is not

a function

now for this one what we're going to do

is we're going to draw a function table

of the relation

and then we're going to determine if the

relation is a function

so in this table we're going to list

the input values

next to the output values

the input values represent the x values

the output values represent the y values

so the input values it corresponds to

the domain and the output values

corresponds to the range

so the lowest input value that we have

is negative three

next

is one

and then we have another one

and then after that is

it's three and five

now for this function table

i'm going to write the input value twice

because that's what we have here

when writing out

the domain and the range

for repeat values we would write repeat

values once

now negative three corresponds to two

for the table

these numbers need to match so i'm not

going to list the output values in

ascendant order

now for this one

we could use either one so i'm going to

use 1 2 for the next one and then 1 4.

now when x is 3 y is 7

and when x is 5 y is negative 4.

so that's the function table

and because we have

two identical x values that correspond

to two different y values

we know that this relation

is not

a function

so that's it for this problem

when you have a graph

the best way to determine if the graph

represents a function is to use the

vertical line test and that's what we're

going to do in this problem

so any way you draw a vertical line

for the first graph notice that

the line only touches the graph at one

point

therefore

this is the answer is yes it represents

a function

for the next one on the right if we draw

a vertical line

notice at

this point or place a line at that

location

we have

three points of intersection between a

graph and a vertical line

if we can get two or more points on a

vertical line then a relation is not a

function so we're going to say no

now if we put the vertical line here

notice that we have five points on that

line

so this relation is not a function

for the next relation

it doesn't matter where we put the graph

we will only get

i mean doesn't matter where we put the

vertical line we're only going to get

one point

if we put it here

it's only going to touch the line once

so we can't draw a vertical line where

it touches two points therefore

this relation represents a function

for the circle

if we put the line here

we can get two points of intersection

so we're going to say yes i mean no not

yes

this is no

the circle does not represent a function

it does not pass the vertical line test

it touches the line

at two points in order for it to pass

the vertical line test the graph must

touch the line only at one point

as we

saw in these two cases

so that's how you can use the vertical

line test to determine

if a relation represented by a graph

is a function