Functions - Vertical Line Test, Ordered Pairs, Tables, Domain and Range

Summary

This video script offers an insightful exploration of functions and the vertical line test. It clarifies that a function is a unique relationship where each x-value corresponds to a single y-value, using tables to illustrate. The script guides viewers in identifying functions from non-functions through examples, emphasizing the importance of no repeated x-values. It also explains how to determine the domain and range of a function from ordered pairs and introduces the vertical line test for graph analysis. The video concludes by promoting an algebra course on Udemy, covering a wide range of topics from basic arithmetic to complex numbers and sequences.

Takeaways

• 📌 A function is a special relationship where each x-value corresponds to exactly one y-value.
• 🔍 To determine if a relation is a function, use the vertical line test: a vertical line should intersect the graph at most once.
• 📊 The domain of a function includes all the x-values of a set of ordered pairs.
• 📈 The range of a function includes all the y-values of a set of ordered pairs, which should be listed in ascending order.
• 🚫 A relation is not a function if there are repeated x-values with different y-values.
• 👍 A relation can be a function even if there are repeated y-values corresponding to different x-values.
• 📐 In a table, a relation is a function if no x-values are repeated, ensuring one-to-one correspondence with y-values.
• ✏️ When given a set of ordered pairs, check for repeated x-values to determine if the relation is a function.
• 📉 For graphs, if a vertical line touches the curve at only one point, the graph represents a function.
• 🔄 A complete circle does not represent a function because it fails the vertical line test by touching a vertical line at multiple points.
• 📚 The script also promotes an algebra course on Udemy, covering various topics from basic arithmetic to complex numbers and sequences.

Q & A

What is the definition of a function in the context of mathematics?

A function is a special relationship between X and Y values where each X value corresponds to exactly one Y value. In other words, a function does not allow an X value to have two different Y values.

How can you determine if a relation is a function by looking at a table?

You can determine if a relation is a function by checking if there are any duplicate X values with different corresponding Y values. If there are, it's not a function; if not, it is a function.

What is the vertical line test, and how does it help in determining if a graph represents a function?

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

How do you identify the domain of a function given a set of ordered pairs?

The domain of a function is the set of all possible X values. Given a set of ordered pairs, the domain is simply the list of unique X values from those pairs.

What is the range of a function, and how can you determine it from a set of ordered pairs?

The range of a function is the set of all possible Y values. To determine the range from a set of ordered pairs, list all the unique Y values and arrange them in ascending order.

What is the difference between a linear graph and a circle in terms of being a function?

A linear graph represents a function because it passes the vertical line test, touching any vertical line at only one point. A complete circle, however, does not represent a function because it touches a vertical line at two points.

Can a function have multiple Y values corresponding to different X values?

Yes, a function can have multiple Y values corresponding to different X values. The key is that the same X value cannot correspond to two different Y values.

What is the purpose of the algebra course mentioned in the script, and where can it be found?

The algebra course mentioned in the script is designed to help students improve their understanding of various algebraic concepts, including functions, vertical line tests, and more. It can be found on Udemy by searching for 'algebra' and selecting the course with a black background image.

How are video quizzes used in the algebra course to enhance learning?

Video quizzes in the algebra course are used to test and reinforce learning. They are multiple-choice and allow students to pause, work on the problems, and see the solutions, providing an interactive way to review and understand the material.

What are some of the topics covered in the algebra course mentioned in the script?

The algebra course covers a wide range of topics including basic arithmetic, fractions, linear equations, order of operations, graphing linear equations, inequalities, polynomials, factoring, systems of equations, quadratic equations, rational and radical expressions, complex and imaginary numbers, exponential functions, logarithms, functions, conic sections, and sequences and series.

Outlines

00:00

📚 Understanding Functions and Vertical Line Tests

This paragraph introduces the concept of a function as a unique relationship between X and Y values, where each X value corresponds to exactly one Y value. It differentiates between functional and non-functional relationships using table examples, emphasizing the importance of no repeated X values for the same Y value. The paragraph also guides viewers on how to identify functions from sets of ordered pairs and discusses determining the domain and range of a function, providing examples to illustrate these concepts.

05:03

📈 Applying the Vertical Line Test to Graphs

The second paragraph delves into the application of the vertical line test to determine if a graph represents a function. It explains that a graph passes the test and is thus a function if a vertical line intersects it at only one point. Multiple examples of graphs, including linear, circular, and others, are used to demonstrate the test, highlighting the difference between functional and non-functional representations. The paragraph reinforces the concept that a function's graph should not allow a vertical line to touch it at more than one point.

10:04

📘 Comprehensive Algebra Course Overview

The final paragraph provides an overview of an algebra course available on Udemy, covering a wide range of topics from basic arithmetic to advanced algebraic concepts. It outlines the course structure, starting with fundamental operations and moving through fractions, linear equations, inequalities, polynomials, and systems of equations. The course also includes sections on quadratic equations, rational and radical expressions, complex numbers, exponential and logarithmic functions, as well as sequences and series. Each section is equipped with quizzes for review, and the paragraph invites viewers to explore the course for further learning.

Keywords

💡Function

A function in mathematics is a special kind of relationship between two sets of numbers, where each input (x-value) corresponds to exactly one output (y-value). This concept is central to the video's theme, as it explains the criteria for determining whether a relationship is a function. For example, the script uses tables to illustrate that a function cannot have two different y-values for the same x-value.

💡Vertical Line Test

The vertical line test is a graphical method to determine if a curve represents a function. If a vertical line intersects the curve at more than one point, the curve does not represent a function. This concept is used in the script to visually demonstrate which graphs are functions and which are not, by showing that only functions pass the vertical line test.

💡Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the video, the domain is identified by listing all unique x-values from a set of ordered pairs, emphasizing that the domain represents all the x-values that can be used with the function.

💡Range

The range of a function is the set of all possible output values (y-values) that result from the domain. The script explains that the range is determined by listing all the y-values from a set of ordered pairs and then arranging them in ascending order, showing how to identify the range of a function.

💡Ordered Pairs

Ordered pairs in the context of the video are pairs of numbers that represent the x and y values of points on a graph. The script uses ordered pairs to illustrate how to determine if a set of points represents a function, by checking if any x-value is repeated with a different y-value.

💡Graph

A graph in the video refers to a visual representation of a mathematical relationship, typically on a coordinate plane. The script discusses how to determine if a graph represents a function by using the vertical line test, where only graphs that pass the test (intersecting the vertical line at a single point) are considered functions.

💡Linear Graph

A linear graph is a type of graph that represents a linear function, where the relationship between x and y can be described by a straight line. The script mentions linear graphs as examples of functions that pass the vertical line test, indicating a one-to-one correspondence between x and y values.

💡Circle

In the context of the video, a circle is a geometric shape that does not represent a function when graphed because it fails the vertical line test. The script uses a circle as an example of a shape that intersects a vertical line at multiple points, indicating that it does not have a one-to-one correspondence between x and y values.

💡Algebra Course

The script mentions an algebra course offered on Udemy, which covers various topics relevant to the understanding of functions, such as basic arithmetic, fractions, linear equations, and graphing. This serves as a resource for viewers who may want to delve deeper into the subject matter presented in the video.

💡Multiple Choice Video Quiz

A multiple choice video quiz is a type of interactive learning tool mentioned in the script, which allows students to pause and work on problems with immediate feedback on their solutions. This concept is used to describe a feature of the algebra course, providing a practical way for students to test their understanding of the material.

Highlights

A function is a special relationship between X and Y values where each X value corresponds to exactly one Y value.

A table on the left with no repeated X values is identified as a function, while the table on the right with repeated X values is not.

Multiple tables are presented to determine which relations are functions based on the absence of repeated X values.

A quick way to identify a function is to ensure no X values repeat in a table.

Having the same X value with different Y values disqualifies a relation from being a function.

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

A graph is a function if a vertical line touches it at only one point.

Examples of graphs are given to apply the vertical line test, including linear and circular graphs.

A complete circle is identified as not being a function due to the vertical line touching at multiple points.

The domain of a function is defined as the set of all X values from a set of ordered pairs.

The range of a function is the set of all Y values, which should be listed in ascending order.

Ordered pairs are used to determine if a relation is a function, ensuring no X values are repeated.

Algebra courses on Udemy are mentioned, covering a wide range of topics from basic arithmetic to complex numbers.

Each section of the algebra course includes a quiz for review and practice.

The course includes topics on inequalities, absolute value expressions, polynomials, factoring, and systems of equations.

Quadratic equations, rational and radical expressions, as well as conic sections are part of the course content.

Arithmetic and geometric sequences and series are also covered in the comprehensive algebra course.

Transcripts

in this video we're going to talk about

functions and vertical line tests so

what exactly is a function a function is

a special relationship between the X and

Y values for

example in a function every x value has

to correspond to exactly one

yvalue you can't have an x value that

has two different y values so let me

give you two tables that will illustrate

that relationship

so determine which of these two

relations is a

function so is it the table on the left

or the table on the

right the table on the left notice that

there are no two x values on the right

look at that these look at those two x

values in one case when X is 2 y is4 in

the other case when X is 2 Y is

7 this is not a

function a single x value has to

correspond to one y value so that's why

it's not a function this one is a

function because you don't have two

similar X values that correspond to two

different y values

now let me give you another example so

I'm going to give you three tables and I

want you to identify which relation is a

function and which ones are not

okay so take a minute and identify which

relation is a function so the first one

on the left is it a function a quick way

to tell is to make sure that you have no

X values that repeat if you don't see

any X values that repeat then this is

going to be a function in the second

example notice that we have two x values

that repeat when X is one y can't be 3

and4 at the same time so therefore this

is not a function now what about the

last example we have two y values that

repeat you can have multiple y values

that correspond to different X values

but you can't have the same X values

that correspond to different y

variables so the third one is is a

function when X is one y could be seven

when X is 2 y still can be

seven so just keep that in mind if you

have two x values that corresponds to

different yv values it's not a function

but if you have two different or two of

the same yv values that correspond to

different X values then it still will be

a

function now sometimes you might be

asked to determine if a function or if a

relation is a function given a set of

ordered

pairs so let's say if you have the

points 1 three

25

-36 and 4 comma 2 so that's the first

set go ahead and determine if it's a

function and I'm going to write two more

sets

so is the first one a

function what would you

say look at the X

values a quick way to tell is to check

if any X values repeat if they don't

then this is going to be a function

looking at the second example there are

no X values that repeat so that's a

function in the third example here we

have two of the same X

values with two different y values so

therefore the third set is not a

function now sometimes you may need to

be able to determine the domain and

range of a

function so let's say if you're

given this set of ordered pairs

what is the domain the domain is

basically all of the X values 1 3 5 and

8 so that's going to be the domain of

that

function now what is the range the range

is a set of all y values which is -4 2 7

and 1 so I'm going to write it in

ascendant order so4 comes first then

it's 1 2 and then seven so that's the

range

so just keep that in mind the domain

represents all of the X values and the

range represents all of the Y

values now the next thing that you need

to be able to do is you need to

determine if a function I mean if a

graph represents a

function so looking at these two graphs

which one is a function and which one is

not a

function there's something called the

vertical line test if you draw a

vertical line and if the curve touches

it at one point it's a function if it

touches it at two or more points it's

not a function so looking at the first

example draw a vertical line the curve

touches it at one point so this is a

function if we draw a vertical line for

the second example notice that the curve

touches the red line at two points so

the second graph is not a function so

let's try some more

examples I'm going to draw a few graphs

and I want you to determine if it's a

function or

not so feel free to pause the

video as you uh work on these

examples so let's draw a vertical line

for the first example it touches it at

one point so it is a function here we

have a linear graph which touches the

red line at one point so that's a

function as well the circle touches it

at two points so a complete

circle is not a function and for the

last example if we draw the vertical

line it touches it at one point so it is

a function so that's a quick way to tell

if something is a function or

not now I want to show you one of my

algebra courses that might be useful to

you if you ever need it so go to

udemy.com

now in the search box just type in

algebra and it should come

up so it's the one with the image with

the black

background so if you select that

option and if you decide to go

to course content you can see what's in

this particular

course so the first section basic

arithmetic for those of you who want to

focus on addition subtraction

multiplication and division and it has a

video quiz at the end it's a multiple

choice video quiz you can pause it work

on the problems and see the

solutions it covers long division

multiplying two large numbers and things

like that the next tutorial is on

fractions adding subtracting fractions

multiplying dividing fractions

converting fractions into decimals and

so forth so you can also take a look at

that next solve the linear equations

which we

covered and just more examples if you

need more help with that the next topic

order of operations which is also useful

uh graph in linear equations you need to

know how to calculate the slope needs to

be familiar with the slope intercept

form standard form and just how to tell

if lines are parallel perpendicular and

so forth and it's a quiz that uh goes

with that as well the next topic is on

inequalities and absolute value

Expressions which are also seen a

typical algebra course and then we have

polinomial and that's a a long section

and then

factoring you just that's another topic

you need to

master and then system of equations you

can solve it by elimination substitution

there's also word problems as well

sometimes you got to solve equations

with three variables X Y and Z so that

could be helpful next quadratic

equations how to use a quadratic formula

how to graph them how to convert between

standard and vertex form and then you

have rational

expressions and radical expressions

solving radical equations simplifying it

things like that and every section has a

quiz so you can always review what

you've learn if you have a test the next

day so here we have complex imagin

numbers you need to know how to simplify

those

exponential functions logs I have a lot

of videos on logs and then this is just

functions in general vertical line tests

horizontal line tests how to tell what

functions even or odd and then conic

sections graph in circles hyperbolas

ellipses parabas and things like that

there's two video quizzes because it's

actually a long section and finally

arithmetic and geometric sequences and

Series so that's my algebra course if

you want want to take a look at it and

uh let me know what you think