Functions - Vertical Line Test, Ordered Pairs, Tables, Domain and Range
Summary
TLDRThis 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
📚 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.
📈 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.
📘 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.
Mindmap
Keywords
💡Function
💡Vertical Line Test
💡Domain
💡Range
💡Ordered Pairs
💡Graph
💡Linear Graph
💡Circle
💡Algebra Course
💡Multiple Choice Video Quiz
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
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