FUNCTIONS | SHS GRADE 11 GENERAL MATHEMATICS QUARTER 1 MODULE 1 LESSON 1
Summary
TLDRThis video script introduces a high school general mathematics lesson for grade 11, focusing on real-life functions. It sets three objectives: understanding functions and relations, illustrating functions through mapping diagrams, sets, and graphs, and representing real-life situations with functions. The lesson uses a crossword puzzle to review key terms and emphasizes that a function is a special kind of relation where each input value is associated with exactly one output value. Examples of mapping, sets, and the vertical line test in graphing are provided to identify functions. The script concludes with real-life function examples and a 10-item assessment to reinforce learning.
Takeaways
- 📝 The lesson is designed for 11th-grade general mathematics, focusing on real-life functions.
- 📚 Students are encouraged to prepare a paper and pen to write down their answers and solutions to problems presented in the video.
- 🔍 The importance of pausing and revisiting the video for mastery is emphasized, allowing for a flexible learning pace.
- 🎯 The lesson has three main objectives: to determine functions and relations, illustrate functions through mapping diagrams, sets, and graphs, and represent real-life situations using functions.
- 🔑 A crossword puzzle is used to recall essential terms from junior high school mathematics, which are crucial for understanding the new material.
- 🔍 The definition of a function is highlighted as a special kind of relation where every element in the domain is paired with exactly one element in the range.
- 📈 Mapping diagrams are introduced as a tool to visually represent how elements of a function are paired, aiding in understanding the concept of functions.
- 📊 The concept of the domain and range in functions is explained, with examples provided to illustrate how to identify them within sets of ordered pairs.
- ✅ The vertical line test (VLT) is introduced as a method to determine if a graph represents a function, where only one point of intersection with any vertical line is allowed.
- 🌐 Real-life examples of functions are given, such as the relationship between the circumference of a circle and its diameter, the length of a shadow and a person's height, and the location of a car as a function of time.
- 📝 A 10-item assessment is provided at the end of the lesson to check students' understanding of the material covered.
Q & A
What are the three main objectives of the lesson on real life functions?
-The three main objectives are to determine functions and relations, illustrate functions through mapping diagrams, sets, and graphs, and represent real life situations using functions.
What is the definition of a 'relation' in the context of this lesson?
-A relation is a rule that relates values from a set of values, known as the domain, to a second set of values, known as the range.
What is an 'ordered pair' and how is it represented?
-An ordered pair is a sequence of two elements, typically enclosed in parentheses and separated by a comma, representing a specific order of elements.
What is the 'domain' of a function and how is it determined?
-The domain of a function is the set of all input values or the first elements of the ordered pairs in the set of ordered pairs representing the function.
What is the 'range' of a function and how is it determined?
-The range of a function is the set of all output values or the second elements of the ordered pairs in the set of ordered pairs representing the function.
What is a 'function' in mathematical terms and what is its defining rule?
-A function is a special kind of relation where every element in the domain is associated with exactly one element in the range, meaning each x value is paired with only one y value.
How can you determine if a given set of ordered pairs represents a function using the mapping diagram?
-In a mapping diagram, check if every element in the domain is associated with only one value in the range. If so, the set represents a function.
What is the 'Vertical Line Test' (VLT) and how is it used to determine if a graph represents a function?
-The Vertical Line Test (VLT) is a method used in graphing to determine if a graph represents a function by checking if any vertical line would touch the graph at more than one point. If it does, the graph does not represent a function.
What is the circumference of a circle as a function of its diameter and how is it represented mathematically?
-The circumference of a circle is a function of its diameter and can be represented as \( C = \pi D \) or as a function of its radius as \( C = 2\pi r \).
Can you provide an example of a real-life function from the script?
-One example from the script is the length of a person's shadow on the floor, which is a function of their height.
What is the significance of the Vertical Line Test in identifying functions from graphs?
-The significance of the Vertical Line Test is that it provides a visual and straightforward method to determine if a graph represents a function by ensuring that no vertical line intersects the graph more than once.
Outlines
📚 Introduction to General Mathematics for Grade 11
This paragraph introduces a video lesson aimed at Grade 11 students studying general mathematics. The focus is on real-life functions, and the students are encouraged to prepare a pen and paper for note-taking. The instructor suggests pausing the video for reflection and review. The lesson's objectives include understanding functions and relations, illustrating functions through various methods, and representing real-life situations with functions. A crossword puzzle is presented as a warm-up activity to recall key terms from junior high school mathematics.
🔍 Understanding Functions and Relations
The paragraph delves into the concept of functions as a special type of relation where each input (x-value) is linked to a single output (y-value). The importance of this one-to-one correspondence is emphasized as the defining characteristic of a function. The instructor uses mapping diagrams to visually demonstrate how elements of the domain are paired with elements in the range, and provides examples to illustrate whether a given set of ordered pairs represents a function based on this rule.
📐 Function Identification through Mapping and Sets
This section uses mapping diagrams and set notation to further explore the identification of functions. The use of rooster notation to list ordered pairs and the importance of distinct x-values in determining whether a set is a function are discussed. The paragraph provides examples that allow students to practice identifying functions by checking for repeated x-values, which would disqualify a set from being a function.
📈 Graphing Functions and the Vertical Line Test
The paragraph introduces the concept of graphing functions and the use of the vertical line test (VLT) to determine if a graph represents a function. The VLT involves drawing imaginary vertical lines across a graph to see if they intersect the graph more than once. If they do, the graph does not represent a function. Several examples of graphs are given for students to practice applying the VLT and identify which are functions and which are not.
🚗 Real-Life Applications and Assessment of Functions
The final paragraph connects the concept of functions to real-life scenarios, such as the relationship between the circumference of a circle and its diameter, the length of a shadow relative to a person's height, and the location of a car as a function of time. An assessment with 10 items is presented for students to test their understanding of the lesson. The instructor reminds students to remember the definition of a function, the methods of illustration, and the prevalence of functions in everyday life.
Mindmap
Keywords
💡Function
💡Domain
💡Range
💡Relation
💡Ordered Pair
💡Mapping Diagram
💡Set
💡Vertical Line Test (VLT)
💡Circumference
💡Shadow
💡Graph
Highlights
Introduction to the lesson on real life functions for grade 11 general mathematics.
Instructions to prepare a paper and pen for note-taking and problem-solving.
Advice on using video features like pause and replay for mastery of the topic.
Crossword puzzle activity to recall essential junior high school mathematics terms.
Definition of a 'relation' as a rule that relates values from a domain to a range.
Explanation of 'ordered pair' and its significance in mathematics.
Clarification of 'domain' as the set of all input values in a function.
Illustration of how to identify the domain from a set of ordered pairs.
Definition of a 'set' as a collection of distinct objects sharing a common characteristic.
Identification of 'range' as the set of all output values in a function.
Explanation of what makes a relation a 'function' based on the one-to-one correspondence of x and y values.
Introduction to mapping diagrams as a method to illustrate functions.
Demonstration of how to determine if a mapping represents a function using the one-to-one rule.
Use of sets and roster notation to identify whether a set of ordered pairs is a function.
Application of the vertical line test (VLT) to determine if a graph represents a function.
Examples of real-life functions, such as the circumference of a circle being a function of its diameter.
Real-life application of functions in determining the length of a shadow based on a person's height.
Practical example of driving a car where location is a function of time.
Introduction of a 10-item assessment to test understanding of the lesson on functions.
Summary of key points to remember about functions, including their definition, illustrations, and real-life applications.
Transcripts
hello there and welcome to
our new lesson this video is for senior
high school
general mathematics for grade 11.
prepare the following a paper and a pen
for you to write your answers or
solutions
for the problems later on and remember
you can always pause and play this video
whenever necessary you can even go back
or revisit the portion of this video to
clarify some things
for mastery purposes i hope that you are
all excited for this so let's hop in
this video presentation is for the first
quarter
module 1 of our subject general
mathematics
for grade 11. the topic is
real life functions to be specific
this is for the first lesson about real
life functions
what you need to know we have three main
objectives for this
session the first one is we are going to
determine functions and relations
second illustrate functions through
mapping diagrams
sets and graphs and finally you're going
to represent
real life situations using functions
what's in what you see on the screen
right now
is a crossword puzzle exactly
we have here five descriptions of
different terms
that is related to your junior high
school mathematics
these are necessary terms for us to
proceed with our new lesson
okay so you can pause this video
and try to recall those important terms
i'll give you time go ahead pause the
video
are you done that sounds great so let's
reveal the answers so for number one
let's have number one down
a rule that relates values from a set of
values which we call as domain
to a second set of values which we call
as range
what do you think the answer is
relation so let's put it in our
crossword puzzle
relation there
number three three down blank
pair pair of objects taken
in a specific order what do you call
this
blank pair the answer is it's an
ordered pair very good so let's put it
in our crossword puzzle now to clarify
about
ordered pairs we have here an example
remember that ordered pairs are a
sequence of two elements
like for this example one and two they
are enclosed in a parenthesis and they
are separated by a comma
okay that's an ordered pair let's
proceed to the next one
how about across number two the set of
all x or input values can you recall
you have there your clues o and i for
the second and the second to the last
letter
and the answer is domain right
brilliant domain let's review about
domain
when we say domain look at the example
we have four sets
or we have four ordered pairs in this
set
one seven two six three five
and four four now what is our domain
here
our domain are the first elements inside
the parenthesis or first element in each
of the ordered pairs
so that means it's 1 2 3
and four which serves as our domain
how about for number four across
collection of well-defined
and distinct objects called elements
that share a common characteristic
you have this when you're still in grade
seven the answer is
well done that's set s84
set last one
across number five the set of all y
or output values what do you call that
your clue there is it ends with letter e
you already have the domain this is the
pair of domain
that means we are referring to the range
okay we have completed our crossword
puzzle but before that let's clarify on
range
now using the same example for the
domain
we have here this set of ordered pairs
we already have one two three four as
our domain earlier right
now this time the range is these values
the second element of each ordered pair
or the y values that will be 7
6 5 and 4 in this example
what's new what makes relation
a function
a function is a special kind of relation
because it follows an extra rule
just like a relation a function is also
a set of ordered pairs however
take note of this every x
value must be associated to
only one y value i repeat
every x value must be associated
to only one y value that's the most
important part of this lesson
remember that that's for the definition
of our function
illustrations will help us a lot to
learn
functions easily so we have here mapping
sets and graphing a function is a
special type of relation
always remember that in which each
element
of the domain is paired with exactly
one element in the range a mapping
shows how the elements are paired
it's like a flowchart for a function
showing the input and the output of
values
like this the domain for the first set
and the range for the second set now in
this
mapping let's identify if this is
a function or not a function how do we
do that
recall every x
value must be associated to
only one y value so basing on that
let's try to check if every element in
our domain
is associated to only one value in our
range
let's focus on this part our domain a
is associated to roman numeral one
so that's one is to one that's the
correspondence
second domain or second element b
that is associated to only one
y value that is roman numeral 2.
here c third element
of our domain is associated to
or is being paired to only one value of
y
that is 3 or roman numeral 3. and lastly
d in our domain is being paired
with roman numeral 4 in our range
so as you can see every element in our
domain
is being paired to only one value
of y in our range so that
means this example is
correct this is a function
let's look at example number two can you
identify
if the given is a function or not a
function
you may pause this video
okay all right so how about this example
this is still a function y
looking at all the elements of our
domain negative 3 is being paired to 0
negative 1 is being paired to 4 2
is being paired to 7 and 4 is being
paired to 4.
so this shows that every element in our
domain
is being mapped or is being paired to
only one
value in our range which means
that if we have an input of negative
three the output is only zero
if we have an input of negative one the
output is
only four we don't have any other y
values
if we have an input of two therefore our
output is seven
if our input is four our output is also
four this type of correspondence shows
many is the one for this part we have
two elements in our domain
here that's negative one and four we
have two elements in our domain
that has the same value in our range
take note
what we are referring to in a function
is we have
every element in our domain is paired
with
one element in our range which means
that for every input there's only one
output this type of correspondence is
considered
as a function i hope that's clear so
this is
a function third example how about this
is this a function or not a function you
may pause this video
and let's reveal this is not a function
why earlier we saw
many is the one correspondence right
this time
you call this type of correspondence
recall your grade seven
and grade eight mathematics in your
junior high school
this is for this element in our domain
which is letter a
it's being paired let's focus on that
here
that's our domain a it's being paired to
one roman numeral 1 in the range at the
same time
the same element in the domain is being
paired to
roman numeral 3. now that means
this element in the domain has two
outputs
one and three which is clearly a
violation of the definition of our
function right
therefore basing on that element
this example is
not a function i hope i made that clear
a is being paired to two values in our
range
we are done with the mapping again these
are illustrations to help us out
understand or identify if the given is a
function
so this time let's move on to sets sets
in this example we will have rooster
notation
so we have a set of four ordered pairs
beginning with two three
four five five six
and we have six seven now can you
identify if this given set
is a function or not a function
now how do we do that let's identify
first the x
and the y elements in each ordered pair
so for the first one here
two is our x sub one three
is our y sub one four is our second
value of x in the second ordered pair
five is our y sub two five
is our x sub three in the third ordered
pair
six is our y sub three here
in the fourth ordered pair this will
serve as our
x of four and this will be our y sub
four which is seven
now why is it important for us to
identify
our x and y's in each of the ordered
pairs
because these values in a domain
are the critical values so identify if
it's a function or not a function
why look at this we have 2 4 5
and 6 in the domain no x value
is repeated so 2 is distinct from the
rest of the domain that's
4 5 and 6. thus we consider
this as a function
this set is a function remember that
when there's no x value that has been
repeated in the given set
then that means it's a function second
example
this set we have three three four five
five five and five four so again
the first step is let's identify the x
and the y
elements like this followed by
yes we are going to identify the domain
so meaning all the x values in our
ordered pairs
we have 3 4 5 and 5.
now notice that here 5
is repeated that's the x value it's
repeated for that element in our x or in
our domain
we do have two different outputs which
is not
anymore the definition of a function so
this is
not a function well done
we are done with the second illustration
for sets
again we are done with mapping and we
are done with sets now this time let's
focus
into another way that's for graphing
how do we identify if the given graph is
a function or not
your clue there is being pointed it's
vlt
that would be our magic keyword to
identify if the graph
is a function or not how what do you
mean by
vlt vlt stands for
vertical line test
yes functions can also be determined in
graphing we can use this vertical line
test which is a special kind of test
using imaginary vertical lines
and to check if these vertical lines
would touch the graph only
once otherwise it is not a function
what do i mean by that if the vertical
line
hits two or more points on the graph
it is not considered a function
let's look at some examples look at this
graph
how would we know if this graph is a
function
or not again what's our magic keyword
we'll be using vlt that's the vertical
line test
right so that's it the blue line that
you see on the screen right now
that's an imaginary line yeah it's not
part of the graph
we just made that line to test if the
given graph
is a function or not i hope you're
following
so the point there is here which means
that the line the vertical line touches
the graph
at that point only once
now let's move the blue line let's move
the blue vertical line
because here you can check if it's a
function
if any point of the graph it would only
touch
the graph or the given graph once so
let's move the vertical line
how about there yes it only hits or it
only touches the graph once
how about there only once and finally
right here yes it only touches the graph
once hence we can say that the given
graph
is a function so that's an example of a
function
basing on the graph
let's look at another example identify
if this graph
is a function or not a function i'll
give you time
you may pause this video to give
yourself more time
all right are you done let's check let's
create
our imaginary vertical line again we
will be using
vertical line test right there
the black dot represents the point where
in the vertical line touches your given
wrath
it's only once right let's move it a
little bit to the right right there
it touches the graph how many times
still once
let's move it there still once
last one over there still it touches the
graph
once basing on that we can conclude that
the given graph
is indeed a function
so that's an example of a function now
let's practice more let's identify if
these given graphs
are functions or not a function again
let's identify
try these graphs you can pause this
video right now
and give yourself more time to
scrutinize each of the graph
and identify if it's a function or not
a function go ahead
[Music]
okay so let's reveal the answers
now based on the sixth graph we can
actually create
two groups and the first group consists
of
these two graphs
now let's focus on this point
right here for the first graph as you
can see
the vertical line touches the given
graph once
how about for the second graph there it
only touches
yes it touches the given graph same with
the first graph
only once let's try to move the vertical
line
to the right right there
it touches still the same once let's
move it more
right there still touch us once
and last one same result
once thus we can conclude that these two
graphs
are considered as
very good we consider this as functions
so the remaining four graphs looks like
this
observe for the first graph we have here
two points
which means that the vertical line
touches the graph or touches the given
graph
at two points however for the second
graph this would be the first one
and that would be the second one still
the same it touches the given graph
twice
the vertical line touches the graph here
at the same time here
so that means there would be two points
right and lastly we have here the last
graph it touches the graph
twice let's move the vertical line
like this well observe
that for the fourth graph you now have
three points
earlier it was only two this time as we
move the vertical line it touches the
graph at three points
now for the first three graphs it's the
same it touches the given graph twice
let's move it there observe
that in all these given graphs the
vertical line
touches the given graph more than once
again that's more than once because for
the first second and third graph
it touches the graph twice for the
fourth graph it touched us earlier the
graph twice
this time thrice it's more than once
yes which makes all of these graphs
not a function so these are examples of
not a function
so those are the illustrations again for
the mapping
sets and graph
how about functions in real life
this is a circle so an example of a
function in real life
is the circumference of a circle the
circumference of a circle
is a function of its diameter it can be
represented as
circumference or c of d is equal to d
pi alternatively we can also use it
as a function of radius which is c of
r is equal to two pi r
[Music]
another example is a shadow the length
of a person's shadow
along the floor is a function of their
height
and the third example is driving a car
when driving a car your location
is a function of time
what's more i prepared here a 10 item
assessment
first to check your understanding for
our lesson for today
let's try to have a closer look you can
pause this video or you can even take a
screenshot and answer it later during
your available time
so we have your items one two and three
again you may pause or take a screenshot
[Music]
okay let's move on the next set is for
items four to six
next we have seven to nine
again you may take a screenshot or pause
this video
and finally we have item number 10.
[Music]
if you're using the same mode you'll do
not forget to submit your answers to
your teacher on your agreed date and
time
[Music]
what you need to remember a relation
is a function when every x value is
associated to only one y value
do not forget that you can illustrate
functions
through graphing mapping or sets
and lastly functions can be seen in our
daily lives like driving a car
wherein your location is a function of
time
the length of your shadow which is a
function
of one's height and a lot more
and that's it we are done with the first
lesson
for this topic functions for our general
mathematics subject
great job for today see you in the next
lesson
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