FUNCTIONS | SHS GRADE 11 GENERAL MATHEMATICS QUARTER 1 MODULE 1 LESSON 1

Summary

This 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

00:00

📚 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.

05:02

🔍 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.

10:04

📐 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.

15:06

📈 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.

20:07

🚗 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.

Keywords

💡Function

A function in mathematics is a relation where each element of the domain is associated with exactly one element in the range. It is a central concept in the video, as it is the main theme discussed throughout. For example, the video explains that the circumference of a circle is a function of its diameter, where for every diameter, there is a unique circumference.

💡Domain

The domain refers to the set of all possible input values for a function. In the context of the video, the domain is illustrated as the first element in each ordered pair, serving as the input values that are mapped to outputs.

💡Range

The range is the set of all possible output values of a function, which are the second elements in the ordered pairs after the domain elements have been mapped. The video clarifies that the range is the set of y-values in a function, ensuring that each x-value in the domain corresponds to a single y-value in the range.

💡Relation

A relation is a set of ordered pairs, where an element from one set is associated with an element from another set. The video uses the term to distinguish between general relations and functions, emphasizing that a function is a special kind of relation with the unique correspondence of each x-value to a single y-value.

💡Ordered Pair

An ordered pair is a sequence of two elements, typically denoted by a pair of numbers enclosed in parentheses and separated by a comma, such as (1, 2). The video explains that ordered pairs are used to represent the input-output relationship in functions, with the first number representing the domain element and the second representing the range element.

💡Mapping Diagram

A mapping diagram is a visual representation used to illustrate the relationship between the domain and range in a function. The video uses mapping diagrams to show how each element in the domain is paired with a unique element in the range, helping to identify whether a given set of ordered pairs represents a function.

💡Set

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. The video discusses sets in the context of identifying functions, ensuring that no x-value is repeated in the set of ordered pairs, which would indicate a function.

💡Vertical Line Test (VLT)

The vertical line test is a graphical method to determine if a curve is a function. If any vertical line intersects the curve at more than one point, the curve does not represent a function. The video demonstrates the use of the vertical line test with various graphs to identify which are functions and which are not.

💡Circumference

The circumference of a circle is the measure of the distance around the circle. In the video, it is used as a real-life example of a function, where the circumference is directly proportional to the diameter or radius, illustrating the concept of a function in a practical context.

💡Shadow

The length of a shadow cast by an object is used in the video as an example of a real-life function. The shadow's length is a function of the object's height and the angle of the light source, demonstrating how functions can be observed in everyday situations.

💡Graph

In the context of the video, a graph is a visual representation of a function or set of data points on a coordinate plane. The video explains how to determine if a graph represents a function by using the vertical line test, with examples of graphs that pass and fail this test.

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