PIECEWISE FUNCTIONS || GRADE 11 GENERAL MATHEMATICS Q1

Summary

The video script discusses the concept of piecewise functions, illustrating their application with various examples. It explains how to calculate values and graph these functions, which are defined by different equations over different intervals. The script also covers real-world applications, such as mobile plans and tricycle fares, demonstrating how piecewise functions can represent costs that vary with usage. The examples help clarify the practicality of these mathematical tools in everyday scenarios, encouraging viewers to appreciate their relevance.

Takeaways

• 📐 Piecewise functions are defined by two or more equations, each applying to a specific interval over the domain.
• 📈 Example: f(x) = 3x + 2 for x >= 0, and f(x) = -x^2 + 3 for x < 0.
• 🧮 To find the value of a piecewise function, use the appropriate equation based on the given x value.
• 🔍 f(0) in the example uses 3x + 2, resulting in 2.
• 📉 f(-3) in the example uses -x^2 + 3, resulting in -6.
• 📊 Graphing piecewise functions involves plotting each equation on its specific interval.
• 🖊️ Example: y = x + 3 for x >= 0 and y = -x^2 + 3 for x < 0.
• 📏 Real-world application: Calculating mobile plan costs with 300 pesos monthly for 100 free texts, and 1 peso per additional text.
• 🚴‍♂️ Another example: Calculating tricycle fare with 20 pesos for the first kilometer and 5 pesos for each additional half-kilometer.
• 🧩 Piecewise functions are useful for modeling situations where a function’s rule changes based on the input value.

Q & A

What is a piecewise function?

A piecewise function is a function defined by multiple equations, each of which applies to a specific interval within the function's domain.

What is the domain of a piecewise function?

The domain of a piecewise function is the set of all possible input values (x-values) for which the function is defined, which may be broken into different intervals.

How do you find the value of a piecewise function at a specific point?

To find the value of a piecewise function at a specific point, you determine which interval the point falls into and then use the corresponding equation for that interval to calculate the function's value.

Can you provide an example of a piecewise function and how to evaluate it?

An example is f(x) = 3x + 2 if x ≥ 0, and f(x) = -x^2 + 3 if x < 0. To evaluate, say, f(0), since 0 is greater than or equal to 0, you use the first equation, resulting in f(0) = 3*0 + 2 = 2.

How do you graph a piecewise function?

To graph a piecewise function, you plot the function according to each equation's interval. You may end up with multiple line segments or curves that connect at the boundaries of the intervals.

What is the value of f(x) at x = -3 for the piecewise function f(x) = 3x + 2 if x ≥ 0 and f(x) = -x^2 + 3 if x < 0?

Since -3 is less than 0, you use the second equation, resulting in f(-3) = -(-3)^2 + 3 = -9 + 3 = -6.

What is the significance of the term 'kappa' used in the transcript?

In the context of the transcript, 'kappa' seems to be a filler word or a mispronunciation that does not have a specific mathematical meaning. It is likely a transcription error.

How can piecewise functions be used in real-life scenarios?

Piecewise functions can model real-life situations where different rules apply to different ranges of values, such as pricing structures with tiered rates or speed limits that change in different zones.

Can you give an example of a piecewise function representing a pricing model?

A mobile plan might have a base charge of 300 pesos for 100 text messages, with an additional charge of 1 peso per message above 100, represented by a piecewise function.

What is the fare for a tricycle ride if the first kilometer costs 20 pesos and each additional half kilometer costs 5 pesos?

The fare can be represented by a piecewise function. For the first kilometer, the fare is 20 pesos. For distances beyond the first kilometer, the fare is 20 + 5 * (additional distance in kilometers / 0.5).

How would you calculate the tricycle fare for a 3-kilometer ride using the given piecewise function?

For a 3-kilometer ride, the fare would be 20 + 5 * (3 - 1) / 0.5 = 20 + 5 * 4 = 20 + 20 = 40 pesos.

Outlines

00:00

📚 Introduction to Piecewise Functions

This paragraph introduces the concept of piecewise functions, which are defined by multiple equations, each applicable to a specific interval within a domain. The script explains that these functions can be used to model various real-world scenarios and provides an example of a piecewise function defined by two expressions, one for values of x greater than or equal to 0 and another for values less than 0. It also discusses how to find the value of the function at specific points and includes a step-by-step explanation of how to calculate f(0) and f(-3) for the given function.

05:02

📈 Graphing Piecewise Functions

The second paragraph delves into the process of graphing piecewise functions, using the previously introduced function as an example. It explains how to determine the value of the function at points such as f(-5) and f(3) and emphasizes the importance of understanding the intervals for which each part of the piecewise function applies. The script also mentions the use of graphing tools like Desmos to visualize these functions and provides a brief guide on how to sketch the graph manually.

10:02

📝 Practical Application of Piecewise Functions

This paragraph discusses the practical applications of piecewise functions, using the example of a mobile plan where charges are determined by the number of text messages sent. The script outlines a scenario where a base charge applies for the first 100 messages, with additional charges for messages beyond that limit. It then describes how to represent this pricing structure as a piecewise function, emphasizing the importance of understanding the breakpoints in the function where the charging rules change.

15:10

🚗 Piecewise Functions in Transportation Pricing

The fourth paragraph explores another real-world application of piecewise functions, specifically in the context of transportation pricing. It uses the example of a tricycle fare that has a base rate for the first kilometer and an additional charge for each subsequent half kilometer. The script explains how to calculate the fare for different distances and illustrates this with a detailed example, showing the mathematical process involved in determining the cost for a trip of three kilometers.

20:13

🎓 Conclusion and Encouragement to Learn

The final paragraph wraps up the discussion on piecewise functions by emphasizing their importance and versatility in various scenarios. It encourages viewers to continue learning about these mathematical concepts and to apply them to solve real-world problems. The script ends with a reminder for viewers to engage with the content by liking, subscribing, and enabling notifications for the channel.

Keywords

💡Piecewise Function

A piecewise function is a mathematical function that is defined by multiple sub-functions, each applicable over a certain interval within the function's domain. In the video, the concept of piecewise functions is central to understanding how different equations can be used to represent a single function over different ranges of values. The script provides examples of piecewise functions, such as one defined by '3x + 2' for x ≥ 0 and by '−x^2 + 3' for x < 0, to illustrate the concept.

💡Domain

In mathematics, the domain of a function refers to the set of all possible input values (x-values) for which the function is defined. The script discusses how a piecewise function has different equations that apply to different intervals of the domain, emphasizing the importance of the domain in determining which equation to use for a given input.

💡Graph

Graphing is the visual representation of a function, showing the relationship between the input and output values. The video script mentions sketching the graph of a given piecewise function, which helps in visualizing how the function behaves over its domain and understanding the transitions between different segments of the piecewise function.

💡Interval

An interval is a set of numbers with a specific range, often used to define the domain of a piecewise function. The script uses intervals to specify the range of x-values for which each sub-function of a piecewise function is valid, such as 'x ≥ 0' and 'x < 0'.

💡Function Value

The function value is the output of a function for a given input. In the context of the video, finding the function value for specific inputs, like f(0) or f(-3), involves determining which sub-function of the piecewise function to use based on the input's interval and then calculating the result.

💡Quadratic Function

A quadratic function is a polynomial function of degree two. The script refers to a piecewise function that includes a quadratic component, '−x^2 + 3', when x is less than zero, to demonstrate how different types of functions can be combined within a piecewise function.

💡Desmos

Desmos is an online graphing calculator used to plot functions and equations. The script mentions Desmos as a tool for graphing piecewise functions, suggesting its utility in visualizing and analyzing the behavior of such functions.

💡Tricycle Fare

The script provides a real-world example of a piecewise function representing the fare for a tricycle ride, where the fare depends on the distance traveled. This example illustrates how piecewise functions can model situations where rates or costs change at certain thresholds.

💡Text Messages

In the context of the video, text messages are used as an example to demonstrate how a piecewise function can represent a pricing model where the cost changes based on the number of messages sent. This example helps to explain how piecewise functions can be applied to practical scenarios.

💡Subscription Plan

A subscription plan is a type of service agreement where a customer pays a fixed amount for access to certain benefits or services. The video script discusses a mobile plan that includes a fixed monthly charge and additional charges for text messages beyond a certain limit, using a piecewise function to represent the total cost.

Highlights

Introduction to piecewise functions and their definition

Explanation of how piecewise functions are composed of multiple equations for different intervals

Example of a piecewise function with two cases based on the value of x

How to find the value of a piecewise function at a specific point

Step-by-step calculation of f(0) and f(-3) for the given piecewise function

Graphical representation and sketching of piecewise functions

Determining the value of a piecewise function at f(-5) and f(3) with a different equation setup

Using Desmos to graph piecewise functions and visualize their behavior

Application of piecewise functions in real-world scenarios like mobile plan charges

Formulating a piecewise function to represent the cost of text messages in a mobile plan

Another real-world example of a tricycle fare as a piecewise function of distance

Calculating the tricycle fare for different distances using the piecewise function

Importance of understanding piecewise functions in various practical applications

The versatility of piecewise functions in modeling different types of situations and problems

Encouragement to practice and explore piecewise functions for better comprehension

Invitation to like, subscribe, and follow the channel for more mathematical content

Transcripts

[Music]

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a lesson wants a general mathematics or

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and about function so an abiding

piecewise function a piecewise function

is a function that is defined by two or

more equations where each equation

applies to a certain interval over

specified domain so chimera by you

Nikita Margarito and that is a piecewise

function so a bag marking and grab that

is a sample of piecewise function so

Parimal alone a teammate in the end how

to find the value and graph of piecewise

function so given a piecewise defined

function f of X is equal to 3x plus 2 if

your X is greater than or equal to 0

negative x squared plus 3 if your X less

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than or equal to 0 so apart

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and so on

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function better by less than 0 or

negative numbers you can keep anybody

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x squared plus 3 let's try to answer F

of 0 and F of negative 3 so

para MacEwan attenion f of 0 a link I

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in say 10 tipping

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it Donald kohonen attend or gamma t

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0 we use f of X is equal to 3x plus 2 so

simply substitute the value of x 3 times

0 plus 2 so the answer is 2 so how about

negative 3 demand poor since that is

negative value long x know it on GOG a

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you know parentheses and then he will

not be negative syllabus so negative 3

times negative 3 is positive 9

so mugging negative times positive 9 is

negative 9 plus 3 so the answer is

negative 6

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three more than three know it though

sketch the graph of the given piecewise

function what is F of negative 5 and

what is f of 3

for letter e f of X is equal to X plus 3

if your X is greater than or equal to 0

and then negative x squared plus 3 if

your X is less than 0 so thing they're

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hannahb see negative vibe there for a

non-functioning agony be nothing you

less than 0 city only so it's a negative

x squared plus 3 so a guy a Sugino a

pnina supreme you slide this up the

chitlin at then

so again again a parenthesis negative 5

times negative 5 is positive 25 times

negative that is negative 25 plus 3 so

that's is negative 22 and if your X is 3

unknown jogger maybe not then it'll

don't function at all

so therefore simply certitude answer is

6 my own and I mean our application of

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saying okay so you learning in a gonna

come grab a pad like paper Center so for

example you know eat an apple

okay so halimbawa Plessy is to get to

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f of X is

also represented as ye so looking at and

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let's take show no I'm excellent at then

the but greater than or equal to zero so

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learn in bracket you canyon ill again yo

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greater than or equal to 0

okay so idiom drop ya know to learn him

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greater than or equal to 0 where

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attic in Gretna so straight line sure no

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and effort so bring in your graph yeah

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tournament so the net in y equals

technolon negative x paper mean the

thing is on top is exponent x squared

plus 3 so it is muggy king-crab yeah

pertaining the not a quadratic function

decision example the quadratic function

you say no literally nettings negative

care down wagon graph you can grab yeah

param a restriction die or a young x

donut in a less than 0 so looking again

X is less than 0 so in a MIDI

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have a learning internet kappa got a

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grab little piecewise function at all

c'était all graphing our next example

tire okay / letter b f of X is equal to

X plus two if your X is less than or

equal to two negative x plus three if

your X is greater than or equal to two

okay since the value or the first one is

negative five so it's a substitute like

naughty negative five data so don't

function you express to so negative five

plus two that is negative three Kampala

why you three f of three so substitute

lambda not in detour so negative three

plus three so the answer is zero so

gamma cloneville it you grab or pedicle

government written in desmos but clean

up nothing n say - um grab Nieto

it won't function at item a gig in crap

yeah so ceremony no saying an oil grab

piecewise function okay so we begin by

nom sample problem no person may apply C

piecewise function a user charge 300

pesos monthly for a particular mobile

plan which includes 100 free text

messages messages in excess of 100 are

charged one person each represent the

amount of customer peace each mind as a

function of the number of messages M

sent in a month so and I'm not in a lab

ko I was Lahab near my cell phone now so

his parents and the rains and needs me

or no cuz I'm Annessa needs to say and

I'm in Garmon yang times of

communication okay sabi down McBee began

and 300 pesos monthly summable plan

pero my particular Xiang number and text

messages okay in 300 pesos monthly more

kasama long de nition 103 text messages

so a bug net exists casa exit card 100

text messages so mikvah biood garden or

a much charge gonna one vessel so doing

some exist mana got me okay so ibig

sabihin any mouth open up at 101 text

messages castle nomad pass condenser so

madad actor gets a 301 peso per my

percent saying yo so we need to press a

good mamita of piecewise function una a

so let T of M bucket p sa text messages

but they'll obtain a variable he'll

again you know running soggy banner

impersonating and even represent the

amount paid by the consumer

each man it can be expressed by the

piecewise function so can it to share

anything any present now in problems at

all sappy my biking and 300 pesos cap on

your number of messages more I greater

than or equal to 100 or your the number

of messages is greater than 0 but less

than or equal to 100

or pedaling amend that your M is greater

than or equal to 100 now and less than

or equal to 100

balem so you need to measure a percent

so compact Magadan and 300 pesos kappa

you number of messages more I hang on

100 text messages llama perro

Copaxone abraca so 100 text messages

magma by tienen el piso halimbawa mob

would can have 105 text messages so

healing you so Bravo 5 so you so Bravo 5

it added so equivalent kills of bypass

or Samba by Erin white 305 pesos so if

you commit an piecewise function another

okay since nasarah Manila tyonna very

particular ties but by Biden tricycle

case a missin like really climate IAP

addressing in Tainan hand I want you to

focus in this but I'm too dumb to saying

ever consumers are again in trying to

care line up a big line and obscene

entertaining on hand so if you Gaga one

piecewise function sabi dito 20 pesos

Inga bites a tricycle for the first one

kilometer then McKagan animal piece or

in every per half kilometer so a

tricycle right cause 20 pesos for the

first one kilometer in each additional

per hop kilometer adds five pesos to the

fair he is a piecewise function to

represent the tricycle fare in terms of

a distance D in kilometers some

Committee nothing later you will be

Cassidy stands so the input value is the

distance in the output is the course of

the tricycle fare sort we're using this

now as a representation of our function

so f of D represent the fare as a

function of these stands the function

can be represented as final so but

nothing in it represent the end twenty

pesos kappa and distance i equals a 1

kilometer so by biking ambient a or 20

pesos kappahd 1 kilometer long distance

and then another map represent

halimbawa umar Buchannon 3 kilometers so

maha kassapa so ma ni or no trainable

mama but now three kilometres so Panna

not in it represent you using piecewise

function Kayla and Tama so it to yan

it will compact limit per second is a

one kilometer mikvah by Jenna 20 plus

five pesos times B minus 1 divided by

0.5 so far nothing much Tecna Tama you

piecewise function not in at all for

example Messina sabi ko marble cannon

that look

in this stance when I travelled more

three kilometers McCarney b'bye it was a

crisis again got an idiom function at

all

Japan ceasing healing can apply security

malam a column or a computer know at a

number in the necklace you see behold

and ganito ordnance much bye-bye and

Kenneth Dhamma so ceasing in Lincoln and

Tama get up Atmos Mahalo I'm a single

sailor predicament complain so halimbawa

class three kilometers sent in opponent

tricycle and three kilometers malai unit

oh so it's a sub the chitlin attendance

Edina ten sub formula so 20 plus 5 by

returning Dino 3 minus 1 divided by 0.5

equals 20 plus 5 so 3 minus 1 is 2

divided by 0.5 and 2 divided by 0.5 is 4

4 times 5 is 20 plus 20 40 pesos sorry

Baba is a tricycle I'm 40 pesos love

kappahd Neha that know the hot 3

kilometers no pero ha Pagani bow-wow

bow-wow

raha handling higher the hot delawanna

killer Metro

Lyndon didn't processor 2-1 so I gotten

1 1 divides ear 0.5 the answer is 2 2

times 5 is 10

that's 20 30 % but I'd miss a tricycle

so Mahalo game piecewise function K say

AHA - don't saying you're not to give

you a formal our they seriously /

represent you know Oh panic in the

computer memory bad day and the problem

operating sample and piecewise function

sod you be sorry about

wholesale retail retaining and so pretty

not in airplane piecewise function

during Holi bhava kappa remod paris on

the midline being a limo I'm pressuring

an individual price no but 200 in the

met by what he said Bharata by some poem

beanie removed emit reading 185 nylon

body sir so in my mug AMARG a new

scenario that is the best example of

piecewise function say they do not eat

the whole agenda applying the piecewise

function so sad Amina to tune in K Raven

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