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in this episode you’ll learn what a piecewise defined function is and we’ll

practice evaluating them and then graphing them let’s do some math [Music]

okay we’re going to begin with what are piecewise defined functions

um and we’re going to do this through a series of examples example one here

so here is an example of a piecewise defined function

so first off it needs to be a function so we’re using function notation here

and we’re going to say it’s piecewise defined because on its domain which is

specified right here we can say how the different pieces are defined so

for example when x is less than zero then we will substitute x into this part

right here when a given x is a greater than or

equal to zero then we’ll substitute into this part right here

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and so let’s see some examples let’s find f of minus one

so here’s how we’ll work this out f of minus one is equal to

so now um i’m substituting in minus one so now i’m going to check which

condition do i uh check over here so minus one is less than equal to one

that’s true and minus one is greater than or equal

to one that’s false so we’re gonna use the one that’s true

so we’re gonna use two times minus one and then plus one and so that’s minus

two plus one and so that’s minus one now what about zero here let’s

substitute in zero so f of zero here zero is less than one

that condition is true so i’m going to plug in 0 here so we’re going to get 2

times 0 plus 1 and then that’s just 0 plus 1 or just 1.

and then for this third example here this part c

now i’m going to substitute in 2 here so we’re going to substitute in 2

and now i check which condition is 2 true so 2 is greater than or equal to 1 so

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i’m going to use this piece right here so 2 times 2 plus 2

and that’s going to be 4 plus 2 which is just six

so there we go there’s three examples of evaluating uh this function here

now um i’m going to um let’s see here we’re going to

work on five examples here and on this example first example here or example two

um we’re gonna we’re gonna evaluate some numbers um

find some points on the graph and then we’re gonna try to sketch the graph

and then towards the end we’ll use computer to sketch the graph if you want

to see that so stick around to the end if you can

alright so let’s see how to sketch the graph

so the first thing i notice is that it’s two lines here

this is a line with y-intercept one and the slope is two

but that piece is only good for when we’re strictly less than one

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so i’ll just graph the line right here it goes through one and it has slope of

two so it looks something like that but at one right here at x equals one

we’re gonna have a whole and this line is only good this piece of

the graph right here is only good for less than one

now when we’re greater than or equal to one now the y intercepts at two

so we’ll need more up here so i’ll put a two right here um actually

the uh that’s the two right there and let’s see here when this is one what

is the height here when this is one the height here is um [Music] you know three

so this is looking like something like um

you know because even though we’re going to have a hole here i want to know what

this height here is going up to so what’s this height right here even

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though there’s a hole there there’s a point missing right so

this one this piece right here was going towards the two plus one which is a

three and this is a one right here so two is going

to be right here in the middle and we’re going to have a y-intercept

for the second piece when we’re greater than or equal to 1

we’re going to have y intercept of 2 and a slope of 2. so it’s going to have the

same slope and then now when we’re greater than or equal to 1

means we’re filled in and then we’re going to take away this part right here

so i just wanted to make a good shape there but the point is is that when we’re

greater than or equal to one and and we’re possibly equal to one that’s why i

filled it in here we’re going to look like this line right

here which has y intercept two and slope two

and when we’re less than or equal when we’re less than one strictly less than

one so since it’s strict it’s open right here that’s not a circle it’s just

representing the fact that there’s there’s a point missing there so the

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graph will look something like that so there’s the graph um for example two

there let’s see if we can go back to example one now so an example one here

um we have two different pieces also this time the um domain is broken up

into two pieces uh using the using zero so let’s look at a here will

be f of minus two and so what will f of minus two be where

where will minus two minus two be less than zero so i’m gonna use a minus two

in here so minus two squared plus one so that’s four plus one is five

and then for part b we have f of minus one

and again minus one is satisfying this so we have minus one with a square plus

one so we got one plus one is two and then we have zero right here so f of zero

so now which piece do we use for zero so

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zero is equal to zero so now we’re gonna use this piece right here we’re gonna

get zero minus one which is minus one and then we have f of 1

and 1 is greater so we’re going to use 1 minus 1 we’re going to get 0

and then for the last part what happens when we substitute in 2 here

so we’re going to get uh 2 using this 2 we’re gonna choose greater

anchor to zero so we’re going to use this piece here

and we’re going to get a 1 here so here’s some examples of evaluating a

piecewise function you just take a look at what you’re plugging in and you see

which condition holds and you use that one now because this is a function only

one of the conditions hold in other words if someone put an equals right

here less than or equal to this is not a function then because when

you plugged in zero you would be getting two different outputs you would be

getting a one and a minus one and you know a function must have unique output

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so that wouldn’t be a function right there so when you substitute in only one

of the conditions will be satisfied and we’ll use one of the pieces that

corresponding piece to get the output there

so let’s take a look and see if we can sketch this graph right here so first

off do we know what this piece right here looks like so if i just consider

this right here on its own what would that look like

so that looks like a y equals x squared but it’s been moved up one so it looks

something like this and what would we what would this right

here look like all on its own so that’s just the line

the slope is positive one it’s gone through the y-intercept of negative one

so it’s just the line going up like that so by themselves we have two different

functions and we’re piecing them together using this so um

when we’re less than zero we’re going to

look like this so here’s right here zero so when we’re less than zero now

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we’re going to have a hole right here because it’s strictly less than 0.

we’re going to erase this part right here now when we’re

greater than or equal to 0 we’re going to look like this part right here so

what’s happening at zero we’re down here at zero minus one

and we’re just lines going up but we’re going to erase this part right here

and just put a tick mark there but make sure and fill that in because

at zero we are equal to -1 which is what we found right there

so this would be the sketch of the graph

and we’ll just erase that right there so this is the sketch of the graph of the

function f right here it’s a piecewise function to the left of zero looks like

this quadratic to the right and equal to zero it looks like this line right here

all right very good so let’s look at our third example now

so for our third example that looks like we have two quadratics

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and we’re broken uh this function uh into two pieces so for the left we’re

going to be looking at this and for the right we’re going to be looking at this

so let’s see if we can sketch uh actually

before we do that let’s practice plug in some numbers

so minus two where does minus two fit in here it’s it’s we’re gonna use this

piece right here so we’re gonna look at minus two squared plus two

which is four plus two which is six and then for this one right here we have

f of one so we’re going to substitute in one

which condition holds this one right here so we’re gonna plug in one right

here so one squared plus two so this is three and then f of two right here

f of two will be where’s two fit in so two is greater than equal to one so

we’re gonna use this piece right here so

this would be two times two squared plus two right just plugging into 2 there so

that’s 4 times 2 which is 8 plus 2 which is 10.

so there’s plugging in some numbers right there

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now let’s see if we can sketch the graph of this just by hand

and so we’re going to be broken up at 1. um so first off i usually like to um

you know make some little sketches over here you know what does this one right

here look like it’s x squared plus 2. so now we’ve shifted up two but looks like

a x squared graph and what happens this one right here

it’s two x plus two so it looks very similar

so it’s also been shifted up to but it’s going up a little bit faster so i’ll

just draw this one a little bit skinnier but they both go through two right here

so we’re breaking this up at one though so where would the one be so at one

where right here on this graph when we plug in one we’re going to get three

on this graph we plug in one we’re going to get something higher

right so we plug in one here we’re gonna get a four out all right so

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there’s the two graphs separately now let’s try to piece them together

um and try to sketch the graph of f now not just the pieces right the whole

graph so we’re broken up at one and less than or equal so that means i’m

going to be filling in the the point at 1 but i’m going to be looking like this

right here which is going to come and have this right here so i’m going to

come in here and have this right here and this point right here is a two and

that’s a three now when we are greater than one we look like this one right here

um which is just you know going up like that but it’s going up like a parabola

it’s not it’s not like a line going up it’s got some some curviness to it so

we’ll come up here like this and now actually that’s not right though

because this is hitting at a four so let’s put a four here

and we’re going to have a hole here and then it’s just gonna go up so

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you know sometimes people like to put these arrows here but uh this is filled

in right here and it looks just like a um i like to make it a little

pointier there something like that there and then this is open right here

so when we substitute in a one where are we going to go we’re going to

go to a three that that’s filled in right there

all right and when we go to a two you know then we get the 10

you know way up here all right so there we go there’s a sketch of the graph

of example for example 3 there there’s the sketch of the function and these

were just you know little doodles on the side we don’t need them anymore

so there’s a sketch of the graph of f all right so there we go let’s look at

another example now um number four so here we go let me erase this real quick

all right so number four here now we have three pieces

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so this looks like a lot of fun this piece looks like a line this piece looks

like just a constant four and this piece looks like a parabola so we got these

different pieces here broken up by these

conditions over here let’s practice with some numbers first so f of two is minus

two we’re going to plug in minus two whereas minus two fit in it’s in right

here you always gotta check that you can’t just assume it’s the first one you

gotta check the condition so we’re gonna

get three times minus two and then minus one

so that’ll be minus six minus one so that’s minus seven

now let’s substitute in here a minus one half so where’s minus one half

fit in that fits in right here minus one half is between minus one and

one so we’re gonna get a four out and then uh what about f of three here

so f of three here is you know substituting in a three we’re

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going to go right here so we’re gonna get out of three squared which is just

nine all right so there’s uh you know how we plug in

but now let’s sketch the graph of this and i’m going to graph each one of these

separately just as a little doodle over here it’s just a little scratch work so

the first one here is we’re looking at the y-intercept at minus 1 so it’s right

about there and then the slope is positive 3 so i’ll just make it go up and now

you know what are some numbers on here this it’s interesting around minus 1

right here so when x is the minus 1 right here what’s going to be going on

down here so minus 1 we’re going to have here we

didn’t plug in minus 1 but if we just look at this graph right here 3x minus 1

all by itself i’m not doing the whole piecewise yet but just this by itself

what happens when we plug in a minus one here we’re gonna get a minus four out

right that’ll be minus one times three so minus three minus one so

minus four so there we are right there so that’s an interesting point on this

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graph so we got two interesting points now what about this graph right here

what does y equals four look like well that’s just a horizontal line very boring

what’s happening here at minus one and one so minus one it’s still height of

four and at one it’s still a height of four okay

and now let’s look at one more piece right here the y equals x squared

this time there’s no shift on it so it just looks like a nice parabola like that

and where is it interesting where’s an interesting point on it what’s happening

at one so at one we just have the point one one

so there’s an interesting point on it all right very good so there’s our three

pieces and i want to know what each piece looks like before trying to sketch

the graph of this let’s sketch the graph of this piecewise function now

so we’re going to be broken up at -1 and 1.

so at -1 what’s happening to the left of

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-1 i look like this right here so to the left i just have this line going down

here so we’re going to be looking like something like this right here this is

right here and it’s the line and the slope is three

and this is going to be here minus one and this is going to be minus four now

the question is do we fill this in or not since it’s strict right here we’re

not going to fill this in so i’m going to make that clear by putting

just a missing point on it it looks like a small little circle

all right so now what about between -1 and 1 well it’s just a constant it’s

just a 4. so i’ll go and draw my 4 across here

and this is let’s say a 1 and that’s a minus 1. and now you know do we fill in

or or put empty spots here right so these are both equal here so these are

both going to get filled in here and let me just label this a height of four

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and that’s my one for my tick mark and here’s my minus one all right so um

what’s happening now greater than one so greater than one i look like this

parabola here it’s coming through here and what’s the

height coming in here it’s coming in here let’s move the one down here again

now so it’s coming in here about a height of one so it’s going to come in

through here it’s going to look like that but we don’t need to graph this

part of it it’s going to start right here and it’s just going to go up

so it’s just coming like that going through there like that just like

this right here it’s just going to go through here like that and now the

question is does it get filled in here or not it’s

strict so i’m going to put an open spot right here

and now maybe we want to like just shape it just make sure and don’t don’t make

it look like a line right so you know we’re going to come in through here like

that and it would look like a parabola all right so there we go there’s the

graph of f and there’s the piecewise graph right there

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all right very good some people like to put arrows there in any case

there’s a sketch of f and i hope you’re getting the hang of it

and well let’s see one more example though on the next example we’re going

to have three pieces also so here we go let’s erase this

and while i’m racing i’d like to mention that this episode is part of the series

functions and their graphs step-by-step tutorials for beginners so check out the

whole series um the link is below in the

description all right so example five so

we got a line here a linear par a linear piece a constant piece again and a

quadratic piece again so i’m going to look at this piece right here four x

four minus five x and i’m going to look at this piece right here and i’m going

to look at this piece right here so these are just my scratch work here

so when i’m looking at this piece right here it’s got a y-intercept of four

and it’s got a slope of minus five so it’s going down right there

and then um y equals zero so that’s just the horizontal axis

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and then this one right here is y equals x squared but it’s been shifted up one

so it looks something like that so now where some interesting

points on these graphs right here so this one is interesting around -2 so

what’s happening here at -2 so minus 2 we’re getting 4 minus 5 times -2

which is 14. so the height up here is 14. so this isn’t just scale right there

because this is a 4 and that’s a 14. but it’s the shape that’s important right

here um now before i go on with those um actually i forgot i wanted to get

started with the fun stuff let’s go ahead and plug these in and just make

sure that we’re okay with the function so -3 that’s satisfied right here so

we’re going to get 4 minus 5 times -3 so that’s 15 so that’s 19. 4 plus 15

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and then for f of 4 4 is going to go right here so we’re going to get

4 squared plus 1 or 17. and then for this piece here we’re going to get

f of minus 1. so where does -1 come in it comes in right here so we’re going to

get out zero here all right so this is an interesting point on this

graph right here the graph of this line is just y

intercept four slope is minus five and because we’re going to stop looking

at it and we’re going to break it up at -2 i want to make sure and understand

that point right there y equals 0 is plain and boring but it’s

only good between -2 and 2’s we’ll we’ll just plot those points there

all right and then this one right here it’s interesting around two what’s

happening around two so two i’m going to plot the point here

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what’s the height on this graph uh it’s a five right 2 squared plus 1.

so there’s an interesting point on this graph all right so now that we’ve done a

little bit of scratch work and we’ve done some evaluating let’s go ahead and

sketch this piecewise function right here here we go

now we’re gonna be breaking it up at minus two and two so let’s go ahead and

see if we can plot those right now let’s

just put a minus two and two and see how it goes

all right so less than or equal to i’m gonna be looking like this line right

here so um we’re gonna be looking at the line to

the left and it’s just this part right here

and so i’ll plot this point right here this will be a height of 14

and we’re just going to put that line up there right there there it is so that’s

the that’s the branch that’s coming from the 4 minus 5x right there

and now between -2 and 2 is just right on top of the axis

i’ll just shade that in a little bit so you can see it

now what about how it’s connected right here both of them cannot be filled in

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because it’s a function so at -2 this piece is going to

be the important part so this will be strict right here so it’s open so i need

to put an open spot right here and same thing for right here this is

open right here and now at two i’m at five for this

piece right here when we’re greater than or equal to two

so two i’m at five let’s put a five down here

and then i’m just shaped like a parabola

again so here’s the 0.25 and i’m filling it in because it’s equals and so it’s 2

5 and i’m just going to be shaped like some kind of like parabola

in fact i might want to flatten it out more just to make it look like it goes

through the origin there or actually goes through through a one but any case

there’s a let me make that a little bit straighter all right so there we go

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there’s the um that’s supposed to be straight and

that’s curvy because that’s a quadratic and then it’s constant right there

all right so there we go so now let’s take a look at how to generate these

graphs using a computer so let’s switch over to python and while

we do that let’s erase this board here real quick all right so let’s get some

um python going here so we’re going to open up a python notebook and here we go

now if you if it’s your first time trying to

use python i recommend using the link below in the description

and that link will take you to a google document where you can open it up and

run a python notebook it’s free and it’ll open up and it’ll save it on

your google drive automatically but you’ll be able to run this code and

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you’ll be able to follow along so let’s see what the setup here is

first so you’ll want to type everything up in here

into the python notebook again the link is below in the description just click

on the link and then that click i think you click on a new notebook and once you

do that then you’ll be able to start typing things in here so the first thing

we want to type in are these packages right here this allows us to do the

following work so i would just type up this right here this import and this

import and this import and once you type those in just like they are then you

just hit shift enter on your keyboard okay so um now i’m going to be making

some sketches some graphs and i want my graphs to look like they

look like in a in a math course not like they look like and by default

for python so i cooked up this uh function right here called pc axes

and so if you just type this in now this is optional you don’t have to use this

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right here but this pc stands for precalculus axes and this allows me to

make my sketches when they come out to look a certain way

all right so let’s look at our first example so this is example one that we

did earlier and so here’s how we would define a

piecewise function or this is one way at least so i’m going to define a function

f of x so again you need to type in things exactly as they are spaces and

everything these tabs or whatever all right so we’re going to define this

function f of x right here and we’re going to say

if the condition x is less than zero then we’re going to return x squared

plus one so that’s an x squared and then a plus one

if the condition is greater than or equal to zero then we’re going to return

the x minus 1. so this is defining this function right

here notice it’s quick and easy doesn’t take very much work at all

in fact the harsh part is probably to get the notebook started

all right and now we’re going to make a plot and so this is how we would make a

plot now i want to run through this here

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very briefly at least for the first time so that you can see what you can

and or what you might want to customize so first i’m going to define a figure

and then i want to define a set of axes to go on my figure

then i’m going to customize my axes now as you can see my axes have arrows right

here and right here and the axes go through the origin

so by default the python doesn’t do that so if you if you like the default way

python gives you graphs then you don’t have to use this line right here

but i like to have my plots looking like this

all right so now we’re going to specify the domain

now this is where there’s some trial and error you know because there’s no

nothing up here that tells you where where to look at this function all

it says is there’s it’s broken into pieces at zero

so i’m going to look a little bit to the left to zero and a little bit to the

right of zero minus 1 is the left and then 1.5 is the right

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then this right here tells me what kind of steps i’m going to take it into so

the 0.001 tells me when i’m sketching this graph

right here or when i’m looking at the domain

i’m going to take it into little tiny broken steps

minus 1.1 and then minus 1.1 plus this little break so it’s going to plot

tiny tiny little points and it’s going to connect them together with line

segments so this way if you use a nice uh increment right here

it’s going to look nice and smooth so if you if you choose this larger then

you’re going to be able to see it’s going to be broken into line segments

however if you choose too many zeros then you’ll make your computer work much

harder than necessary but any case these

values right here you may want to change depending upon what function you have

we’re going to do some other functions down here and we’ll change our domain

what we’re going to be looking at all right and so then after that we’re

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going to say what the y’s are we’re going to use the function f right here

right here to make the y’s and then we’re going to plot all the x’s and y’s

and then we’re going to show that plot so that’s how this code works

so i’m going to execute that and then we

get this sketch right here so this looks like the parabola over here

and this looks like the line over here so that’s we got right here we got the

line right here and we got the parabola right here and

it all looks great except for the fact that python doesn’t

put in the holes like we did so this was strict right here so we

actually put a hole in our graph right here so we put a circle right here

so by default python doesn’t do that so and then this right here is equal so

this would be filled in right here so python didn’t do that the way we did

that by hand moreover python actually tries to

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connect these two points you can see just a slight tint of green so python is

actually trying to connect those two points in fact if you change our

step increment right here so now it’s going to be much more coarse

so now it actually tries to connect them and that just

gives a bad representation of this of the graph

so i’m going to throw in a couple zeros here you don’t want to throw in too many

or your machine will get bogged down all right so there’s a sketch that we’re

going to end up with and this gives us a reasonable

approximation to the sketch that we drew by hand in the sense that

yeah it’s the same formulas here but it’s not going to tell us the

inequalities where it should be filled and not filled

all right so now let’s look at example 2 that we did

so here’s the function right here we’re broken up at 1 and greater than or equal

to 1. so that’s my conditions right here

if x is less than one or if it’s greater than or equal to one

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and for this piece it’s two times x plus one and for this piece it’s two x

plus two so i define my function and then i use the exact same code here

i didn’t even bother to change the window because for this window right

here it turned out to be nice also um so to the left of one we have this

line here and to the right of one we have this line right here

and you see it tries to connect these two so it shouldn’t be connected number

one number two the lower line should have a open

right here and it should be closed right here

and that part right there shouldn’t be there so the drawback of using a

computer program is sometimes you have to interpret the results on the other

hand it gives you the plots pretty fast doesn’t it i mean it doesn’t take long

to type that up and to execute the cell and then you get the graph in a split

second for more complicated graphs this could be of great value

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all right so let’s look at example three so we had x squared plus two

when we’re less than or equal to one so x less than or equal to one we have x

squared plus two and greater than or equal to one

we have two times x squared plus two and then i use the same window minus one

to one point five and i use the same step and so let’s execute again

there we go we look like this nice parabola coming through here and then

it should be and then we have this nice parabola

going right here now it may look like a line but it’s not it’s a parabola you

can tell that by the equation up here at one we’re broken up into pieces

greater than one we’re strict so this should be open right here

and this should be closed right here all right and then for the fourth

example right here we’re broken into three pieces

so we’re gonna have if x is less than minus one

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if x is between minus one and one if x is greater than one and then i just

typed in the formula three x minus one that’s what we’re going to return two x

times plus two oh actually that should be a four right there and then

and then the last one is the x squared if we’re greater than one so let’s hit

shift enter and execute that and here we go now

this domain here minus one isn’t great choice because we don’t get to see this

branch at all so let’s change this to say a minus two and see how that works

all right so then we can see a little bit more of it we got this line uh right

through here um and the slope is one y intercept is

minus one right so it’s coming right through here it’s going to go through

your minus one but it has to stop right here and it’s strict so this is open

right here and these are equal so at four this is

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going to be filled these are going to be filled in right here and again you just

have to ignore this part right in here that’s not part of the graph

and then we have a height of this is filled in also and then now right here

this is strict so this is open right here and so again this is not part of the

graph so this will be the graph if we were graphing this by hand like we

already did and then this looks like a parabola

coming through here right there so that’s the graph that we we did by hand

right there earlier all right so number five the last one

all right so uh less than or equal to minus two we have four minus 5 times x

between minus 2 and 2 and it’s strict here so we have it strict here

and then greater than or equal to 2 and then we have here the x squared plus 1

and now let’s hit shift enter on that and let’s see if our domain worked out

i use minus four to four so minus four to four

00:34

and so here’s where the four minus five x is coming in

and here’s where we get the x squared plus one coming in right here and then

in between it’s just a zero so it’s just right on top of the axes so this one

right here would be filled in this part right here

and this right part right here would be filled in

and then it’s on top of the axis so it’s

it’s too hard to see by computer so i’ll just draw that in for us right here

and then this was open and this was open and that was the graph that we had

before coming in here like here like a parabola and then this part right here

coming in here like a line right there so there’s how we would sketch the graph

uh using computer and by hand and uh i hope you enjoyed this video and

i’ll look forward to seeing you in the next episode

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