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in this episode you’ll learn what it means for a function to be increasing on

an interval and what it means to be decreasing on an interval we’ll practice

lots of examples let’s do some math [Music]

all right so here we go we’re going to talk about what it means to be

increasing on an interval and what it means to be decreasing on an interval so

before we get started with some examples let’s look at a generic case to

see what we’re going to be looking for so

here i’ll just draw a function right so we’re dealing with the function

and so here’s a function right here in fact let’s make it a little bit more

curvier let’s make it something that looks like that now

right here now when we’re going to be looking at increasing and decreasing

so we’re always going to be moving from left to right so we’re going to be

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saying less than here and less than here so we’re going to be reading from left

to right so as we’re reading from left to right it looks like we’re rising and

then we get to a height and then we start to fall

and then we start to rise again so this is where we’re increasing this is where

we’re decreasing and then this is where we’re increasing again so how do you

specify that in mathematical notation though or how do you write that down

mathematically so i’m going to say that we’re increasing right here at this um

on this interval right here so i’ll call that um [Music]

i’ll say we have an interval right here and i’ll say this here is an x1 and this

right here is an x2 and if i choose any point in here right in between the two

right so now we’re going to be reading from left to right

and so and what i mean by that is so if x1 is less than x2 like i drew it

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right here then that implies that the height here

so this will be the height right here this will be the f1 f of x1

and for the x2 right here will be higher and so this will be the f of x2 here so

the output right here so we don’t need this

point here in the middle here or this one here so the

uh x1 which is smaller than the x2 the height is smaller in other words

we’re going up so when we pick an x1 here

and then we choose an x2 we’re going to be higher the f of x2 is higher than the

f of x1 so in other words the f of x2 is

greater than the f of x1 so this is what is what it means to be decreasing

so we can go over here now and look at a decreasing one so i’ll put that over

here so let’s say we’re decreasing now and so

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i’m going to pick an x1 and an x2 here and if i go up here

and i get a height say f of x one here and i’m right here at x two

and i come over here and and get the height f of x two

so now again we’re reading from left to right so if if x1 is less than x2 if x1

is less than x2 then the height of at x1 should be

greater than the height here at f of x2 so that means we’re decreasing right

there and so we’re always decreasing on an interval

and so like if this was say a little bit

uh better here so i’ll make it look like this and then

down like that right so now right here we can tell that we’re increasing over

here we’re increasing and then we get to this

point here wherever it is and then we’re decreasing and then right here

we’re decreasing and then we’re still decreasing right here right and so we

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can be decreasing forever so from this point on right here let’s call it say a

and so for this function right here we can say that we’re increasing on

minus infinity to a this will be our interval here and

it’s uh always an open interval right these are strict inequalities here

so it’s increasing on here minus infinity to a and then after a it’s going to be

decreasing so this will be from a to all the way the rest of the way

so it just keeps decreasing decreasing decreasing decreasing decreasing

and so um yeah so you know this is the decreasing and

increasing decreasing now i’m going to leave these definitions up here as we

work through lots of examples here so you can keep trying to make sense of

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this and keep this in front of you so i think that’s uh always a good idea

when you’re trying to learn how to read uh like you are in a say precalculus

course when you’re trying to read learn how to read technical material

all right so anyways let’s go on to another example let’s look at

something that looks like this let’s say f of x is um 3 over 2x so you know

where is it increasing or decreasing so let me give you the graph so it’s coming

through right here and where is it increasing and

decreasing so as we move from left to right this graph is always increasing no

matter which x1 and x2 i choose the f of x1 is less than strictly less

than the f of x2 right here always happening no matter which x1 and

x2 you choose the second one is always greater than the first one here so

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this will be increasing on sometimes you can be increasing on

the whole real line minus infinity to positive infinity

so this is an example of something that’s increasing everywhere and

similarly if i had like a minus three over two and the line was going down

then that would be decreasing on the whole real line

so there’s our first example there let’s look at a second one let’s go here with

this one here number two let’s look at this function right here um f of x is

x squared minus four x so what does this look like let me give

you the graph right here and so it looks something like this

so it’s going to go through the origin so let’s make it go through the origin

and it comes down a little bit and it goes up and this point right here is two

minus four and so we’re going to be increasing

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where are we increasing and where are we decreasing that’s the question

so decreasing on and it’s always an interval so we’re

going to be increasing on here and 2 to positive infinity

not including the 2. so you never include the 2 the

definitions here are strict so you never include the x’s so let’s go here to

decreasing is minus infinity to 2. so it’s decreasing here until you get to

two and then at two it’s neither increasing nor decreasing it’s just a

point so you can’t be you can’t be doing

either one of these at a single point so you always need it to be on an interval

here all right let’s look at another example

let’s say this is the example 3 here so let’s say here this is f of x is

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x to the third minus three x squared plus two and let me give you the sketch

so it’s going to look something like this it’s gonna come up like that it’s

gonna hit this high point and it’s going to come back down and then go back up

right here and so this is the point right here 2 minus 2 and 0 2 right here

and so where is it increasing and where is it decreasing now

as you move further along into precalculus you get

you get more ability to sketch these graphs on your own so right now you can

use a computer or a calculator or something like that to make this sketch

for you but you know as you move along through

precalculus and then calculus you’ll get really skilled at making these graphs

so for right now i’m we’re giving the function and the graph and we’re asking

where is it increasing where is it decreasing so it’s increasing here on

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minus infinity up to zero right here so on this interval right here

but not including the zero so it’s increasing on minus infinity to zero

and then it starts to increase some more so right here at two

at x equals two on this interval right here all the way

to the end on this interval right here the graph is increasing so i’m going to

say union and then well i’m going to say union and then

2 to positive infinity there and so it’s going to be decreasing on

where is it decreasing so it’s decreasing between zero and two so

between these two right here and it’s not going to be

doing either at two so it’s decreasing on minus two to two

and so make sure you understand that that’s an interval right there oh sorry

it’s actually uh zero to two so zero to two

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but make sure that you understand that this is an interval so we’ll say

decreasing on and when you use the word on that means

you’re you’re saying that that’s an interval not a point

and so this is increasing on this union right here

which consists of this part right here and this part right here

all right let’s do another so let’s look at number four here which will be

f of x is square root of x squared minus one and let’s say here we have um

the sketch right here let’s put the sketch right here let’s see

and it’s gonna come in like here like this and it’s gonna be symmetrical so i’ll

try to make it symmetric like that all right and then this is the point

right here um minus 1 0 right when x is minus 1 we get out 0

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and this is the point right here 1 0. all right so increasing on

minus infinity to minus one and make sure that’s open decreasing on

one to infinity and so you might say what is it doing in

between well it’s just not defined there all right so make sure you use the word

on to represent it using an interval and make sure you have the right

parentheses the right way open this as opposed to closed intervals

and there we go there’s some more examples let’s do let’s do a couple more

um so here we go so a let’s look at this function right

here this looks like a real real nice function here so it’s going to say x

plus 1 absolute value plus and then absolute value of x minus one

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and so let’s see what that looks like so we’re going to um between

minus one and one we’re going to be constant

and then we’re going to have a slope of 1 and we’re going to be coming down here

and there i think that looks okay this is 1 and -1

and the height here is 2. so this right here point right here is -1 to 2 and

this point right here is 1 2. all right so there’s the sketch now

where is it increasing and where is it decreasing

okay so where’s it increasing so here’s decreasing

and then here’s where it’s constant it’s just a fixed value of 2

and now it’s increasing so it’s going to be increasing on 1 2 positive infinity

and it’s going to be decreasing on minus infinity to -1 so it’s decreasing

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on this interval right here so minus infinity to -1

all right good um so let’s call that one five and let’s call this one six

and so let’s look at something that looks like this here let’s look at a

rational function so we’ll make a series on

rational functions here coming up soon but for now here’s what the here’s what

the graph looks like so it’s going to be coming through here

at the point here 0 1 this is what happens when you plug in

zero everywhere you get one over one and this is the point right here

it’s there’s going to be actually an isotope at minus one i’ll dash that in here

and so then it’s going to come through here like this well let’s make it look a

little bit better it’s going to come in like that and there’s going to be a

highest point right there and this is the point right here um let’s call that

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minus 2 and minus 3. so that’s that point right there

um yeah so there’s that point right there i think you can see that

all right so we can call it point a and we’ll just say this is point a right

here all right any case what is where is it increasing and where

is it decreasing increasing on and decreasing on so it’s increasing um

so it’s going to be increasing up until we get to the point right here what’s

the x value is minus two so it’s going to be increasing on minus two sorry

minus infinity to minus two so we’re going to be increasing so if we

could sketch it out some more right so it’s increasing until we get to here and

then it starts to decrease and then the graph jumps over here and

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it’s decreasing until we get to right here and then at 0

it starts to increase again so i’m going to say union 0 to positive infinity

there and you can’t see that again so i’m going to say union to 0 to

positive infinity union to [Music] positive infinity there

there we go that’s a minus 2. all right so it’s increasing on this

and it’s going to be decreasing on so here we’re going to be decreasing

right here so the x value is -2 so -2 to all the way to this vertical isotope

which is at minus one [Music] union to minus one to

and then it’s going to be decreasing again on the other side of minus one but

not including the minus one and then all the way to zero

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and so we’ve got the whole line covered except for of course the minus one and

the two and the zero at those three points there all right so

i use that symbol right there for decreasing you use that symbol a lot for

decreasing and we could have used this symbol right here for increasing

like that up arrow slanted up arrow and any case so there’s two more examples

let’s see if we can find some more interesting ones here

i do have some for you here let’s look at for example a piecewise function

yeah so we did a whole uh episode on piecewise functions and so

let’s kind of remember what they are so we have these pieces right here x

plus three and three and two x plus one and we have x is less than or equal to

zero and this is between uh zero and 2 including 2

and then this is piece is good for greater than greater than 2.

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all right so here’s what the graph looks like

now you could probably sketch this graph

because you’ve seen the previous episode

but just in case here is the graph right here so we’re going to be looking at um

between 0 and 2 we’re going to be flat at a 2 out of 3. the height is going to

be a 3 right there so we’re going to be nice and flat at three

and then this is gonna be open right here because it’s strict right here

and then at two we’re gonna be closed all right so i graph i graph the easy

part first the the constant part in the middle now

when we’re less than or equal to zero when we’re on this side of of zero

uh we’re going to look like this line right here

um which goes through a height of one right so sorry a height of three

and which is already this right here is already three so it’s going to get

filled in and then it’s going to come in through here at a minus three

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and it’s going to keep going so this has slope of 1

and it’s going to go through the y-intercept at 3.

all right so there’s that graph so far and then on greater than 2 so on this

part right here greater than equal to 2 now we’re going to look like this line

so this has a y-intercept of 1 right here and it’s going to go through

that’s going to have a slope of 2 and so it’s going to go up so when the x

is actually the 2 it’s going to give me an output of 5

but i’m not going to include that as part of the graph so the slope is 2.

so it’s going to look something like that right there all right so again this is

coming through this this this line i’ll make it a little bit more

look like it’s going to go through one right there because the y intercept is

one so it’s going to look like so i’ll put it maybe a little bit more

uh inclined all right so there we go there’s our sketch there and

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now we’re going to ask the question it’s increasing on

so it’s increasing on minus infinity and then this is 0 right here

uh so i’m gonna say minus infinity to zero now remember increasing always on

an interval and then does it start to increase

anywhere else so here’s the constant and then uh after two it starts to increase

again so i’ll say union 2 to positive infinity and then decreasing on

on this interval right here decreasing so here’s increasing here it’s constant

and then here it’s increasing okay so it’s it’s not increasing on nothing all

right so there we’ll just erase that so we could say this function is not

increasing on any interval um or we could just leave it blank

actually leaving it blank may communicate to somebody that we didn’t

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do it so let’s write it f is not decreasing [Music] decreasing on any interval

yeah let’s do that f is not increasing on any interval

all right so there we go there’s there’s

another example uh what do you say we do one more one more piecewise

all right one more piecewise here we go let’s call this one here f of f f of x

again and let’s say it’s 2x plus one and let’s

do a quadratic x squared minus two and i’m going to say this is less than

or equal to minus 1 and this is greater than minus 1.

so this graph looks like a parabola but on the only on the right side of minus 1

so when i’m right here at minus 1 what’s going on with these two graphs

right here so minus 1 this is going to be a minus 2 plus 1 so that would be -1

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and when i use a minus 1 in here that’s going to be minus 1 squared so

that’s 1 minus 2. so that’s also minus 1. both of these graphs individually go

through this point here but the whole function here f

well it’s going to look like a line right here slope of 2

and over here it’s going to look like this parabola right here so it’s just

going to come through here like that and so it hits right here at 2

and this point right here is -1 2. all right so i’ll put a minus 2 here

all right so there’s a rough sketch and so where is it increasing

is increasing on uh minus infinity to minus one open

and then is it increasing again yeah so it hits the minimum right here at minus

two and then it starts to increase again so right here it’s decreasing

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so i’m going to say union and then let’s put it all in one line if we can union

and then -2 to infinite positive infinity there we go i’ll just use infinity

all right and so let’s say decreasing on [Music] minus one to zero

and actually that’s wrong right there right because it’s the x value right here

where where the graph is housing in an interval right here

and so it should be 0 right here so 0 to positive infinity

now squeeze in a positive sign all right so it’s decreasing um right

here in this part right here you can see it’s decreasing

and on this part right here from 0 to infinity it’s increasing there all right

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very good so i lost count what number that was but there’s another example there

and let’s do one where we got lots of them

in fact um you know sometimes you just have a graph you don’t even have an

equation so let’s look at something like this right here

we just have a graph so it’s coming up right here

and then it’s going down here like this and then it comes down here like this

so i’ll give you some points here so this will be the point here 3 4

and this will be the point right here 5 4 that’s a 5

and this will be the point right here so let’s call this one here four minus two

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all right and so what if we’re given this right here so where are we increasing

so i’ll say increasing [Music] on so we’re increasing here to a three

so minus infinity to the three union we’re increasing again

so we’re increasing now we’re decreasing

now here at four we’re going to start to increase again until we get to the five

so four to five union and then once we get to five and

afterwards we’re decreasing right so there’s where we’re increasing we’re

increasing here and we’re increasing here

so where are we decreasing so decreasing on

so between two and four we’re going to be decreasing i’m sorry between three

and four in this interval right here between three and four

and then are we decreasing anywhere else we’re decreasing here

and we’re decreasing five and afterwards

so union five to positive infinity there all right so there’s where we’re

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increasing there’s we’re decreasing and sometimes you just don’t have the

function rule you just have a graph right so you can determine where you’re

increasing and where you’re decreasing all right so there’s

some examples if you want to see more examples of where something is increasing

decreasing by looking at a graph let me know in the comments below

i hope you enjoyed this video and i look forward to seeing you in the next

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