Introduction to Increasing and Decreasing Functions (According to Who?)

Video Series: Functions and Their Graphs (Step-by-Step Tutorials for Precalculus)

(D4M) — Here is the video transcript for this video.

00:00
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

00:01
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

00:02
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

00:03
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

00:04
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

00:05
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

00:06
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

00:07
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

00:08
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

00:09
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

00:10
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

00:11
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

00:12
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

00:13
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

00:14
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

00:15
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

00:16
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.

00:17
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

00:18
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

00:19
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

00:20
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

00:21
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

00:22
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

00:23
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

00:24
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

00:25
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
episode if you enjoyed this video please like and subscribe to my channel
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About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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