Finding the Zeros of a Function (Zeros or Roots?)

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 the zeros of a function are and why they’re
important but to be honest finding them can be very hard depending upon the
function let’s do some math [Music] okay so i want to begin with a brief
review about what a function is this will just be a quick second here
so a function from a set a to a set b is a relation
and for each input we’re going to have exactly one output the set of outputs
are called the range and the set of inputs are called the domain
now in the previous uh episodes in this series uh that’s right this episode this
video is part of the series functions and their graphs step-by-step tutorials
for beginners so in the previous episodes um by the
way the link is below in the description i hope you find that

00:01
so yeah we talked about introduction to functions we talked about piecewise to
functions we did a whole video over finding the domain and range of a
function and then very recently we did uh the vertical line test
excuse me so i hope you check out those videos and we’re going to continue on
with our function with our study of functions and in this video we’re going
to concentrate on finding the zeros of a
function all right so here we go what is the zero of a function so um
you know the zero of a function are those inputs there could be multiple
inputs but the output is a zero and so we call those these inputs the zeros of
the function so it’s kind of strange because you’re
calling something that’s not necessarily zero but you’re calling it a zero right
so those inputs whose output is a zero so ask the question what are all the
inputs that you get out zero those are the zeros

00:02
all right so in other words if f is a function
then all the x values the inputs for which f of x equals 0 are called the
zeros of f in other words you just solve this
equation right here and the solution set will be the set of zeros now sometimes
they’re also called the roots of f all right so
let me show you some graphs here so here’s a graph
and tell me if you can spot the zeros so this is the parabola going up right
here so there’s one where we have no zeros
um what about if we draw it upside down like this
well there this graph has zeros uh to see it though we have to you know draw
more of it so let’s draw more of it and so there right there would be the zeros
um how about if we have something that looks like this
we’re going up and down like a roller coaster

00:03
and so if we look something like that and then now let’s just put the arrow down
to making it look like it’s just going to go down forever so how many zeros do
we have here and it’s going to go down forever so we have
0 right there we have a 0 right here we have a 0 right here and we have a 0
right here so because this function is going to keep decreasing and this
function is just increasing on this whole interval here there’s only going
to be four now you can have functions that have
infinitely many zeros in fact if you’ve seen a little trigonometry before and
you might recall the sine graph it keeps repeating over and over again
it has infinitely many zeros or cosine graph right so there’s all
kinds of graphs whether or not you’ve seen them yet or not but you know you
could have all different types of functions and some functions will have
zero some will not some will have lots of zeros some will even have infinitely
many zeros and some functions will just have

00:04
one zero in fact let’s take a look at lines what if you have a function a
linear function like ax plus b are there any uh zeros so if we graph this right
let’s say mx plus b if we go graph this well it may not have any zeros at all
because um the slope may be zero in other words
we may just be a constant function right here let’s say this is four y equals
four right so the slope is zero and the b is four
right just y equals four right straight across so this would be a linear
function with no zeros but if your slope here is not zero
then it’s slanted and then it’s going to cross somewhere and then it’s going to
have exactly one zero so linear functions are pretty easy to
to work with because they’re either going to have no zeros or they’re going
to have exactly one zero even if the line is going down or if the line’s

00:05
going up you’re only going to have exactly one zero
unless of course you have a horizontal line even if you have a vertical line
let’s say this line is x equals two how many zeros do we have we have
exactly one all right so there’s the different scenarios for linear functions
and let’s start looking at some more examples though besides linear examples
so let’s look at some examples real quick
so in this first example here i’m going to sketch some graphs for you and i want
you to point out the zeros just by looking at a graph okay so here we go
let’s call this number one here and it’s going to look like this right here so
it’s coming in and then it goes like that and then number two
it’s going to come in here like this and then like this
so can you point out the zeros and then this one right here and then it

00:06
comes in here like this goes up like that and this one right here
has just been shifted down here and then number five here oops
try to draw it again and then number six um something like a hyperbola
so we’ve seen these graphs before in the previous video
when we’re trying to decide if it passes the vertical line test
so find the zeros on the graph if the graph represents a function
and we saw before that these failed the vertical line test these were not
functions right here so we’re not going to ask the question
what are the zeros on five and six although you can clearly see they cross
the x-axis there but these are not functions so we’re not going to really
entertain the question there for functions that’s what we’re really
interested in um finding the zeros of a function right so

00:07
where do you see the zeros pause the video and point them out on your piece
of paper if you’re following along all right so here we go um we have a
there we go let’s move it down all right so here we go we got a zero here
two and three that’s got three zeros there this one’s got two zeros right
there and right there and this one right here has no zeros
and this one right here has one unique zero now let me make these graphs a
little bit better to help you in case you were confused this graph is
going to continue going up and in fact this graph just goes up all the way just
from left to right it’s just always increasing now this one right here
to give better guidance or better graph i’m going to dash this in and call it an
isotope and what that means is in fact i’m going to do both axes what that
means is is that this graph is going to keep decreasing but it’s never going to
cross and this graph right here is going to

00:08
it’s going to it’s going to never cross the y-axis here either
so now that we have a better sketch of the graph or a better understanding of
the graph looks like we can say right here confidently that there are no zeros
because it doesn’t cross here and it doesn’t cross here this is a horizontal
isotope which we’ll study in greater detail later but the fact is right now
all we need to do is realize that a graph can be decreasing but not cross
and that’s a pretty exciting idea how can something be decreasing forever and
ever and ever but never cross so we’ll come up with uh and we’ll study that
kind of behavior in a whole chapter when we get to rational functions right so
that’ll be real exciting right so this has no zeros
and this one has two this one has three this one has one and these are not
functions here so you see different uh kind of scenarios right there
all right so let’s do um six more if we can oops uh yeah so let’s do six more

00:09
let me get this out of the way here real quick all right so
let’s look at six more so here we go one
it’s going to be coming in like this and then it’s gonna bounce right there
number two it’s going to come in here like this and then like this
and then number three it’s going to uh looked like a flower and
we studied these graphs like this when we get to polar coordinates later on but
it’s a really fun topic and then something looks like this
and then look this and then number five um we’re going to look at a step
function this is open and then it’s filled in and closed
and then open and then filled in and closed and then open and then filled in and
closed and i’ll just look at three of them and then six here
for six here we’ll look at something like it’s going straight across and then it

00:10
becomes a parabola like that all right so let’s look at those six
right there and pause the video if you need to and uh i’ll move up here
just so that you can see where to pause the video
uh after i move i’m gonna start talking here we go all right so here’s the zero
right here there’s one zero right there it’s in red
now this one right here again it’s got some isotopes and i want to sketch those
and i just want to pay attention and put some good detail in the graph right
there so there’s isotopes right there and what that means is this graph
continues to go down but it doesn’t cross and the same idea right here it keeps
getting closer and closer but it doesn’t cross
these graphs right here are incredibly important when you get to calculus
but for right now i’m just going to dash in the lines and call them isotopes and
so what we’re going to see is that this doesn’t cross and this doesn’t cross so

00:11
this one has no zeros this one has one zero and this one from
the previous video you might remember that this one right here does not pass
the vertical line test it’s not a function
number four right here is not a function also it fails the vertical line test
because it crosses twice or more so this one is not a function not a function
and by that i mean the graph of it does not represent a function
all right this right here has one zero so let’s mark it right here
there’s the one zero right there this uh number five here has infinitely
many zeros all these numbers right here that are right on top of the axis
let’s call it the x axis all these right here are zeros so
there’s infinitely many zeros right here
and this right here is uh also fails the vertical line test so this graph right
here does not represent a function right here
all right so there you go we got one we got none
we got two graphs that don’t represent functions we’ve got another one that’s

00:12
got one and here’s one that has infinitely many zeros
infinitely many zeros right there that are crossing the x-axis right there
all right so there’s some graphs there to help us
um get the idea let’s get this out of the way and move on
so in the next example here we’re going to um
work algebraically now so now let’s work algebraically so here number one
we’re going to start with this function right here we’re going to find the zeros
and so using algebra is going to be much more difficult
now for the examples that i’m going to give you they’re going to be workable
but i just want you to know that strictly speaking if you’re given a
a rule it can be very difficult to find the zeros in fact you may have to
approximate to find the zeros but for the exercises that we’re going to work
on it’s going to be pretty straightforward so

00:13
to find the zeros means we need to solve this equation right here
right we need to solve this right here to find those x’s the inputs where we
get out zero so i need to solve this right here i’m going to say this right
here is equal to zero so i’m going to say i’ll do it over here so 3x
squared plus x minus 10 is zero so i need to find those x’s so i’m going to
try to factor this so i’m looking at factors of 10 how about like 2 and 5
so i’ll go here with 5 and 2 and one of them needs to be negative one
of them needs to be positive we want a positive x so i’m going to go positive
6x and then minus 5x and that’ll give us a 5x in there so we’re going to get a 3
minus 5x equals 0 and we’re going to get an x plus two equals zero so we’re
looking at x equals minus two and here we’re looking at three x equals
five or in other words x equals five thirds so those would be the zeros

00:14
these are the zeros minus two if you plug in minus two
into this right here you get out zero if
you plug in five thirds into all of this you get out zero now if you plug in
minus two here then you’re just kinda wasting your time because this is equal
to this so it’s better to plug in your minus 2 here and then you can quickly
see that in fact you’re going to get out 0 here so
that’s good let’s move on to the next one right here g
so now we need to solve the problem here that g of x is equal to zero
so how do we solve this right so we have square root of minus 16x squared equals
zero that’s what g is so we need to solve this so in order for the square
root to be 0 whatever is underneath the radical must be 0.
so we’re going to go with 16 minus x squared is equal to 0

00:15
or if we move the x squared over 16 equals x squared or as i usually like to
write it x squared equals 16 and so we’re going to get x is plus or minus 4.
now another way to have done this would be to just have
multiplied through by a negative and said x squared minus 16 is zero
and then you just have factored this so x equals four and x equals minus four
so in either case you get the same zeros so there are the
zeros of g four and minus 4 are the zeros are very good so now let’s look at
h here so h here has um interesting we need to um solve this
equation here h of x equals zero to so to find the zeros we’re just solving

00:16
this equation we’re solving that equation we’re going to solve this
equation right here so we’re going to get 2x minus 3 over
x plus 5 equals to zero okay now this equation for um some people if
it’s been a while since you’ve been work solving equations especially with
rational expressions like this right here so this may be um a little bit
challenging for you but let me just kind of remind you of some things
what is 0 over 10 and so that is just 0. what is 0 over 100 that’s also 0
what is 0 over negative one thousand well i think you’re starting to get the
idea now if you’re going to have zero over here
you have to have zero up here it doesn’t matter what this is as long as it’s not
zero so in order to have zero right here in
order to have zero right here what do we need we need this numerator to be zero

00:17
so i’m gonna go find out where that is and if i move the three over
and simply divide by the two then well i’ll move it over here
so x equals in fact i’ll just put it right here x equals 3 over 2.
right so 3 over 2 makes the numerator 0 and if the numerator’s 0 then we’re
going to get out 0. that’s great except for the caveat that the
denominator cannot be 0 right because 0 over 0 is not 0. so i need to make sure
that that makes the numerator 0 but this also does not make the denominator zero
well three halves plus five is not zero so in fact we’re done this is the zero
this is a zero right here because it makes the numerator zero
which makes the whole function zero but it does not make the
denominator zero all right very good so there’s three examples there
there’s f there’s a g and there’s the h there’s three good examples there of

00:18
finding the zeros of a function uh you know using some algebra using some
problem solving skills but wait there’s more let’s do let’s do a couple more
maybe at least two more right so let’s look at something like
something maybe a little bit more challenging so we use f g and h let’s
use say a capital f and let’s say this is x over nine x squared minus four
all right so we need to solve so we’re going to find the zeros
so we need to solve this equation right here in other words we need to solve
the x over 9x squared minus 4 equals 0. we need to solve this right here
now in order to have zero over here the way to get that
is this numerator has to be zero well that’s actually easy to solve the
numerator which is just an x has to be zero

00:19
but we have the caveat of checking that the denominator um isn’t zero right here
so if we plug in zero we get minus four down here so in fact if we plug in this
zero right here we’re going to get 0 over
minus 4 and that is 0. so we got it this is our 0 right here
all right so i guess that wasn’t any more challenging but i think that’s
fun to fun to do um so let’s see how about this one right here
let’s say g of x is let’s look at something with a higher
power perhaps so let’s look at it next to the third and
yeah let’s look at this one right here so this right here is a third power so it
may be difficult to to work with but we’re going to find the zero so in other
words we need to solve this equal to zero we’re trying to find the zeros
so out of these first two right here i’m going to factor an x squared

00:20
and so that’ll be an x minus four and then out of these two right here i’m
going to factor a minus nine and then that will be an x minus 4 also
now let’s check that i factored that right x squared times x x to the third x
squared minus 4 minus 4x squared and then this will be minus 9x and then
minus 9 times -4 good now the reason why i did that is because
these factor right here out x minus 4 and then x squared minus 9.
so this is called factor by grouping right both of these terms right here
i’ll underline them this term and that term both have an x
minus 4 in them so we factor out the x minus 4. now we can continue factoring
this is a perfect square x minus three x plus three
and so now we’re left with the zeros x equals four three and minus three

00:21
so these are the zeros all right that was fun um [Music]
now before we do another one or or let me just end on this right here i guess
um if we’re given a function let’s say um i don’t know i’ll call it s um
or how about let’s call it capital h for hard and let’s say here this is an x
and let’s say this is x to the 107th power minus 2x to the
31st and then plus a 7x to the fifth and then minus let’s go 31.
so how do you find the zeros of that function there
well you just set it equal to zero and then you solve right when you say and
then you solve right that there’s a lot of difficulty behind that
that could be very difficult to solve now i’m not saying i can’t solve it

00:22
but let’s say i can solve that all right well let’s just put a 1 there then all
right or even let’s raise this to the 31st
power there so now the exponent is x to the
1071 to the 31 power right there that’s a huge number right there
so the fact of the matter is it’s all right let’s just put a 51 there so the
fact of the matter is it’s very easy to cook up a function
that it can be very hard to find the zeros of now because of that we’re going
to actually come up with another episode and we’re going to
see how to approximate the zeros using a
computer so even something like this can be done pretty quickly if we’re willing
to go to approximations and not have the exact answers so um
if we look at something like this right here so i’ll call it h of x

00:23
and i’ll say this is x squared minus seven right so what are the zeros of this
so we set this equal to zero and then we just solve so we move the 7 over
or i’ll say yeah move the 7 over and then we’re going to say plus or minus
square root of 7. so this is an exact solution we have a square root of 7
which is an irrational number and so you know we’re not going to be able to
write out the digits of this number right here like we can with these over
here right so but we can approximate them so we can
f for some functions we can find exact uh
the exact zeros but for other functions it’s very difficult to find the exact
zeros and you’re going to be wanting to approximate to find the zeros
so for these examples here yeah we’re able to find the exact zeros but for um
arbitrary functions if you just put down a function it can be very difficult to

00:24
find the zeros of that function all right so there we go i hope that you
enjoyed this video and i’ll see you in the next episode but i want to say thank
you for watching and i’ll see you then if you enjoyed this video please like
and subscribe to my channel and click the bell icon to get new video updates

About The Author
Dave White Background Blue Shirt Squiggles Smile

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.

Let's do some math together, today!

Contact us today.

Account

Affiliates

About Us

Blog

© 2022 DAVE4MATH.com

Privacy Policy | Terms of Service