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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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