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in the last episode i forgot to talk about the range of a function when

working through some examples in this episode i asked the question what about

the range let’s do some math [Music] okay we’re going to begin with a quick

review of what a function is a function from a set a to a set b is a relation

that assigns to each and every input x one and only one value output value y in

the set into sub b so in yeah in the last episode um i

forgot to mention uh the range on some of the examples we are working through

so i want to fix that real quick so on this episode we’re going to concentrate

on the range so the set a is called the domain and

the set b is called the codomain and the range are the set of output values so

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you may illustrate it something like this so here we got some set a and we

got some numbers in here or some elements and these are all getting mapped over

here to into a set b with this function here so f is mapping each one of these

uh elements over here in the set a and the outputs are over here

and so they’re getting mapped into into this set here and this right here is

called the range so the range of the output values

there may be more output val there may be fewer output values than input values

um so what’s important is that this goes to only one

but this one could also go to the same one and also the set b here is called the

codomain it could be larger than the range so the range is just a set of

output values and the set a is all of these are input values so each

input value everything in a has to get mapped to something

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and only one thing and those outputs values are collected together called the

range there all right so in this video um yeah in the

uh last video domain and range functions um we talked about um

finding the domain and so now i want to go through some examples and concentrate

on the range so let’s do that now let’s get rid of this here

and let’s find the uh range so uh we might remember this from the previous

episode we said what the domain was now let’s say what the range is so the range

of f range of f is the set and it’s going to be the output values

so zero four and two and so for number two um the range here

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is the set of output values and so what are the output values here

when we’re looking at g so when we look at the domain we are

looking at the inputs what can we input and so now the question will be what are

the outputs what is it possible to output so that’s a little bit hard to do

when you don’t have a graph or when you’re just trying to guess the

domain here so the range just takes a little bit more thinking because it’s

not just about the operations and what you could output

sorry what you can input now it’s collectively together what is the

possible output values so as x here is ranging through all real numbers

the minus 5 is also ranging through all real numbers

but then we’re going to say 2 divided by any real number except of course 0 down

here so this right here will be able to take on all real real numbers

except for the fact that we will not be able to get 0 as an output

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now you may ask how do you know you can’t get zero as an output well i know

this from two different ways so the first way is i’ll try to refresh your

memory on properties of zero so for example if we have zero over ten that’s zero

zero over a hundred that’s zero what is zero over minus one thousand

that’s still zero in other words to have zero you have to have zero up here on

the numerator and so this will never output the whole

thing will never be equal to zero because the numerator is not zero

their numerator cannot be zero no matter what the input is this numerator cannot

be zero here so the range is going to be um all real numbers

um except zero so i’ll put that in interval notation here

or you could put it in words all real numbers except zero

and that’s the range that’s the set of all output values now another way i know

this is true is just simply by looking at the graph of this so if we sketch the

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graph of this it’s been shifted by five units so we’re going to have an isotope

a vertical isotope at five and then we’re going to have some um

you know when when x is zero this is going to be minus two fifths so it’s

going to cut through there minus two fifths but anyways we’re also going to

have a horizontal isotope and so if you were given this graph of the function g

then you could more easily determine what the range is because the range is

going to be found just by squishing all of this so you want to

take all of this and squish it to the y-axis what part is covered so when i

project onto the y-axis all the whole y-axis is getting covered except for the

fact that there’s no y equals zero on this graph there’s nowhere where you can

input an x and output y zero okay so let’s go to number three so number three

now is uh the square root square root also is interesting or even roots

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so the range of h is and i’m going to say um

zero or sorry not zero but i i want the uh the outputs have to be um

greater than or equal to zero right because we’re taking the square root of

something that has to be zero or positive and so the square root of

something that’s zero or positive is also just zero or positive so i’m going

to say zero to positive infinity but is it possible that we could output

a zero yes in fact we could actually find what that input is

by solving this right here you know move the four over divide by three

so when x is four thirds if we input four thirds into here we get out the zero

and so the range is going to actually include 0 so i want to make that a

square bracket right there so that’s the range of h

now the range of v here so you might remember in the previous episode we

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interpreted this to be the formula for the volume of uh and

r is to be the radius and so we said last time that r was going to be

something that’s greater than or equal to 0 that was what the domain was going

to be so if we calculate all this up the

output values since r is greater than or equal to zero

if you cube anything that’s a greater than or equal to zero you get something

greater than or equal to zero and if you multiply by a constant four

pi over three times something that’s zero or greater you’re also going to get

something that’s zero or greater so the range on number four here the range of v

is also zero to positive infinity or if you wanted to put it in

that’s a v or if you want to put it in set notation you could say you know this

is equal to the set of real numbers such that um r

let’s use an r because i like to keep the variable the same

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r greater than or equal to zero so you could write it in set notation or

interval notation in fact we could have written these two in set notation also

okay so um let’s look at maybe a couple more examples here okay so

on this uh next example we’re going to look at

something like this let’s say here this is number five here let’s call this h of

t h of t is uh let’s go with four over t okay so now last time we talked about

the domain all real numbers except t equals zero

but now what about the range so this is also going to have the same problem that

we had in number uh two here with this function g of x with this function g

so here the t cannot be zero but the output cannot be zero either so the

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range of of h is so minus infinity uh to zero and then

union zero to positive infinity so in other words the output values can be

anything you want um except that you cannot get out 0 for

all this so for example if you say oh i don’t think the output

can be 1.2 can that be an output and you say oh yeah i can find the t for that

for this to be the output you’ll just simply you know multiply both sides by t

and then divide so this is the input right here that you

input to get out the 1.2 right if you input h of 4 over 1.2

is going to be 4 over t which is 4 over 1.2 and you simplify

this fraction you just get out 1.2 so this is the input for that output and so

in fact you can do this for any number that you choose any number y that you

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choose i’m just going to multiply by t on both sides and so that’ll just give

me yt and then i’ll divide by the y here and get four over y and so you can do

this as long as the y is not as long as the t is not zero and as long

as the y not zero so that’s why the range here is everything

except for the zero right there you can always find an input to get any

output you want as long as you’re trying to not get zero as an output you

will never get zero as an output all right so let’s look at another one

let’s say here g of y is square root of y minus 10

okay so um you know when we talk about the domain it’s going to be um you know

we’re we’re going to need everything under the square root to be greater than

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or equal to 0 so we’re going to need the y to be greater than or equal to 0

and so we’re going to say that this is 0 sorry 10

right move 10 over so this will be 10 to positive infinity that’s the domain any

if you put in 10 you’ll get an output and it’ll be a zero and if you put

anything greater than 10 in here this will be positive and you’ll be able to

take the square root of it so that’s the domain so the range here

is going to be so we can get a 0 as an output namely the 10 and

if if x is getting larger than 10 then we’ll be able to take the square

root of it so we’re going to go to positive infinity here so this will be

the range now in some upcoming episodes we’re

going to learn how to sketch the graph of these functions so if you could

visualize this graph it’s a lot easier to find the domain and range when you get

some graphs in your mind so for example i know exactly what this graph looks

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like it looks like a square root and then it’s been shifted because we have a

minus 10 here so we’ll cover that in some upcoming episodes so for right now

we’re just trying to purely look at a rule and try to figure out the domain and

range just by looking at what’s possible for inputs and what’s uh

you know what’s what has to happen as an output

all right so let’s look at one more here let’s go with f of x is equal to

square root of x plus 6 and then 6 plus x here

so let’s look at the domain and range for this function right here so

you know again we’re taking the greedy approach this is something i talked

about in the last video that whenever we write down this being a

function like this right here we’re making the implicit assumption that the

domain is the largest set possible of real numbers so what’s not possible

so minus six is not possible because we’re not going to be able to divide by

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zero so minus six is not possible what about

anything else for example can we try minus seven so minus seven works down

here because we’ll get a minus 1 and you

can always divide by minus 1 but minus 7 does not work here in the numerator

because minus 7 plus 6 that’s a negative 1 and that’ll be the square root of

negative 1 and this is a pre-calculus course in which we’re only going to be

interested in functions for of the real numbers here so

we need x plus 6 to be greater than or equal to 0

and so we need x to be greater than or equal to minus 6.

and can we include minus six well if we just look at the numerator we’ll answer

yes but when we look at the whole thing right we cannot divide by minus six

right so the domain will not include minus six

so the domain will be greater than minus six so i’ll say minus six open

and then rain and then infinity and then now what about the range right what

about the range so now i’m looking at the numerator the

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numerator here has to be zero or positive because we’re taking the square

root of it that’s always going to be positive but the denominator can be

negative or positive for example um well actually let’s see if that’s

true right if this is our domain right here we have to say negative 5 right

here negative 5 is certainly in this domain can we plug negative 5 into here

right so we have negative 5 here and then we have a plus 6 so that’ll be a

positive so we’ll be able to take the square root of it

what what about 6 minus five well that’ll be positive also so in fact this

whole thing will always be positive on this domain right here

given this domain right here this will always be positive right so

if we think about this right here is minus -6 and the domain here is

is open right so we’re not going to include -6 but it’s everything else

so now if you pick anything in the domain here even if you’re really close

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but we’re going to add 6 to it like minus 5.9 but we’re going to add 6 to it

that’s going to make it positive so the denominator is always positive and the

numerator is always positive so the whole expression there is always

positive so i’m going to say 0 to positive infinity and then can we

be actually equal to 0 no because the numerator would have to

be 0 but the only way to make that happen is that minus 6 but minus 6 is

ruled out by the denominator so this is going to be open right here

so there’s the domain and there’s the range right there

all right good so now let’s look at a couple more examples all right so

i think some examples we looked at last time were some equations

and we looked at something like y equals x squared which we know what it looks

like so it looks like this right here and we said what the domain was the

domain is all real numbers any any pick any x you want you can go

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plug it in you can find an output pick this x you can go find an output

any x you choose you can go find an output and you can only find one output

this is a function and what’s the range of this function right here so the range

is it starts here at zero and goes up and so think about the ranges projecting

onto the y-axis so if if you know any output that you

have project it to the y-axis what part of the y-axis is being covered so from

zero on up so the range is zero to positive infinity

so this is not part of the range right here minus two because there’s there’s

no corresponding part on the graph right this right here say uh a 2 4

and so four is part of the range here four has to be in the range zero has to

be in the range right okay so what about something like

this though y equals x to the third so that’s a cubic

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and that looks something like this it goes right through through the origin

right there and now when we project to the y-axis it’s

the whole y-axis is going to get covered now so if i take this point right here

on the graph i project it to the y-axis so boom there’s one number in the domain

if i go here and i project onto the y-axis there’s another number in the

domain if i take this on the graph and project to the y-axis the outputs so

there’s another number in the domain in fact if you just keep projecting onto

the y-axis you’ll find all the outputs and you realize it’s the whole real line

so the range here is all real numbers of course the domain is all real numbers

also you can pick any x you want and you’ll

get an output pick any x you want and you’ll get an output so for this

function right here the domain and range is all real numbers

all right so let’s look at something like um

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how about a line something like say y equals let’s say 2x minus 5.

so what about this function uh what about this right here so it is a linear

function and it’s going to represent a function the graph

and so let’s see here this is going through minus five right here and the

slope is two so it looks something like that roughly

and we can ask the question what is the domain

and what about the range right so if you

pick any x you’ll get an output pick any x you want you get only one output this

line keeps going right pick any x you’ll get only one output

so the domain is all real numbers so what about the range so

now i’m going to look at an output and see where it goes

here and i’m going to you know this line keeps going right so i’ll get another

one and i’ll get another one and i’ll get another one and if you keep looking if

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you keep thinking about it no matter which point you choose on the

line you’ll go over here and find the y value and the whole y axis is going to

get covered so all the outputs the range is all real numbers all right

um so maybe let’s look at um something we haven’t covered

uh how about something that looks like this how about something with the

negative on it yeah how about this one right here

so what about this function right here so this is a square root

but it has a negative sign in front of it so so the graph is going to look

something like this and it really helps to see the graph so

we’re going to we’re going to really focus on in the upcoming videos on how

to how visualize these graphs without having to plot points

but it does go through 0 0 right here the square root of 0

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is 0 and minus 0 is just the same thing as 0.

now the domain is all x is greater than or equal to 0 right so the domain is

i’m going to include 0 and i’m going to call it a function

let’s just give it a name f and what would be the range

and let me move out of the way here i’ll go up here and get small all right

so the range is um so the outputs are going to be negative

this will be the square root will be positive but this will be a negative

times a positive so the outputs will be negative and it’ll be

zero we could hit a zero right when we input a zero we get out of zero so we’re

going to include zero there so there’s the range right there the range is minus

infinity to zero and the domain is zero to positive infinity and we have a

function right here so if someone were to write this

equation down you can interpret a function out of it you can infer an a

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function out of it by using this rule right here and by using the greedy

approach for the domain the domain is the largest possible input values into

the into the rule all right so and then and then then you can figure out what

the range is all right so um that’s it for this video i hope that helps you

with a better understanding of the range and if you like this video maybe

make some comments let me know below that really helps out the channel i look

forward to seeing you in the next episode and have a great day

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