What About the Range of a Function? (Sometimes Forgotten)

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

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

00:00
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

00:01
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

00:02
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

00:03
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

00:04
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

00:05
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

00:06
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

00:07
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

00:08
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

00:09
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

00:10
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

00:11
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

00:12
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

00:13
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

00:14
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

00:15
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

00:16
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

00:17
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

00:18
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

00:19
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

00:20
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

00:21
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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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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