Finding the Domain and Range of a Function (The Greedy Approach)

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 i explain what the domain and range of a function is
and we’ll practice finding the domain and range for a given rule let’s do some
math [Music] okay well so we’re gonna begin um where
we were in the last video uh introduction to functions
um and we’re gonna quickly explain what a function is just to kind of review it
and then we’re going to concentrate on finding the domain and range
so a function is a relation between two sets a and b
and the assignment works from assigning each element in x
to exactly one element in the set b which we’re going to call y
and the set a is called the domain or the set of inputs of the function
and the set b contains the range or the set of outputs

00:01
so for each input there’s exactly one output the inputs are
denoted usually by an x and the outputs are denoted by y’s
although sometimes you’ll use different letters here for the variables but you
have an independent variable and you have a dependent variable all right so
this is sort of the definition but what i wanted to point out for us is the
fact that whenever you’re defining a function you um need to specify the domain
and that’s important because you have to check that each and every input in the
domain so you cannot specify a function without also specifying the domain
so you know it’s like you try to define a function but if you don’t say what the
domain is then you really haven’t defined what a function is you have to
define both of these at the same time in order to have a function for example if

00:02
i have this function right here x squared and then i say the domain is
all of the real numbers or if i say this is my function here
and then i say this the domain is say minus one to one
um and this is enclosed interval notation but these are two different functions
because they have different ranges oh sorry because they have different
domains the domain of this one is all real numbers the domain of this function
is all real numbers now if someone just writes down this right here and they
don’t explicitly say exactly what the domain is
then they’re using an implicit inception that the domain is as large as possible
and so we would look at something like this a rule like this and then we would
say oh the domain is all real numbers and it’s the domain is only something
else unless you explicitly say what the restriction is and so then we can
consider this a different function and you would try to use a different letter

00:03
like a capital left or something like that so that you could specify the
difference so that you could talk about the two different functions okay so
let’s start looking at the domain and range and
find the domain of the function so the first one we’re going to look at is a
set of ordered pairs by the way before we get started you
know if you’re not clear on the definition of what a function is then
you should definitely check out the last video so
this video or this episode is part of the series functions and their graphs
step-by-step tutorials for beginners so you should definitely check out the link
below in the description for the full series all right so here we go find the
domain of this function so this function is pretty small it’s just containing
three different points so if you were to graph it it would just be three points
that’s all it would be but to find the domain and since it’s

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finite and since it’s just three the domain is actually very easy to find so
i’m going to say the domain of f is and now i’m going to use set notation to
collect together all the domains so you have to remember the domain is a set so
i can’t just list them so minus three minus one minus two that would just be a
list that’s not the domain the domain is a set
consisting of the inputs minus three minus one and minus two
all right next example so now this is g of x now this one is given by rule
and so in order to call this a function right here what we’re doing is we’re
implicitly assuming the domain is largest possible so i’m going to say the
domain of g is and now i’m going to make a set
now the set that i’m going to make up here is can be
communicated in different ways we can just use plain old set notation or we

00:05
can use interval notation or in fact we could use graphical notation so i’ll
show you all three here but to find the domain
what we need to realize is that we’re taking a greedy approach the direct the
domain is as large as possible now when i input an x into here what is
is it that we cannot do what is it that we can and cannot do
well you can subtract anything from 5 any number you want you could subtract a
5 from it but the thing that you cannot do is
divide by zero you probably heard that expression before so
the x value of 5 would not be acceptable but any other value will not give us 0
down here so any other value we can divide by it and we can sim
uh certainly multiply by 2 any number by 2. so i’m going to say the domain is um
so we can put this in words so i’m going to give different

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expressions of saying that i’ll say here’s a all real numbers except x equals 5
or perhaps just five another way to say this would be
using set notation so we could say all real numbers except five
or we could say um the set of all real numbers
with the condition that x is not five that’s a little five there
or d so so all these are exactly the same meaning
um or we could use interval notation so we could say minus infinity to 5
union and then 5 to positive infinity now whenever we use interval notation
we’re going to make sure that if we have infinity we don’t include it
so these are rounded here and we don’t include the five so these

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are rounded here and this and this is the symbol for
union set union so another way to do this would be to a graphical so let’s
say this is the real line and somewhere on here is a five so i’ll
put a five right here and i’m going to put a circle through
five representing the fact that five is not part of the domain and then i’m
going to shade this part in here and we can shade it in let’s say with the blue
and let me get out of the way here so we can shade this in with the blue
right here there we go so that would be like a graphical
representation of the domain so the domain is and then you can make
your choice about which one you would like to respond with all right so for
part c or part three so now our function is h and let’s see here
for h here we’re going to notice that we cannot take the square

00:08
root of a negative number um you know because we’re assuming that
we’re only going to be working with the real numbers this is a precalculus
series so we’re going to assist that 3x minus 4
is greater than 0 so that we can take the square root of it but in fact we can
also take the square root of 0 so i’m going to include a 0 here
and so now we can just kind of isolate the x we can move the 4 over
we can divide by 3. and so now what we’re looking at is in
order for this to be true we need x to be greater than or equal to four thirds
and so we can represent that in a variety of ways we could represent it
graphically or you know each of these ways we could
represent the domain so i’m going to say over here um actually let’s go
i’ll move down here and let’s put h up here so i’ll say h of x is

00:09
uh the domain of h the domain of h is and so
we can say all real numbers except four thirds so that would be exactly oh sorry
all real numbers greater than or equal to four thirds so i’ll just write that
out all real numbers greater than or equal to four thirds
all real numbers greater than or equal to four thirds
all right uh and then that should say four thirds here
and then uh okay so let’s just get rid of this uh f right here and
in fact i’ll just get rid of all this over here so that would be one way to say
um so another way is just to write it all real numbers
such that x is greater than or equal to four thirds
you know one of my personal favorites um or we can write it in interval

00:10
notation so we can say so since it’s equal we can say four thirds
and then positive infinity and this one is open and this one is closed make sure
this one’s close because we can include four thirds uh some people don’t use the
positive symbol in front of that but i always do
um so positive infinity all right so there we go so there’s three different
ways you can write the the domain for h here let’s look at another one here so
let’s look at this right here so now i’m using r as my independent variable and
i’m looking at this expression right here and again we’re going to take the
greedy approach what is the domain for this rule right here it’s just a rule
even though we’re using function notation of a rule what that’s telling
us is that the domain is as large as possible so what are all the possible r
values that make sense for this now v is going to represent volume so v is

00:11
volume and r is radius so in this example 4 here we have a
physical interpretation of the rule and that’s going to give us boundary
conditions that’s going to give us a boundary so we want the volume to be
positive we want the radius to be positive perhaps the radius could be
zero in which case we would have zero volume so that makes sense so i’m going
to say the domain of v is uh all real numbers uh greater than or equal to zero
and here i’m using an x and probably should match it with an r
just to make just to make it look nice let’s just say all real numbers are such
that r is greater than or equal to zero and this is because i’m assuming that
this rule right here is uh has a physical interpretation and
i’m using that interpretation to restrict the domain if you don’t have
that interpretation if this is just a regular formula that has no

00:12
interpretation at all then actually r can be any real number so it just
depends on how you interpret this rule and what and how greedy do you want to
be so the domain could be written like this right here so
for these examples up here i think it’s pretty clear that we just go as big as
possible for the domain this one right here though
it could be interpreted this way right here all right so let’s look at some more
examples so let’s look at this function right
here so i’ll call this one right here example 5.
let’s look at this function right here we have 5x squared and then plus 2x and
then minus 1. and so what’s the domain so i’m going to give the domain here in
interval notation and set notation just so that you get familiar with those two
and i’m not going to go through the whole complete list every time because
you can always write the domain in words and you can always make a graphical

00:13
representation of your domain but for these examples here i just want to give
the set notation and the interval notation now when we input an x there’s
no restriction at all about what you can and cannot input and the reason why i
say that is because if i input an x well i can square any number at all any real
number that you give me i can square it i can multiply 5 by any real number i
can multiply by 2 by any real number i can add to any real numbers i can
subtract 1 from any real number so there’s no restriction at all so the
domain of f is all real numbers or we could write it like this or say
minus infinity to positive infinity if we want to write it in interval notation
so let’s look at another example let’s look at something like h of t equals
4 over t so what would be the domain of this function here h

00:14
so again we’re using function notation because we’re
explicitly assuming that the domain is as large as possible now i have division
right here and we know that we cannot divide by zero so t cannot be zero so
i’m going to say the domain of h is all real numbers so x all real numbers
or let’s match our variable here let’s say t
technically you could use x or t or any any variable here that you want
but i usually like to match so t is a real number but t is not zero
all right and then we can write it in in interval notation so let’s say that’s
minus infinity but do not include zero so i’m going to curve these
all right there we go so let’s do another one um let’s do one where

00:15
we have something to worry about in the numerator and the denominator
so let’s look at x plus 6 and then 6 plus x
so how about this function right here f so let’s say the domain of f is so
when i’m looking at the numerator here i
have to say that it has to be minus 6 or greater
and the reason why i say that is because i know that x plus 6 has to be greater
than or equal to zero whatever is under the radical whatever it is all of it has
to be greater than or equal to zero so x has to be greater than or equal to
minus six now in the denominator we have to rule out the case of minus 6
because -6 will make division by zero right so we have to include we have to
exclude the equal sign here so i’m going to go with this right here
so i’m going to say the domain of f is all real numbers x such that or such that

00:16
x is greater than minus 6. or in interval notation minus interval um
minus infinity to minus six but do not include minus six
minus six to positive infinity here all right there we go
so there’s three more examples there um yeah so you know when you’re working
with uh finding the domain there’s two big things you need to look out for is
uh square roots and it’s not just square roots though it’s really even roots
right if this had been a fourth root here uh the domain is really not going to
change because you still need whatever is under the radical to be greater than
or equal to zero but if this had been a cube root
um then we would not have to have that uh worry all right so
uh there’s three more examples there um let’s see what’s next

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all right so um now uh in this next example here um let’s erase this first here
now let’s go back to equations here and let’s ask the question
is is this even a function and if it is a function then we’ll find
the domain and the range all right so let’s look at our first example right
here so let’s call this example a and it’s just going to be something
simple like y equals x squared right so is this equation does this equation
represent a function and if it is what is that function
now when i ask that question i’m going to be assuming again the greedy approach
if the equation does represent a function the domain of that function
we’re implicitly assuming is as large as possible so when i look at this when i
look at something like this i’m going to say yes this is a function and the
reason why is because any input that i give there’s only going to be one y

00:18
output and the reason why i know that is because it’s already solved for y
and whatever you plug in here you calculate it all up and you get one y out
so this is going to represent a function and i’m going to name that function and
we can name it whatever we’d like because we’re given the equation and not
a function and so we’re going to ask this question
if the relation defines y as a function of x yes it does so i name my function
and i’m going to state the domain so the domain is all real numbers domain is
all real numbers here all right so that that’s a easier
example let’s look at something more a little bit more interesting
how about let’s look at something like let’s switch it up and say x equals
so is this equation right here going to define a function and so the question is
for each input and we’re saying right here the x’s or the inputs

00:19
um is their unique output so for example we’re looking at the sixth power here
right um so let’s see here um what is the all right so let’s say here we have um
two to the sixth and uh so let’s see here we have 64 and
let’s say here we have y to the sixth so how many so x is the input
so 64 is our input and the question is how many outputs how many
outputs i chose 64 because it’s a easy power of six it’s two to the six right
two to the fifth is 32 two to the six is 64. in any case um
how many y’s are there how many outputs are there for this input how many

00:20
outputs are there so i could say 2 y equals 2 to the 6 is 64.
so 2 to the 6 is 64. but so is minus 2 to the sixth
if you do minus two minus two times minus two times minus two six times
you’re gonna still get sixty-four right so there’s gonna be two outputs and so
we’re gonna say this is uh does not represent a function so does not
represent a function and the way that you show that it does
not represent a function is you come up with one input and it can be whatever
you want you don’t have to choose 64. but you come up with one input where you
get two outputs so the output would be um you know the 2 or 64. now
let’s go here to another one let’s look at something like how about
just a regular old line right so minus 6x plus 4 and this is a function

00:21
so it is a represents a function it’s not a function yet
to be a function we need to um specify the domain so i’m going to use
function notation i’m going to say just a regular f f of x is minus six x plus
four and what’s the domain so the domain here is all real numbers
so the domain is all real numbers and the reason why i
know that’s because you can multiply minus 6 by anything and you can add 4 to
anything any real number you can multiply minus 6 times any real number
and you can add any real number you can add a 4 to it so there’s no restriction
upon what you can input in other words the domain is all real numbers here
all right so there’s three more examples let’s look at uh let’s see let’s see we
can fit in another example here let’s say let’s look at something like this
this equation right here so this the question is does this
equation right here represent a function can we turn it can we get a function

00:22
from it so we’re going to look at all these
points right here x and y’s that satisfy this all the ordered pairs that satisfy
this this is a relation or this represents a relation
um but the only thing that we cannot do here is five we cannot input a five if x
is five we’ll get minus seven over zero which is division by zero right so here
we’re going to say our function and just to give it a different name i’ll just
say it’s g of x so our function is minus 7 over x minus 5 and the domain is
all real numbers x such that x is not 5. or interval notation
will not include the 5. all right so that’s a 5. all right so
there we go this equation allows us to define a function um

00:23
each input gives one output no matter what input i give for x i calculate it
up and i’ll get an output except for five five is the only one where you not
get a unique output in fact you would not get any output at all
all right so let’s see here let’s just do one more here e
and then this will say be y equals and let’s go with square root of 7 minus 2x
so does this right here define a function
or can we use this equation to define a function
and so the answer is yes if i input an x i calculate the whole right side up and
we’ll get a unique output right there and so what is the domain of this
function right here so we’re going to go 7 minus 2x needs to be greater than or
equal to 0. again we’re taking square roots so everything under the root
has to be greater than or equal to zero so now i’m going to solve for the x
because i want to know what x’s can i input that’s why we know we need to

00:24
solve for x because i’m asking the question what is the domain in other
words what are all the x’s that we can substitute in all right so here what we
get minus 2x greater than or equal to minus 7 and then we’re going to say
2x is less than or equal to 7 in other words x is less less than or equal to
seven over two for example zero zero is less than seven over two we
can substitute in zero we get square root of seven right anyways here’s the
domain so domain of let’s call it h is so we’re going to say h of x is
7 minus 2x so we’re defining a function here and
we’re going to say what the domain is so it’s all real numbers such that
x is great less than or equal to 7 over 2 or
we can write it in interval notation 7 two

00:25
can we actually include seven over two yes if we input seven over two the twos
will cancel we’ll get seven minus seven square root of zero that’s fine square
root of zero is zero and there we go oops make sure it looks like it’s um
open here all right there we go so that’s open right there i guess i should
make that uh readable 7 over 2 and then plus infinity all right so
hopefully you can read that there all right so i hope that’s helpful uh
finding the domain and range and uh i hope you like this video hope you see
you in the next one and have a great day if you enjoyed this video please

00:26
like and subscribe to my channel and click the bell icon to get new video
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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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