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[Music]
hello everybody welcome to another video
hope ur ready to flex those brain
muscles today we're going to talk more
about composite functions but this time
we're specifically talking about how to
find the domain of a composite function
so this is a very common issue that
students have with this cuz it's a
little bit tricky but basically we're
often given two functions F and G we're
asked to find the composite function but
also find the domain so let's go and get
into how to do this first how to find
the domain of a composite function our
first step identified domain
restrictions of the inner function

00:37

remember composite functions we have an
outer function and an inner function so
first we look at the domain of our inner
function and let's go ahead and do this
example so the domain of our inner
function is what we have no restrictions
here this domain is all real numbers so
I'm going to go and put that here okay
all real numbers second find the
composite function so basically I'm
evaluating this I'm plugging G of X into
F and I can simplify it and yeah through
that stuff so I'll go ahead and plug G
of X into F so I have F of well what is

01:09

G of X 2x minus 3 and we did this a lot
in the last video that's why I'm gonna
move a little quickly when I'm doing
this in this video because we did this
in the last video this video were just
focusing on domain so basically I square
this because I'm plugging this into my F
function so 2x minus 3 is getting
squared all right there's a shortcut for
this if you know how to do that in fact
I'm going to go ahead and do it a
squared is my first term 4x squared plus
2 a B which is negative 12x

01:41

let's see plus B squared that is 9 okay
so now where are we at we found the
composite function we identified domain
restrictions of this composite function
are there any domain restrictions in
this case there are not so my domain for
this function is all real numbers
or should I put that
and that means what we include both
restrictions well we don't have any
restrictions either way so our final
domain is all real numbers as well and
the reason I started with this example
is because it's the simplest of all the

02:13

cases right we have two functions that
both have a domain of all real numbers
so any composition function we make
using these two functions will also have
a domain of all real numbers okay so in
general if you have polynomials
quadratics linear functions anything
like that and you're composing domain
will be all real numbers it's other
stuff where we have problems that we'll
get in so there's those examples right
now all right let's go ahead and try
another example we're gonna find f of G
of X and we're gonna find its domain as
well so think about it the reason we
identify the restrictions of the inner
function first is because this inner

02:44

function happens first right I plug in a
number it has to go through this inner
function then it goes through the outer
function right
so if there's some number where this is
not defined or where this has issues
like you know negatives under square
root or zeros in the denominator like
this case right if I plug in zero to
this function it's undefined so zero is
not in the domain of this function so
that means I cannot plug zero into f of
G of X as well and that's why we do this
because every number has to go through

03:15

the inner function first so if there's
some values being excluded from the
domain and the inner function we need to
exclude them from the composite function
as well hopefully that makes sense so
what are the restrictions we've just
said X cannot equal zero now let's go
ahead and find this composite function
so I'm plugging G of X into F let's see
what does that give me so I replace
everywhere I see X I replaced with 2
over X right so I have 2 over X over 2
over X plus 3 all right if I'm plugging

03:48

G of X into F so what is this equal I
gotta do a little algebra let's see what
algebra can I
you I'll take away this equals so I can
multiply top and bottom by something and
there's multiple things you can do you
can combine these that's one thing you
can do I'm gonna multiply by x over 1
over x over 1 and the reason is because
that's going to get rid of all my
fractions basically look how this works
x over 1 over x over 1 that gets rid of
this X I'm left with two at the top x

04:20

over 1 times and again this is a
parenthesis so this is going to
everything x over 1 times 2 over X that
just gives me to my XS cancel again plus
X over 1 time times 3 that gives me 3x
and this is really my final function I
have 2 over 3x plus 2 so again I have
what I have potentially have 0 in the
denominator so I set the denominator
equal to 0 right and I solved for X and
those are the values that have excluding
from the domain of this composite

04:51

function along with X cannot equal 0 ok
so let's go ahead and set this equal to
0 2 plus 3x equals 0 subtract 2 from
both sides I have 3x equals negative 2
now I can divide both sides by 3 and I
have x equals negative 2 which means
what in our domain of our final
composite function X cannot equal 0 and
X cannot equal negative 2 over 3 so if I
were to write this in interval notation

05:24

it may get a little crazy here in
interval notation I have domain goes
from negative infinity all the way up to
negative 2/3 then it starts back up at
negative 2/3 is not including negative
2/3 and it goes up to what to 0 then it
starts back up as 0 and it goes all the
way up to infinity so this is the domain
in interval notation and it may be
acceptable as well just write X such
that X does not equal 0 and x2

05:55

equal negative two over three let's do
one more example I really encourage you
to pause the video try it on your own
we're gonna find f of G of X and its
domain so it's going to get started
first we identify domain restrictions of
the inner function our inner function G
of X we have a square root with X under
it which means what everything under the
square root we set greater than or equal
to zero we don't want to have any
negatives under our square root so I'm
gonna do this off to the side X minus 1
greater than equal to zero so I add 1 to

06:26

both sides of the inequality and I have
X is greater than equal to positive 1 so
this is my restriction from the domain
ok so whatever my final composite
function is and whatever its domain is
it has to include this restriction as
well as any other additional
restrictions from the domain of the
composite function itself so now what do
we do next find the composite function
so I plug G of X into F and what does
that give me well let's see I plug
square root of x minus 1 into this X so

06:57

I have square root of x minus 1 squared
plus 5 square root and squared those
kind of cancel each other out right I'm
left with X minus 1 plus 5 X minus 1
plus 5 hopefully I'll see how those
cancel each other out and now I can
combine these like terms and I'm left
with what X plus 4 and this is the most
common mistake made on these kind of
problems is people ignore that first
step they look at this they say oh
linear function it's clear that the

07:30

domain is all over the numbers and they
put all real numbers as their answer but
that's not true remember we plug in an
input this is a composite function so
that input has to go through the inner
function first this inner function is
restricted to values such that X is
greater than equal to 1 so that means my
composite function as well has to be
restricted so the domain is X such that
X is greater than equal to 1 that's one
way to write it
my currently brackets are a little ugly
sorry about that if I want to do
interval notation I do a bracket right

08:01

bracket because we're including negative
1 all the way up to infinity alright
last example just to get some more
practice I encourage you to pause it try
it on your own it's very similar to the
first example so let's go ahead and do
it first thing we do what we identify
the domain restrictions of the inner
function right
and again we're finding F of G of X as
well as its domain this is what we're
finding as well as its domain so our
inner function is G so we're gonna look
at what domain restrictions we have here
well we have X's in the denominator so I

08:32

can set my denominator equal to zero and
that would be the value that I've
excluded from the domain so when x
equals 2 I get 0 in the denominator so
for this domain of G of X I have X such
that X does not equal positive 2 ok
now I can move on to finding the
composite function I'm plugging this
function in to F so I have 1 over X over
X minus 2 plus 3 this may get a little
messy and again there are many different
ways you can do this as far as

09:04

simplifying it you could combine these
what I'm personally gonna do is I'm
gonna multiply top and bottom by let's
see X minus 2 over 1 over X minus 2 over
1 this is just another form of 1 right
so I'm still allowed to do this this is
legal and what this will do is get rid
of that denominator so let's go ahead
and do this X minus 2 times 1 that's
just X minus 2 up top X minus 2 cancels
with this X minus 2 I'm left with X X

09:34

minus 2 times 3 so I have plus 3 times X
minus 2 make sure to be real careful
with the parentheses and stuff this 3
has to go to everything so I can
simplify this again maybe I'll rewrite
it down here X minus 2 I'm basically
putting this 3 in that's my next step is
multiply this 3 out so I have 3x minus 6
so on the bottom I have X plus 3x minus
6
and on the top I still have X minus 2
and I just have one more step and that's

10:07

to combine the like terms and this is my
function f of G of X I have 4x minus 6
in the bottom so what is the domain of
this function let's see and you can even
factor out a 2 from the bottom will that
help it all that'll leave you with 2x
minus 3 no that will help you it also I
wouldn't even really bother the domain
of this function I basically set the
denominator equal to zero right same as
we've been doing 4x minus 6 equals 0 I
saw 4x I can add 6 to both sides and I

10:41

get 4x hopefully I can see this equals 6
now I can divide both sides by 4 and
I'll rewrite this up here I get x equals
6 over 4 which is 3 over 2 so X cannot
equal 2 because that's what we
restricted from an inner function and
from our final composite function we
have a restriction of x cannot equal 3
over 2 so our final domain if I even
have room to write it where can I write
it I'm going to put it into this box

11:11

domain X such that X does not equal 2
and X does not equal 3 over 2 hopefully
I can read that sorry it's a little got
a little messy but alright that's our
last example I'm hope this video helped
everyday hit the like button hit
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questions see in the next vid video keep
making those brain games

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