Good day students welcome to mathgotserved.com in this clip were going to be going over how
to find the domain of composite functions right so let's go ahead and write down the
all instructions we are to find find the domain of F compose with G of X where F of X is equal
to three divided by X this to and G of X is for over X okay now before we go ahead and
solve this problem is about domain real quick Om domain basically involves the set of all
inputs that guarantees is defined or real outputs okay so any impact value in a function
that results in an imaginary or undefined outputs those values are normally excluded
from the domain of the function okay so for example if you have the square roots of a
negative number square of negative that yields an imaginary number right let's imaginary
and this is normally excluded from the domain also when you divide by zero what's your answer
division by zero result in undefined outputs so that is also on excluded from the domain
so if you have the square roots let's see how the square root of the function of functions
you a family have the situation you want to take erratic X you and stated to be greater
than or equal to zero F on so for the domain involving radicals if you have irrational
is expression let's see how Kubrick you what you do is you take the denominator and said
it not to be equal to zero I so these are have basically what some of the way to find
the domain of functions the other function out there to have restriction under domains.com
use of two common ones were going to be focusing on okay so let's take a look at this problem
now when you finding the domain of a composite function what you going to do is you going
to find the union on of two domain okay so there may procedure is to confine that you
neon of two things union the input function G and that of G and union of the domain started
domain of G union of the domain of G of G and the domain of the composite function F
of G of X right so basically have to problems in one first there were going to do is find
the domain restrictions G and there were going to compose this two functions and find the
domain of that that unites to two results and that will give us our domain our right
so let's break it up this is one in this is to so part one we going to find the domain
of the inner function domain of G now on earth so I do that G is the inner function well
you look at this notation right here at composing G of X weather that mean F compose with G
of X simply means that we taking G and plug-in it into F so this is F of G of X so we find
a of domain of the composition of two functions you take the inner function G of X and find
that domain verse okay so the inner function in this problem is former X so we need to
find the domain of over X G of X is former X now when you dealing with a rational function
which is a function the have the variable in the denominator what you simply do as indicated
here is you set the denominator equal to zero so the denominator here is X equal to zero
so this is a restriction this value results in undefined outputs of X cannot be equal
to zero right so this is one of the restrictions of our domain we done with the first parts
which is finding the domain of G what are we going to do next we going to find the domain
of F of G of X okay that's what were going to do next right so in other for us to do
that we need to find what F of G of X is okay so one of find the domain of the compose function
F of G of X okay so what is the, F of G of X is go ahead and figure out what that is
F of X the outer function is three divided by X plus two now to find F of G with that
substitute the inputs with another functions of the out X three divided by parenthesis
is plus two now let's take a minute a look at what just happened here the outer function
at X is here taking of the axes why did I do that while because I want to find F of
G of X which is the same unit F compose with G of X in the same thing so G is former X
so that the name of the function on the left in the volume the function on the right okay
so this is your compose function right here three divided by four over X plus two okay
now I can simplified is further but that's the message remember one we want to find the
domain so what is the domain of this function right here we have a rational function so
what you do is you take the denominator for over X this to the denominator cannot be what's
the denominator cannot zero okay so let's all this equation that will tell of the restriction
of X that we need to consider so here we going to subtract from both sides so subtract to
you have four over X equals negative two and then were going to set this over one so we
have to fractions it across multiply multiply what sites by X you going to end up with negative
X equals for divide both sides by negative to and you final answer is X equals negative
two write X equals negative two so this is the second constraint on your domain right
so X cannot be negative to or a to have zero in the denominator okay I so what is our final
answer remember we are to unites the domain of G and the domain of the compose function
okay so that X cannot be zero and X cannot be negative to so our domain is set X such
that X cannot be negative to and we dividing the union here so is of four and X cannot
be zero okay so this is the domain of our composite composite function F of G of X thanks
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