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Finding the domain of a composite function involving radicals

Find the domain of

( f g ) ( x )   where f ( x ) = x + 2   and   g ( x ) = 3 x

Because we cannot take the square root of a negative number, the domain of g is ( , 3 ] . Now we check the domain of the composite function

( f g ) ( x ) = 3 x + 2

The domain of this function is ( , 5 ] . To find the domain of f g , we ask ourselves if there are any further restrictions offered by the domain of the composite function. The answer is no, since ( , 3 ] is a proper subset of the domain of f g . This means the domain of f g is the same as the domain of g , namely, ( , 3 ] .

Find the domain of

( f g ) ( x )   where f ( x ) = 1 x 2   and   g ( x ) = x + 4

[ 4 , 0 ) ( 0 , )

Decomposing a composite function into its component functions

In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function , so we may choose the decomposition that appears to be most expedient.

Decomposing a function

Write f ( x ) = 5 x 2 as the composition of two functions.

We are looking for two functions, g and h , so f ( x ) = g ( h ( x ) ) . To do this, we look for a function inside a function in the formula for f ( x ) . As one possibility, we might notice that the expression 5 x 2 is the inside of the square root. We could then decompose the function as

h ( x ) = 5 x 2  and  g ( x ) = x

We can check our answer by recomposing the functions.

g ( h ( x ) ) = g ( 5 x 2 ) = 5 x 2

Write f ( x ) = 4 3 4 + x 2 as the composition of two functions.

Possible answer:

g ( x ) = 4 + x 2
h ( x ) = 4 3 x
f = h g

Access these online resources for additional instruction and practice with composite functions.

Key equation

Composite function ( f g ) ( x ) = f ( g ( x ) )

Key concepts

  • We can perform algebraic operations on functions. See [link] .
  • When functions are combined, the output of the first (inner) function becomes the input of the second (outer) function.
  • The function produced by combining two functions is a composite function. See [link] and [link] .
  • The order of function composition must be considered when interpreting the meaning of composite functions. See [link] .
  • A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function.
  • A composite function can be evaluated from a table. See [link] .
  • A composite function can be evaluated from a graph. See [link] .
  • A composite function can be evaluated from a formula. See [link] .
  • The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function. See [link] and [link] .
  • Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions.
  • Functions can often be decomposed in more than one way. See [link] .

Section exercises


How does one find the domain of the quotient of two functions, f g ?

Find the numbers that make the function in the denominator g equal to zero, and check for any other domain restrictions on f and g , such as an even-indexed root or zeros in the denominator.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
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Berger describes sociologists as concerned with
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Source:  OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1
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