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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   or ( f g ) ( x ) = 5 x

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 ] .

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Find the domain of

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

[ 4 , 0 ) ( 0 , )

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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
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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

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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

Verbal

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.

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Questions & Answers

what are you up to?
Mark Reply
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
hi Miranda
jai
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Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
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Nokwanda
C'est comment
Abdel
Hi
Amanda
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SORIE
Hiiii
Chinni
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ANSHU
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Chinni
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Abdel
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Yaona
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
Practice Key Terms 1

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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