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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

why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
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Kanniyappan Reply
Hey
Rightspect
hi
chesky
Give me algebra questions
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Vandna
What does this mean
Michael Reply
cos(x+iy)=cos alpha+isinalpha prove that: sin⁴x=sin²alpha
rajan Reply
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
is there any case that you can have a polynomials with a degree of four?
victor
***sscc.edu/home/jdavidso/math/catalog/polynomials/fourth/fourth.html
Oliver
can you solve it step b step
Ching Reply
give me some important question in tregnamentry
Anshuman
what is linear equation with one unknown 2x+5=3
Joan Reply
-4
Joel
x=-4
Joel
x=-1
Joan
I was wrong. I didn't move all constants to the right of the equation.
Joel
x=-1
Cristian
Adityasuman x= - 1
Aditya
y=x+1
gary
what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
Shadow Reply
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Please see ***imgur.com/a/lpTpDZk for solutions
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Marjun Reply
factor or use quadratic formula
Wilson
what is algebra
Ige Reply
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
Martin Reply
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
Yanah Reply
hi
John
hi
Grace
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John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
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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