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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

find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
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Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
Give me the reciprocal of even number
Aliyu
The reciprocal of an even number is a proper fraction
Jamilu
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
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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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