<< Chapter < Page Chapter >> Page >

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 ] .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the domain of

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

[ 4 , 0 ) ( 0 , )

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
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
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
Practice Key Terms 1

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask