<< Chapter < Page Chapter >> Page >

Using [link] , evaluate g ( f ( 2 ) ) .

g ( f ( 2 ) ) = g ( 5 ) = 3

Got questions? Get instant answers now!

Evaluating composite functions using formulas

When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.

While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition f ( g ( x ) ) . To do this, we will extend our idea of function evaluation. Recall that, when we evaluate a function like f ( t ) = t 2 t , we substitute the value inside the parentheses into the formula wherever we see the input variable.

Given a formula for a composite function, evaluate the function.

  1. Evaluate the inside function using the input value or variable provided.
  2. Use the resulting output as the input to the outside function.

Evaluating a composition of functions expressed as formulas with a numerical input

Given f ( t ) = t 2 t and h ( x ) = 3 x + 2 , evaluate f ( h ( 1 ) ) .

Because the inside expression is h ( 1 ) , we start by evaluating h ( x ) at 1.

h ( 1 ) = 3 ( 1 ) + 2 h ( 1 ) = 5

Then f ( h ( 1 ) ) = f ( 5 ) , so we evaluate f ( t ) at an input of 5.

f ( h ( 1 ) ) = f ( 5 ) f ( h ( 1 ) ) = 5 2 5 f ( h ( 1 ) ) = 20
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Given f ( t ) = t 2 t and h ( x ) = 3 x + 2 , evaluate

  1. h ( f ( 2 ) )
  2. h ( f ( 2 ) )

a. 8; b. 20

Got questions? Get instant answers now!

Finding the domain of a composite function

As we discussed previously, the domain of a composite function such as f g is dependent on the domain of g and the domain of f . It is important to know when we can apply a composite function and when we cannot, that is, to know the domain of a function such as f g . Let us assume we know the domains of the functions f and g separately. If we write the composite function for an input x as f ( g ( x ) ) , we can see right away that x must be a member of the domain of g in order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see that g ( x ) must be a member of the domain of f , otherwise the second function evaluation in f ( g ( x ) ) cannot be completed, and the expression is still undefined. Thus the domain of f g consists of only those inputs in the domain of g that produce outputs from g belonging to the domain of f . Note that the domain of f composed with g is the set of all x such that x is in the domain of g and g ( x ) is in the domain of f .

Domain of a composite function

The domain of a composite function f ( g ( x ) ) is the set of those inputs x in the domain of g for which g ( x ) is in the domain of f .

Given a function composition f ( g ( x ) ) , determine its domain.

  1. Find the domain of g .
  2. Find the domain of f .
  3. Find those inputs x in the domain of g for which g ( x ) is in the domain of f . That is, exclude those inputs x from the domain of g for which g ( x ) is not in the domain of f . The resulting set is the domain of f g .

Finding the domain of a composite function

Find the domain of

( f g ) ( x )     where f ( x ) = 5 x 1   and  g ( x ) = 4 3 x 2

The domain of g ( x ) consists of all real numbers except x = 2 3 , since that input value would cause us to divide by 0. Likewise, the domain of f consists of all real numbers except 1. So we need to exclude from the domain of g ( x ) that value of x for which g ( x ) = 1.

4 3 x 2 = 1 4 = 3 x 2 6 = 3 x x = 2

So the domain of f g is the set of all real numbers except 2 3 and 2. This means that

x 2 3 or x 2

We can write this in interval notation as

( , 2 3 ) ( 2 3 , 2 ) ( 2 , )
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 1

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask