<< Chapter < Page Chapter >> Page >

Draw graphs of the functions f   and   f 1 from [link] .

Graph of f(x) and f^(-1)(x).
Got questions? Get instant answers now!

Is there any function that is equal to its own inverse?

Yes. If f = f 1 , then f ( f ( x ) ) = x , and we can think of several functions that have this property. The identity function does, and so does the reciprocal function, because

1 1 x = x

Any function f ( x ) = c x , where c is a constant, is also equal to its own inverse.

Access these online resources for additional instruction and practice with inverse functions.

Visit this website for additional practice questions from Learningpod.

Key concepts

  • If g ( x ) is the inverse of f ( x ) , then g ( f ( x ) ) = f ( g ( x ) ) = x . See [link] , [link] , and [link] .
  • Only some of the toolkit functions have an inverse. See [link] .
  • For a function to have an inverse, it must be one-to-one (pass the horizontal line test).
  • A function that is not one-to-one over its entire domain may be one-to-one on part of its domain.
  • For a tabular function, exchange the input and output rows to obtain the inverse. See [link] .
  • The inverse of a function can be determined at specific points on its graph. See [link] .
  • To find the inverse of a formula, solve the equation y = f ( x ) for x as a function of y . Then exchange the labels x and y . See [link] , [link] , and [link] .
  • The graph of an inverse function is the reflection of the graph of the original function across the line y = x . See [link] .

Section exercises

Verbal

Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?

Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that y -values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no y -values repeat and the function is one-to-one.

Got questions? Get instant answers now!

Why do we restrict the domain of the function f ( x ) = x 2 to find the function’s inverse?

Got questions? Get instant answers now!

Can a function be its own inverse? Explain.

Yes. For example, f ( x ) = 1 x is its own inverse.

Got questions? Get instant answers now!

Are one-to-one functions either always increasing or always decreasing? Why or why not?

Got questions? Get instant answers now!

How do you find the inverse of a function algebraically?

Given a function y = f ( x ) , solve for x in terms of y . Interchange the x and y . Solve the new equation for y . The expression for y is the inverse, y = f 1 ( x ) .

Got questions? Get instant answers now!

Algebraic

Show that the function f ( x ) = a x is its own inverse for all real numbers a .

Got questions? Get instant answers now!

For the following exercises, find f 1 ( x ) for each function.

f ( x ) = x + 3

f 1 ( x ) = x 3

Got questions? Get instant answers now!

f ( x ) = 2 x

f 1 ( x ) = 2 x

Got questions? Get instant answers now!

f ( x ) = x x + 2

f 1 ( x ) = 2 x x 1

Got questions? Get instant answers now!

f ( x ) = 2 x + 3 5 x + 4

Got questions? Get instant answers now!

For the following exercises, find a domain on which each function f is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of f restricted to that domain.

f ( x ) = ( x + 7 ) 2

domain of f ( x ) : [ 7 , ) ; f 1 ( x ) = x 7

Got questions? Get instant answers now!

f ( x ) = x 2 5

domain of f ( x ) : [ 0 , ) ; f 1 ( x ) = x + 5

Got questions? Get instant answers now!

Given f ( x ) = x 2 + x and g ( x ) = 2 x 1 x :

  1. Find f ( g ( x ) ) and g ( f ( x ) ) .
  2. What does the answer tell us about the relationship between f ( x ) and g ( x ) ?

a.   f ( g ( x ) ) = x and g ( f ( x ) ) = x . b. This tells us that f and g are inverse functions

Got questions? Get instant answers now!

Questions & Answers

how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
what is cell
faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
ok
Innocent
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
en français
Adama
which site have a normal flora
ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
Safaa
skin
Asiina
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
all
Tesfaye
by fussion
Asiina
what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
Micheal
what is inflammation
Shelly Reply
part of a tissue or an organ being wounded or bruised.
Wilfred
what term is used to name and classify microorganisms?
Micheal Reply
Binomial nomenclature
adeolu
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, 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