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f ( x ) = 9 + 2 x 3

f −1 ( x ) = ( x 9 2 ) 3

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f ( x ) = 2 x + 8

f −1 ( x ) = 2 8 x x

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f ( x ) = x + 3 x + 7

f −1 ( x ) = 7 x 3 1 x

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

f −1 ( x ) = 5 x 4 4 x + 3

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f ( x ) = 5 x + 1 2 5 x

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f ( x ) = x 2 + 2 x ,   [ −1 , )

f −1 ( x ) = x + 1 1

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f ( x ) = x 2 + 4 x + 1 ,   [ −2 , )

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f ( x ) = x 2 6 x + 3 ,   [ 3 , )

f −1 ( x ) = x + 6 + 3

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Graphical

For the following exercises, find the inverse of the function and graph both the function and its inverse.

f ( x ) = x 2 + 2 , x 0

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f ( x ) = 4 x 2 , x 0

f 1 ( x ) = 4 x

Graph of f(x)=4- x^2 and its inverse, f^(-1)(x)= sqrt(4-x).
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f ( x ) = ( x + 3 ) 2 , x 3

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f ( x ) = ( x 4 ) 2 , x 4

f 1 ( x ) = x + 4

Graph of f(x)= (x-4)^2 and its inverse, f^(-1)(x)= sqrt(x)+4.
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f ( x ) = 1 x 3

f 1 ( x ) = 1 x 3

Graph of f(x)= 1-x^3 and its inverse, f^(-1)(x)= (1-x)^(1/3).
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f ( x ) = x 2 + 4 x , x 2

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f ( x ) = x 2 6 x + 1 , x 3

f 1 ( x ) = x + 8 + 3

Graph of f(x)= x^2-6x+1 and its inverse, f^(-1)(x)= sqrt(x+8)+3.
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f ( x ) = 1 x 2 , x 0

f 1 ( x ) = 1 x

Graph of f(x)= 1/x^2 and its inverse, f^(-1)(x)= sqrt(1/x).
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For the following exercises, use a graph to help determine the domain of the functions.

f ( x ) = ( x + 1 ) ( x 1 ) x

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f ( x ) = ( x + 2 ) ( x 3 ) x 1

[ 2 , 1 ) [ 3 , )

Graph of f(x)= sqrt((x+2)(x-3)/(x-1)).
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f ( x ) = x ( x + 3 ) x 4

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f ( x ) = x 2 x 20 x 2

[ 4 , 2 ) [ 5 , )

Graph of f(x)= sqrt((x^2-x-20)/(x-2)).
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f ( x ) = 9 x 2 x + 4

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Technology

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y -coordinates given.

f ( x ) = x 3 x 2 , y = 1 , 2 , 3

( 2 ,   0 ) ;   ( 4 ,   2 ) ;   ( 22 ,   3 )

Graph of f(x)= x^3-x-2.
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f ( x ) = x 3 + x 2 , y = 0 , 1 , 2

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f ( x ) = x 3 + 3 x 4 , y = 0 , 1 , 2

( 4 ,   0 ) ;   ( 0 ,   1 ) ;   ( 10 ,   2 )

Graph of f(x)= x^3+3x-4.
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f ( x ) = x 3 + 8 x 4 , y = 1 , 0 , 1

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f ( x ) = x 4 + 5 x + 1 , y = 1 , 0 , 1

( 3 ,   1 ) ;   ( 1 ,   0 ) ;   ( 7 ,   1 )

Graph of f(x)= x^4+5x+1.
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Extensions

For the following exercises, find the inverse of the functions with a , b , c positive real numbers.

f ( x ) = x 2 + b x

f 1 ( x ) = x + b 2 4 b 2

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f ( x ) = a x + b 3

f 1 ( x ) = x 3 b a

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Real-world applications

For the following exercises, determine the function described and then use it to answer the question.

An object dropped from a height of 200 meters has a height, h ( t ) , in meters after t seconds have lapsed, such that h ( t ) = 200 4.9 t 2 . Express t as a function of height, h , and find the time to reach a height of 50 meters.

t ( h ) = 200 h 4.9 , 5.53 seconds

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An object dropped from a height of 600 feet has a height, h ( t ) , in feet after t seconds have elapsed, such that h ( t ) = 600 16 t 2 . Express t as a function of height h , and find the time to reach a height of 400 feet.

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The volume, V , of a sphere in terms of its radius, r , is given by V ( r ) = 4 3 π r 3 . Express r as a function of V , and find the radius of a sphere with volume of 200 cubic feet.

r ( V ) = 3 V 4 π 3 , 3.63 feet

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The surface area, A , of a sphere in terms of its radius, r , is given by A ( r ) = 4 π r 2 . Express r as a function of V , and find the radius of a sphere with a surface area of 1000 square inches.

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A container holds 100 mL of a solution that is 25 mL acid. If n mL of a solution that is 60% acid is added, the function C ( n ) = 25 + .6 n 100 + n gives the concentration, C , as a function of the number of mL added, n . Express n as a function of C and determine the number of mL that need to be added to have a solution that is 50% acid.

n ( C ) = 100 C 25 .6 C , 250 mL

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The period T , in seconds, of a simple pendulum as a function of its length l , in feet, is given by T ( l ) = 2 π l 32.2 . Express l as a function of T and determine the length of a pendulum with period of 2 seconds.

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The volume of a cylinder , V , in terms of radius, r , and height, h , is given by V = π r 2 h . If a cylinder has a height of 6 meters, express the radius as a function of V and find the radius of a cylinder with volume of 300 cubic meters.

r ( V ) = V 6 π , 3.99 meters

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The surface area, A , of a cylinder in terms of its radius, r , and height, h , is given by A = 2 π r 2 + 2 π r h . If the height of the cylinder is 4 feet, express the radius as a function of V and find the radius if the surface area is 200 square feet.

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The volume of a right circular cone, V , in terms of its radius, r , and its height, h , is given by V = 1 3 π r 2 h . Express r in terms of h if the height of the cone is 12 feet and find the radius of a cone with volume of 50 cubic inches.

r ( V ) = V 4 π , 1.99 inches

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Consider a cone with height of 30 feet. Express the radius, r , in terms of the volume, V , and find the radius of a cone with volume of 1000 cubic feet.

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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
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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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