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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

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
how do I attempted a trig number as a starter
Tumwe Reply
cos 18 ____ sin 72 evaluate
Het Reply
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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