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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

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
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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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