1.51.5 Inverse Functions and Modeling. Quick Review

1.51.51.51.5

Inverse Functions and ModelingInverse Functions and Modeling

Quick Review

2

Solve the equation for .

1. 0.1 10

2. 1

33.

2

14.

2

5. 2, 2

y

x y

x y

xy

yx

y

x y y

Quick Review Solutions

2

2

Solve the equation for .

1. 0.1 10

2. 1

33.

2

14.

10 10

2

5. 2, 2

0

1

32

1 2

1

2, 2 a nd 0

y x

y x

yx

xy

x

y x y

y

x y

x y

xy

yx

x y x

y

y

What you’ll learn about• Inverse Relations• Inverse Functions

… and why Some functions and graphs can best be

understood as inverses of functions we already know.

Using a function to model a variable under observation in terms of another variable often allows one to make predictions in practical situations, such as predicting the future growth of a business based on data.

Inverse RelationThe ordered pair (a,b) is in a relation

if and only if the pair (b,a) is in the inverse relation.

Horizontal Line TestThe inverse of a relation is a function

if and only if each horizontal line intersects the graph of the original relation in at most one point.

Inverse Function

-1

-1

If is a one-to-one function with domain and range , then the

, denoted , is the function with domain and range

defined by ( ) if and only if ( ) .

f D R

f R D

f b a f a b

inverse

function of f

How to Find an Inverse Function Algebraically

-1

-1

Given a formula for a function , proceed as follows to find a

formula for .

1. Determine that there is a function by checking that is one-to-one.

State any restrictions on the domain of .

2. S

f

f

f f

f

-1

-1

witch and in the formula ( ).

3. Solve for to get the formula for ( ). State any restrictions on

domain of .

x y y f x

y y f x

f

Example Finding an Inverse Function

Algebraically-1 2

Find an equation for ( ) if ( ) .1

xf x f x

x

Example Finding an Inverse Function

Algebraically

-1 2Find an equation for ( ) if ( ) .

1

xf x f x

x

-1

2 Switch the and

1

Solve for :

( 1) 2 Multiply by 1

2 Distribute

2 Isolate the terms

( 2) Factor out

Divide by 22

Therefore ( ) .2

yx x y

y

y

x y y y

xy x y x

xy y x y

y x x y

xy x

xx

f xx

The Inverse Reflection Principle

The points (a,b) and (b,a) in the coordinate plane are symmetric with respect to the line y=x. The points (a,b) and (b,a) are reflections of each

other across the line y=x.

The Inverse Composition Rule

A function is one-to-one with inverse function if and only if

( ( )) for every in the domain of , and

( ( )) for every in the domain of .

f g

f g x x x g

g f x x x f

Example Verifying Inverse Functions

3 3Show algebraically the ( ) 2 and ( ) 2 are inverse functions.f x x g x x

Example Verifying Inverse Functions

3 3Show algebraically the ( ) 2 and ( ) 2 are inverse functions.f x x g x x

3

3 3

3 3 333

Use that Inverse Composition Rule:

( ( )) ( 2) 2 2 2 2

( ( )) ( 2) 2 2

Since these equations are true for all , and are inverses.

f g x f x x x x

g f x g x x x x

x f g

Example A Maximum Value Problem

A square of side inches is cut out of each corner of an 8 in. by 15 in. piece

of cardboard and the sides are folded up to form an open-topped box.

(a) Write the volume as a function of .

(b) Find th

x

V x

e domain of as a function of .

(c) Graph as a function of over the domain found in part (b) and use

the maximum finder on your grapher to determine the maximum volume

such a box can hold.

(d) How b

V x

V x

ig should the cut-out squares be in order to produce the box of

maximum volume?





x

V x





x

V x

(a) The width 8 2 and the length 15 2 . The depth is when the

sides are folded up.

8 2 15 2

x x x

V x x x




(b) Find the domain of as a function of .

x

V x




(b) Find the domain of as a function of .

x

V x

(b) The depth of must be nonnegative, as must the side length and width.

The domain is [0,4] where the endpoints give a box with no volume.

x




(c) Graph as a function of over the domain found

x

V x in part (b) and use






(c) Graph as a function of over the domain found

x

V x in part (b) and use



3

The maximum occurs at the point (5/3, 90.74).

The maximum volume is about 90.74 in. .




(d) How big should the cut-out squares be in order to

x

produce the box of

maximum volume?




(d) How big should the cut-out squares be in order to

x

produce the box of

maximum volume?

(d) Each square should have sides of one-and-two thirds inches.

Example Finding the Model and Solving

Grain is leaking through a hole in a storage bin at a constant rate of 5 cubic inches per minute. The grain forms a cone-shaped pile on the ground below. As it grows, the height of the cone always remains equal to its radius. If the cone is one foot tall now, how tall will it be in one hour?

Example Finding the Model and Solving

Grain is leaking through a hole in a storage bin at a constant rate of 5 cubic inches per minute. The grain forms a cone-shaped pile on the ground below. As it grows, the height of the cone always remains equal to its radius. If the cone is one foot tall now, how tall will it be in one hour?

2

3 3

The volume of a cone is 1/3 . Since the height always equals the radius,

1/3 . When 12 inches, the volume will be (1/3) 12 576 in. .

One hour later, the volume will have grown by (60 mi

V r h

V h h V

3 3

3

3

3

3

n)(5 in. / min) 300 in .

The total volume will be 576 300 in .

1/3 576 300

3 576 300

3 576 300

12.63 inches

h

h

h

h

Functions

Functions (cont’d)

Chapter Test

2

2

2

Find the (a) domain and (b) range of the function.

1. ( ) ( 2) 5

12. ( )

93. Is the following function continuous at 0?

2 3 if 0( )

3 if 0

4. Find all (a) vertical asymptotes and (b)

h x x

k xx

x

x xf x

x x

horizontal asymptotes

3of the function .

4

xy

x

Chapter Test

3

2

5. State the interval(s) on which is increasing.6

6. Tell whether the function is bounded above, bounded below or bounded.

6( )

17. Use a grapher to find all (a) relative maximum values and (

xy

xg x

x

3

2

-1

2

b) relative

minimum values. 3

8. State whether the function is even, odd, or neither.

3 4

69. Find a formula for . ( )

4

10. Find an expression for given ( ) and ( ) 4.

y x x

y x x

f f xx

f g x f x x g x x

Chapter Test Solutions

2

2

2

Find the (a) domain and (b) range of the function.

1. ( ) ( 2) 5

12. ( )

93. Is the following functio

(a)

n continuous at 0?

2 3 if 0( )

3

, (b) [5, )

(a) 3,3 (b) [1/3, )

if 0

h x x

k xx

x

x xf x

x x

4. Find all (a) vertical asymptotes and (b) horizontal asymptotes

3of the function .

yes

(a) 4 (b 34

) x yx

yx

Chapter Test Solutions

3

2

5. State the interval(s) on which is increasing. 6

6. Tell whether the function is bounded above, bounded below or bounded.

6( )

17. Use a grapher to find a

,

b

ll (a) relative max

ounded

xy

xg x

x

1

3

2

-1

imum values and (b) relative

minimum values. 3

8. State whether the function is even, odd, or neither.

3 4

69. Find a formula for

(a) 2 (b) 2

ev

.

en

( ( ) 6 / 4) 4

10. Find

y x x

y x x

f f xx

f x x

22an expression for given ( ) and ( ) 4. 4f g x f x x g x xx

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1.51.5 Inverse Functions and Modeling. Quick Review