Warm UpWarm Up

Suppose that andSuppose that and find and find and

Suppose f x x( ) and g x x( )1

2. Findf g .

1

2 1)( 2 xxf xxg 3)( xfg 2g f

f g x f g x 12

fx

12x

12x

Suppose f x x( ) and g xx

( )1

2. Findf g .

Suppose that andfind

1)( 2 xxfxxg 3)(

xfg

xfgxfg

12 xg 13 2 x

33 2 x

Suppose that andfind

1)( 2 xxfxxg 3)(

2g f

2 2g f g f

2(2) 1g 3g

(3)(3) 9

5

Operations on FunctionsREVIEW Perform the indicated operation.

7 1 1f x x g x x .a h f g

Addition:

Subtraction:

Multiplication:

Division:

h x f x g x

h x f x g x

h x f x g x

f xh x

g x

7 1 1h x x x

8h x x

.b h f g 7 1 1h x x x 6 2h x x

c. h fg

7 1 1h x x x 27 6 1h x x x

. fd hg

7 11

xh xx

Composition: h x g f x

Inverse Functions Inverse Functions

Section 7.8 in textSection 7.8 in text

Many (not ALL) actions are Many (not ALL) actions are reversible reversible

That is, they undo or cancel each other That is, they undo or cancel each other

• A closed door can be opened A closed door can be opened • An open door can be closed An open door can be closed • $100 can be withdrawn from a savings $100 can be withdrawn from a savings

account account • $100 can be deposited into a savings $100 can be deposited into a savings

account account

NOT all actions are NOT all actions are reversible reversible

Some actions can Some actions can not be undone not be undone

• Explosions Explosions

• Weather Weather

Mathematically, this basic Mathematically, this basic concept of reversing a concept of reversing a calculation and arriving at calculation and arriving at an original result is an original result is associated with an associated with an INVERSE. INVERSE.

Actions and their inverses Actions and their inverses occur in everyday life occur in everyday life

Climbing up a ladder Climbing up a ladder Inverse: Climbing down a ladderInverse: Climbing down a ladder

Opening the door and turning on the Opening the door and turning on the lights lights

Inverse: Turning off the lights and Inverse: Turning off the lights and closing the door closing the door

A person opens a car door, gets in, and A person opens a car door, gets in, and starts the engine. starts the engine.

Inverse: A person stops the engine, Inverse: A person stops the engine, gets out, and closes the car door. gets out, and closes the car door.

Inverse operations can be Inverse operations can be described using functions. described using functions. Multiply x by 5 Multiply x by 5 Inverse: Divide x by 5Inverse: Divide x by 5

Divide x by 20 and add 10 Divide x by 20 and add 10 Inverse: Subtract 10 from x and multiply Inverse: Subtract 10 from x and multiply

by 20by 20

Multiply x by -2 and add 3 Multiply x by -2 and add 3 Inverse: Subtract 3 from x and divide by -2Inverse: Subtract 3 from x and divide by -2

NotationNotation

To emphasize that a function is an To emphasize that a function is an

inverse of said function, we use the inverse of said function, we use the same function name with a special same function name with a special notation. notation.

Function, f(x) Function, f(x) Inverse Function of f(x) = f Inverse Function of f(x) = f -1-1 (x) (x)

As we noted earlier, not every As we noted earlier, not every function has an inverse. So when function has an inverse. So when does a function have an inverse? does a function have an inverse?

In words: Each different input In words: Each different input produces its own different output. produces its own different output.

Graphically: Use the Horizontal Line Graphically: Use the Horizontal Line test. test.

15

Line Tests

Vertical Line Test on f:determines if f is a function

Horizontal Line Test on f:determines if f -1 is a function

f Function f Not a Function

Glencoe – Algebra 2Chapter 7: Polynomial Functions

How about if the function is given How about if the function is given numerically or symbolically,how do you numerically or symbolically,how do you determine its inverse?determine its inverse?

Numerically: Numerically: interchange domain interchange domain (x) and range ( f(x) ) (x) and range ( f(x) )

Symbolically:Symbolically: • Interchange x and y Interchange x and y

and solve for the and solve for the new y to obtain fnew y to obtain f-1-1(x) (x)

3 5y x 3 5x y

5 3x y 1 53 3

x y

3 5f x x

1 1 53 3

f x x

InterchangeSolve

Putting It All Together with Putting It All Together with Examples Examples

x f(x)

-1 -2

0 1

1 2

2 1

3 -2

4 -6

Does this table Does this table represent a represent a function? function?

Does this function Does this function have an inverse? have an inverse?

Find the inverse Find the inverse

Example Example x f(x)-3 10

-2 6

-1 4

0 1

2 -3

3 -10

Does this table Does this table represent a represent a function? function?

Does this function Does this function have an inverse? have an inverse?

Find the inverse Find the inverse

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

Does this equation Does this equation represent a function? represent a function?

How do you know? How do you know? Does this function have Does this function have

an inverse?an inverse? How do you know? How do you know? Find the inverse Find the inverse Confirm the inverseConfirm the inverse

f(x) = xf(x) = x33 + 1 + 1

Does this equation Does this equation represent a function? represent a function?

How do you know? How do you know? Does this function have Does this function have

an inverse?an inverse? How do you know? How do you know? Find the inverse Find the inverse Confirm the inverseConfirm the inverse