obj. 10 inverse functions (presentation)

8/3/2019 Obj. 10 Inverse Functions (Presentation)

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Obj. 10 Inverse Functions

Unit 3 Rational Functions


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Concepts & Objectives

Inverse Functions (Obj. #10)

Review composition of functions Identify 1-1 functions

Find the inverse of a 1-1 function


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Composition of Functions

Iffandg are functions, then the composite function, or

composition, ofg andfis defined by

The domain ofg

fis the set of all numbersxin thedomain offsuch thatf(x) is in the domain ofg.

( )( ) ( )( )=g f x g f x


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Example: Let and .

a) Find

b) Find

( ) = 2 1 f x x ( ) =

4

1g x

x

( )( ) 2f g

( )( ) 3g f

( ) = =

42 4

2 1

g ( )( ) ( )( ) ( )= = 2 2 4 f g f g f

( )= =2 4 1 7

( ) ( ) = = 3 2 3 1 7f

( ) = = =

4 4 17

7 1 8 2g


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Example: Let and .

c) Write as one function.

( ) = 2 1 f x x ( ) =

4

1g x

x

( )( ) f g x

( )( ) ( )( )f g x f g x =

42 1

1x

=

81

1=

8 1 8 1

1 1 1

x x

x x

+= =

9

1x

=


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One-to-One Functions

In a one-to-one function, eachx-value corresponds to

only oney-value, and eachy-value corresponds to onlyonex-value. In a 1-1 function, neither thexnor theycan

repeat.

We can also say that f(a) = f(b) implies a = b.

A function is a one-to-one function if, forelements a and b in the domain off,

a b implies f(a) f(b).


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Example: Decide whether is one-to-one.

We want to show thatf(a) = f(b) implies that a = b:

Therefore,f is a one-to-one function.

( ) = +3 7 f x x

( ) ( ) f a f b=3 7 3 7a b + = +

3 3a b =

a b=


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Example: Decide whether is one-to-one.

This time, we will try plugging in different values:

Although 3 3, f(3) does equalf(3). This means thatthe function is notone-to-one by the definition.

( ) 2 2 f x x = +

( ) 23 3 2 11f = + =

( ) ( )2

3 3 2 11f = + =


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One-to-One Function

Another way to identify whether a function is one-to-one

is to use the horizontal line test, which says that if anyhorizontal line intersects the graph of a function in more

than one point, then the function is not one-to-one.

one-to-one not one-to-one


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Inverse Functions

Some pairs of one-to-one functions undo one another.

For example, if

and

then (for example)

This is true for any value ofx. Therefore,fandg are

called inverses of each other.

( ) 8 5 f x x = + ( )5

8

xg x

=

( ) ( )8 10 810 5 5f = + =

( )85 5 80

8

8

8

105g

= = =


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Inverse Functions

More formally:

Letf be a one-to-one function. Theng is the inverse

function off if

for everyxin the domain ofg,

and for everyxin the domain off.

( )( ) f g x x =

( )( ) g f x x =


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Inverse Functions

Example: Decide whetherg is the inverse function off.

yes

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

( )( ) ( )3

3 1 1 f g x x = +

1 1x= +

x=

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

=

x=


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Inverse Functions

Ifg is the inverse of a functionf, theng is written asf-1

(read finverse).

In our previous example, for ,( ) 3 1 f x x =

( )1

3 1 f x x

= +


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Finding Inverses Since the domain offis the range off-1 and vice versa, if

a set is one-to-one, then to find the inverse, we simplyexchange the independent and dependent variables.

Example: If the relation is one-to-one, find the inverse of

the function.

( ) ( ) ( ) ( ) ( ){ }2,1 , 1,0 , 0,1 , 1,2 , 2,2F = not 1-1

( ) ( ) ( ) ( ){ }3,1 , 0,2 , 2,3 , 4,0G=

1-1( ) ( ) ( ) ( ){1 1,3 , 2,0 , 3,2 , 0,4G =


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Finding Inverses In the same way we did the example, we can find the

inverse of a function by interchanging thexandyvariables.

To find the equation of the inverse ofy=f(x):

Determine whether the function is one-to-one.

Replacef(x) withyif necessary.

Switchxandy.

Solve fory.

Replaceywithf-1(x).


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Finding Inverses Example: Decide whether each equation defines a one-

to-one function. If so, find the equation of the inverse.

a) one-to-one

replacef(x)

withyinterchangexandy

solve fory

replaceywithf-1(x)

( ) 2 5 f x x = +

2 5y x

= +

2 5y= +

2 5y x=

5

2y

=

( )11 5

2 2 f x x =


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b) not one-to-one2 2y x= +


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Finding Inverses Alternative Method

We can also use the definition to find the inverse:

Set up the composition of . By definition,

this must equalx.

Solve for f-1(x).

( )( )1 f f x


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c) one-to-one( ) ( )3

2 f x x =

( )3

2y x=

( )3

2x y=

3 2x y=

3 2y x= +

( )1 3 2 f x x = +

( )( ) ( )( )

31 1

2 f f x f x

=

( )( )3

12 x f x =

( )13 2 x f x =

( )1 3 2 f x x = +


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Graphing Inverses Back in Geometry, when we studied reflections, it turned

out that the pattern for reflecting a figure across the liney=xwas to swap thex- andy-values.

Therefore, if we were to graph our inverse functions, we

would see that the inverse is the reflection of the original

function across the liney=x.

( ) ( ), , y y x


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Graphing Inverses

( ) 2 5 f x x = + ( ) ( )3

2 f x x =

( )11 5

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


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Homework College Algebra

Page 411: 3-21 (3s), 30-69 (3s) Homework: 42, 48, 57, 60a, 66a

obj. 10 inverse functions (presentation)

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