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Section 1.8Inverse Functions

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Inverse Functions

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The function f is a set of ordered pairs, (x,y), then the

changes produced by f can be undone by reversing

components of all the ordered pairs. The resultingrelation (y,x), may or may not be a function. Inverse

functions have a special undoing relationship.

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Relations, Functions & 1:1

1:1 Functions are a subset of Functions. They arespecial functions where for every x, there is one y,

and for every y, there is one x.

Relations

Reminder: The definition of

function is, for every x there isonly one y.

Functions

1:1 Functions

Inverse Functions

are 1:1

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Inverse Functions

Let's suppose that

f(x)=x-300 and g(x)=x+300 then

f(g(x))=(x+300)-300

f(g(x))=x

Notice in the table belowhow the x and f(x) coordinates

are swapped between the two functions.

x f(x)1200 900

1300 1000

1400 1100

x g(x)900 1200

1000 1300

1100 1400

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Example Find f(g(x)) and g(f(x)) using the

following functions to show that

they are inverse functions.

x-2f(x)=3x+2 g(x)=

3

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Example

Find f(g(x)) and g(f(x)) using the following functions to

show that they are inverse functions.

x+3

f(x)=5x-3 g(x)= 5

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Finding the Inverse of a Function

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How to Find an Inverse Function

2

2

2

2

-1 -1

Find the inverse function of f(x).f(x)=x 1, 0

Replace f(x) with y: y=x 1

Interchange x and y: x=y 1

Solve for y: x+1=y

x+1 y

Replace y with f ( ) : f ( ) x+1

x

x x

=

=

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Example

Find the inverse of f(x)=7x-1

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Example 3Find the inverse of f(x)=x 4+

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Example 3Find the inverse of f(x)= 5

x

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The Horizontal Line Test

And

One-to-One Functions

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Horizontal Line Test

b and c are not one-to-one functions because they dont pass the

horizontal line test.

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Example

Graph the following function and tellwhether it has an inverse function or not.

( ) 3f x x=

9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10

7

6

5

4

3

2

1

1

2

3

4

5

6

x

y

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Example

Graph the following function and tell

whether it has an inverse function or not.( ) 1f x x=

9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10

6

5

4

3

2

1

1

2

3

4

5

6

x

y

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Graphs of f and f-1

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There is a relationship between the graph of a

one-to-one function, f, and its inverse f-1.

Because inverse functions have ordered pairs

with the coordinates interchanged, if the point(a,b) is on the graph of f then the point (b,a) is

on the graph of f-1. The points (a,b) and (b,a) are

symmetric with respect to the line y=x. Thus

graph of f-1

is a reflection of the graph of f aboutthe line y=x.

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A function and its inverse graphed on the same axis.

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Example

If this function has an inverse function,then graph its inverse on the same graph.

1( )f x x=

9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10

7

6

5

4

3

2

1

1

2

3

4

5

6

x

y

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Example

If this function has an inverse function,

then graph its inverse on the same graph. ( ) 3f x x=

3 2 1 1 2 3 4 5 6 7 8 9 1

3

2

1

1

2

3

4

5

6

x

y

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Example

If this function has an inverse function,

then graph its inverse on the same graph.3( )f x x=

4 3 2 1 1 2 3 4 5

4

3

2

1

1

2

3

4

x

y

A li ti f I F ti

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Applications of Inverse Functions

The function given by f(x)=5/9x+32 converts x degrees Celsius to

an equivalent temperature in degrees Fahrenheit.

a. Isfa one-to-one function? Why or why not?

b. Find a formula forf-1 and interpret what it calculates.

F=f(x)=5/9x+32 is 1 to 1 because it is a linear function.

5( ) 32

9

f x x= + The Celsius formula converts x

degrees Fahrenheit into Celsius.

9( 32)5 x y =

532

9x y =

5

329x y= +

532

9y x= + Replace the f(x) with y

Solve for y, subtract 32

Multiply by 9/5 on both sides

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(a)

(b)

(c)

(d)

-1 4x-5

Find an equation for f ( ) given that f(x)= 2x

-1

-1

-1

-1

5 2f ( )

4

2 4f ( )5

2 5f ( )

44 2

f ( )5

xx

xx

xx

xx

+=

+=

+=

+=

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(a)

(b)

(c)

(d)

-1 3Find an equation for f ( ) given f(x)=(x-3)x

-1 3

-1 3

-1 3

-1 3

f ( ) 3

f ( ) 3

f ( ) 3

f ( ) 3

x x

x x

x x

x x

= +

= +

=

=