One-to-one

[Pages:10]IInveersse FFuunnccttionss

OOnnee--ttoo--oonnee

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fA B

f one-to-one

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A

B

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llaanngguuaaggee::

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f

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hhhaavvee

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aa,,cc a, c

2e

AA

A

wwitithh

aa =~6= cc,,, wwwee aarree gguuaarraannttteeeedd tthhaatt ff((a)) =6=$ ff((c))..

ac

fa fc

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fff

:::

RRIR

!--*

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wwhheerree

ff((x)) fx

===

xx222 x

iiss

nnnooottt

oonnee--ttoo--oonnee

bbeeccaauussee

33

=6 ~

--33

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ff

f

f

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tthheenn

ff(((zx))

fx

iiiss

nnoottt

oonnee--ttoo--oonnee..

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oooff tthhee ttaarrggeeettt..

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oonnee--ttoo--oonnneee..

\~. )L2

669660

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Onto

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of

f

equals

In other words, f is onto if every object in the target has at least one object

from the domain assigned to it by f.

What an inverse function is

Suppose : ! is a function. A function : ! is called the

fA B

gB A

of if

= and

=.

inverse function f f g id g f id

If is the inverse function of , then we often rename as 1.

g

f

gf

Examples.

?

Let

f

:

R

!

R

be

the

function

defined

by

() fx

=

+ 3, x

and

let

g : R ! R be the function defined by g(x) = x 3. Then

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

Because ( ) = and ( ) = , these are the same function. In symbols, f g x x id x x

f g = id. Similarly

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

so

= . Therefore, is the inverse function of , so we can rename

g f id

g

f

g

as 1, which means that 1( ) = 3.

f

fx x

?

Let

f

:

R

!

R

be

the

function

defined

by

() fx

=

2 + 2, x

and

let

g

:

R

!

R

be

the

function

defined

by

g(x)

=

1

x

1. Then

2

1 1

f

( ) = ( ( )) = gx fgx f

2x

1 = 2 2x

1 +2= x

Similarly

1

g

() fx

=

( ( )) gf x

=

(2 gx

+

2)

=

2

2 +2 x

1= x

Therefore, is the inverse function of , which means that 1( ) = 1 1.

g

f

fx x

92

2

The Inverse of an inverse is the original

If 1 is the inverse of , then 1 = and

1 = . We can see

f

f

f f id f f id

from the definition of inverse functions above, that is the inverse of 1.

f

f

That is ( 1) 1 = .

f

f

Inverse functions "reverse the assignment"

The definition of an inverse function is given above, but the essence of an

inverse function is that it reverses the assignment dictated by the original

function. If assigns to , then 1 will assign to . Here's why:

f

ab f

ba

If ( ) = , then we can apply 1 to both sides of the equation to obtain

fa b

f

the new equation 1( ( )) = 1( ). The left side of the previous equation f fa f b

involves function composition, 1( ( )) = 1 ( ), and 1 = , so

f fa f fa

f f id

we are left with 1( ) = ( ) = . f b id a a

The above paragraph can be summarized as "If ( ) = , then 1( ) = ." fa b f b a

Examples.

? If (3) = 4, then 3 = 1(4).

f

f

? If ( 2) = 16, then 2 = 1(16).

f

f

? If ( + 7) = 1, then + 7 = 1( 1).

fx

x

f

? If 1(0) = 4, then 0 = ( 4).

f

f

? If 1( 2 3 + 5) = 3, then 2 3 + 5 = (3).

fx x

xx

f

In the 5 examples above, we "erased" a function from the left side of the equation by applying its inverse function to the right side of the equation.

When a function has an inverse

A function has an inverse exactly when it is one-to-one and onto. both

This will be explained in more detail during lecture.

*************

93

Using inverse functions

Inverse functions are useful in that they allow you to "undo" a function. Below are some rather abstract (though important) examples. As the semester continues, we'll see some more concrete examples.

Examples.

? Suppose there is an object in the domain of a function , and that f

this object is named a. Suppose that you know f (a) = 15.

If has an inverse function, 1, and you happen to know that 1(15) = 3,

f

f

f

then you can solve for as follows: ( ) = 15 implies that = 1(15). Thus,

a

fa

af

a = 3.

? If is an object of the domain of , has an inverse, ( ) = 6, and

b

gg

gb

1(6) = 2, then g

= 1(6) = 2 bg

? Suppose ( + 3) = 2. If has an inverse, and 1(2) = 7, then

fx

f

f

+ 3 = 1(2) = 7

x

f

so =7 3=4

x

*************

The Graph of an inverse

If is an

function (that means if has an inverse function), and

f invertible

f

if you know what the graph of f looks like, then you can draw the graph of

1. f

If ( ) is a point in the graph of ( ), then ( ) = . Hence, 1( ) = .

a, b

fx

fa b

fb a

That means 1 assigns to , so ( ) is a point in the graph of 1( ).

f

b a b, a

fx

Geometrically, if you switch all the first and second coordinates of points

in R2, the result is to flip R2 over the "x = y line".

94

New

How points in graph of f(x)

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ffuunnccttiioonn

bbeeccoommee ppooiinnttss ooff nneeww ggrfraappxhh

visual effect vviissuuaall eeeecctt

f 11f(('x())x) fx

(a, b) (b, a) ((a, b)) 7! ((b, a)) a, b b, a

flip over the "x = y line" flfliipp oovveerr tthhee ""x == y lliinnee""

xy

Example. EExxaammppllee..

(3, s),

c(z)

(s, 3')

(a,-a)

(-3~.-5)

** ** ** ** ** ** ** ** ** ** ** ** **

7915

How to find an inverse

If you know that f is an invertible function, and you have an equation for

( ), then you can find the equation for 1 in three steps.

fx

f

Step 1 is to replace ( ) with the letter .

fx

y

Step 2 is to use algebra to solve for x.

Step 3 is to replace with 1( ). x fy

After using these three steps, you'll have an equation for the function

1( ). fy

Examples. ? Find the inverse of ( ) = + 5. fx x

Step 1.

= +5

yx

Step 2.

=5

xy

Step 3.

1( ) = 5

fy y

? Find the inverse of ( ) = 2x . gx

x1

Step 1.

= 2x

y

x1

Step 2.

=y

x

y2

Step 3.

1( ) = y

gy

y2

Make sure that you are comfortable with the algebra required to carry out

step 2 in the above problem. You will be expected to perform similar algebra

on future exams.

You should also be able to check that

1 = and that 1 = .

g g id

g g id

96

Exercises

In #1-6, is an invertible function. g

1.) If (2) = 3, what is 1(3)?

g

g

2.) If (7) = 2, what is 1( 2)?

g

g

3.) If ( 10) = 5, what is 1(5)?

g

g

4.) If 1(6) = 8, what is (8)?

g

g

5.) If 1(0) = 9, what is (9)?

g

g

6.) If 1(4) = 13, what is (13)?

g

g

For #7-12, solve for x. Use that f is an invertible function and that

1(1) = 2 f

1(2) = 3 f

1(3) = 2 f

1(4) = 5 f 1(5) = 7 f

1(6) = 8 f 1(7) = 3 f

1(8) = 1 f

1(9) = 4 f

Remember that you can "erase" by applying 1 to the other side of the

f

f

equation.

7.) ( + 2) = 5 fx

8.) (3 4) = 3 fx

9.) ( 5 ) = 1 fx

10.) ( 2 ) = 2

f

x

11.) ( 1 ) = 8 f

x 97

12.) ( 5 ) = 3 f

x1

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