1.4c Inverse Relations and Inverse Functions

Homework: p. 129 39-61 odd

Definition: Inverse Relation

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

Consider the “Do Now”:

y



x

2 Which of these relations are functions???

x



y

2 These relations are inverses of each other!

(the x- and y-values are simply switched!)

The HLT!!!

The Horizontal Line Test

The inverse of a relation is a function if and only if each horizontal line intersects the graph of the original function in at most one point.

Fails the HLT miserably!!!

So, its inverse is not a function…

Practice Problems

Is the graph of each relation a function? Does the relation have an inverse that is a function?

Not a function A function Has an inverse that is a function Has an inverse that is a function

Practice Problems

Is the graph of each relation a function? Does the relation have an inverse that is a function?

A function Not a function Has an inverse that is not a function Has an inverse that is not a function

More Definitions

A relation that passes both the VLT and HLT is called

one-to-one

.

(since every

x

is paired with a unique

y

with a unique

x

…) and every

y

is paired If

f

inverse function of f

, denoted

f

and range D defined by   

b f

 1 , is the function with domain R 

a

if and only if

Finding an Inverse Algebraically

1. Determine that there is an inverse function by checking that the original function is one-to-one. Note any restrictions on the domain of the function.

2. Switch

x

and

y

in the formula of the original function.

3. Solve for

y

to obtain the inverse function. State any restrictions of the domain of the inverse.

Finding an Inverse Algebraically

Find the inverse of the given function algebraically: 

x x

 1 Check the graph  is the function one-to-one?

x



y y

 1 

y y



x



x

 1

x xy xy y x y

 1 

x x

Finding an Inverse Graphically 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

.

Finding an Inverse Graphically

The graph of a function is shown. Is the function one-to-one?

Sketch a graph of the inverse of the function.

y

=

x

Yes!!!

f

 1

And one more new tool: The Inverse Composition Rule

A function

f

is one-to-one with inverse function

g

if and only if

f

(

g

(

x

)) =

x

for every

x

in the domain of

g

, and

g

(

f

(

x

)) =

x

for every

x

in the domain of

f

We can use this rule to algebraically verify that two functions are inverses… observe…

More Practice

Show algebraically that the given functions are inverses.



x

3  1  3

x

 1       3  3

x



x

3  1 3    1 1  

x

 3

x

3  

x

1 

x

More Practice

Show that the given function has an inverse and find a rule for that inverse. State any restrictions of the domains of the function and its inverse.



x

 3 Check the graph  Is

f

one-to-one?

x y



x

2 

y



x

 3

y

 3 3

x

2  3 where where where where

x

  3,

y

 0

y

  3,

x

 0

y

  3,

x

 0

y

  3,

x

 0

Let’s graph the inverse together with the original function…

Whiteboard Problems…

Find a formula for . Give the domain of , including any restrictions “inherited” from f.

f

 1  2

x

 5

f

 1  1 2

x

 5 2

D

: (  3

x

 2

f

 1 

x

3  2

D

: (

Whiteboard problems…

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

 

x

 3 4

x

4  3