1.4c Inverse Relations 1.4c Inverse Relations and Inverse Functionsand Inverse Functions

Definition: Inverse Relation

2y x

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

Consider the “Do Now”:

2x y

These relations are inverses of each other!(the x- and y-values are simply switched!)

Which of these relationsare functions???

The HLT!!!A tasty ham, lettuce, and tomato sandwich???

No!!! The Horizontal Line TestThe inverse of a relation is a function if and only if eachhorizontal line intersects the graph of the original functionin at most one point.

Fails the HLT miserably!!!Fails the HLT miserably!!!

So, its inverse is not a function…So, its inverse is not a function…

Practice ProblemsIs the graph of each relation a function? Does therelation have an inverse that is a function?

A functionA function

Has an inverse thatHas an inverse thatis a functionis a function

Not a functionNot a function

Has an inverse thatHas an inverse thatis a functionis a function

Practice ProblemsIs the graph of each relation a function? Does therelation have an inverse that is a function?

Not a functionNot a function

Has an inverse thatHas an inverse thatis not a functionis not a function

A functionA function

Has an inverse thatHas an inverse thatis not a functionis not a function

More Definitions

1f b a

A relation that passes both the VLT and HLT is called one-to-one.

(since every x is paired with a unique y and every y is paired with a unique x…)

If f is a one-to-one function with domain D and range R, then theinverse function of f, denoted f , is the function with domain Rand range D defined by

–1

if and only if f a b

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

1

xf x

x

Find the inverse of the given function algebraically:

1

yx

y

Check the graph isthe function one-to-one?

xy y x xy x y

( 1)y x x

1

xy

x

1

xy

x

1x y y

Finding an Inverse Graphically

The Inverse Reflection PrincipleThe points (a, b) and (b, a) in the coordinate plane are symmetricwith respect to the line y = x. The points (a, b) and (b, a) arereflections of each other across the line y = x.

Finding an Inverse Graphically

f x

The graph of a function is shown. Is the function one-to-one?Sketch a graph of the inverse of the function. Yes!!!Yes!!!

y = x

1f x

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 twoWe can use this rule to algebraically verify that twofunctions are inverses… observe…functions are inverses… observe…

More Practice

3 1f x x Show algebraically that the given functions are inverses.

3 1g x x

f g x 33 1 1x 1 1x x

g f x 33 1 1x 3 3x x

More Practice

3f x x

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

Check the graph Is f one-to-one?

2 3x y

3x y

3y x

2 3y x

3, 0y x

3, 0x y where

where

3, 0y x where

3, 0y x where

Let’s graph theLet’s graph theinverse togetherinverse togetherwith the originalwith the original

function…function…

Whiteboard Problems…

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

1( )f x 1f

( ) 2 5f x x

3( ) 2f x x

1 1 5( )

2 2f x x

1 3( ) 2f x x

: ( , )D

: ( , )D

Whiteboard problems…

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

Homework: p. 129 39-61 odd

3( )

4( ) 4 3

xf x

g x x