Inverse Relations and Functions

Lesson 7-7

Algebra 2

Algebra 2

Using the Table Find the Following: (a.)

f(3)

and explain its meaning.

(b.)

g(4)

and explain its meaning.

(c.) At what time is the

VELOCITY

you obtained your answer.

of the object 160 feet/second? Explain how (d.) At what time is the

DISTANCE

you obtained your answer.

(height) of the object 64 feet? Explain how

t (sec) f(t)- Velocity g(t)-Distance 0 1 0 32 0 16 2 3 4 5 64 96 128 160 64 144 256 400

Algebra 2

Inverse Relations

In other words…If

the ordered pairs of a relation the new set of ordered pairs is called the ’

R’

are reversed, then

inverse relation

of the original relation.

Inverse Relations and Functions Lesson 7-7 Additional Examples a.

Find the inverse of relation

m

.

Relation

m x y

–1 –2 0 –1 1 –1 2 –2 Interchange the

x

and

y

columns.

Inverse of Relation

m x y

–2 –1 –1 0 –1 1 –2 2

Algebra 2

Inverse Relations and Functions Lesson 7-7 Additional Examples (continued) b.

Graph

m

and its inverse on the same graph.

Relation

m

Reversing the Ordered Pairs Inverse of

m

Algebra 2

Inverse Relations and Functions Lesson 7-7

Function - A function is like a machine: it has an input value that results in a single output.

Algebra 2

A function is often denoted f (x).

No two “x” values can be the same!

Vertical Line Test – If for every vertical line on a graph you draw: • It goes through only 1 point, y is a function of x. • It goes through 2 points (or more), y is not function.

Inverse Relations and Functions Lesson 7-7 Algebra 2

In mathematics, an inverse function is a function that undoes another function: A function ƒ that has an inverse is called invertible; and it

denoted by ƒ

−1

: (read f inverse, not to be confused w/exponentiation).

Inverse Relations and Functions Lesson 7-7

Inverse Functions

Definition: The

inverse

,

f

-1 (

x

),

reverses

the operations of

f

(

x

). If

f

-1 (

x

)

exists

for a certain function

f

, then

f

-1 (

f

(

x

)) =

x

.

Algebra 2

Inverse Relations and Functions Lesson 7-7

Is it Invertible or Not?

Algebra 2

One to One (1-1) - A function is called one-to-one if no two values of x produce the same y. No y-values are repeated.

So, a function is one-to-one if whenever we plug different values into the function we get different function values.

Horizontal Line Test - If every horizontal line you can draw passes through only 1 point, then the function is 1-1. If you can draw a horizontal line that passes through 2 points, then the function is

NOT 1-1.

Inverse Relations and Functions Lesson 7-7

y



x

2

f

 1  

x

Algebra 2

• • • • •

Inverse Relations and Functions Lesson 7-7 Algebra 2 Function

- A relation in which each input has

only one output.

Often denoted

f

(

x

).

• •

Vertical Line Test

- If for every vertical line on a graph you draw: It goes through only 1 point,

y

is a function of It goes through 2 points (or more),

y x

. is not function.

One to One (1-1)-

A function is called

one-to-one

if no two values of

x

produce the same

y

.

• •

Range

The set of

y

-values

Domain

The set of

x

-values • •

Horizontal Line Test

- If for every horizontal line on a graph you draw: It passes through only 1 point,

then the function is 1-1

. It passes through 2 points (or more),

then the function is NOT 1-1.

FACT: A function has an inverse if and only if it is

One-to-One (1-1).

To graph an inverse of a function you REFLECT the graph of

f

over the line

y

= x

Inverse Relations and Functions Lesson 7-7

Change in Domain and Range!

Algebra 2

Inverse Relations and Functions Lesson 7-7

Use the Table to answer the following:

x -2 1 4 7 10 13 f(x)

6 4 -2 -5 -13 -28

g(x)

2 4 6 8 9 10

Algebra 2

Inverse Relations and Functions Lesson 7-7

Use the Table to answer the following:

x -2 1 4 7 10 13 f(x)

6 4 -2 -5 -13 -28

g(x)

2 4 6 8 9 10

Algebra 2

Inverse Relations and Functions Lesson 7-7

Inverse Rule

Find an invertible functions inverse: • SWAP the variables “x” and “y” • SOLVE for “new y”. • Using the CHECK Step: 𝑓 −1 ( 𝑓 ( 𝑥 ))= 𝑥 to check your work.

I like to use x=1 or 0 because the math is simpler

Algebra 2

Inverse Relations and Functions Lesson 7-7 Algebra 2

Inverse Relations and Functions Lesson 7-7 Algebra 2

Inverse Relations and Functions Lesson 7-7 Algebra 2

Algebra 2

Inverse Relations and Functions Lesson 7-7 Algebra 2

Inverse Relations and Functions Lesson 7-7 Additional Examples

Find the inverse of

y

=

x

2 – 2.

y

=

x

2 – 2

x

=

y

2 – 2 Interchange

x

and

y

.

x

+ 2 =

y

2 ±

x

+ 2 =

y

Solve for

y

.

Find the square root of each side.

Algebra 2

Inverse Relations and Functions Lesson 7-7 Additional Examples

Graph

y

= –

x

2 – 2 and its inverse. The graph of

y

= –

x

2 – 2 is a parabola that opens downward with vertex (0, –2). The reflection of the parabola in the line

x

=

y

is the graph of the inverse.

You can also find points on the graph of the inverse by reversing the coordinates of points on

y

= –

x

2 – 2.

Algebra 2

Inverse Relations and Functions Lesson 7-7 Additional Examples Algebra 2

Consider the function

ƒ

(

x

) = 2

x

+ 2 .

a.

Find the domain and range of

ƒ

.

Since the radicand cannot be negative, the greater than or equal to –1.

domain is the set of numbers Since the principal square root is nonnegative, the range is the set of nonnegative numbers.

b.

Find

ƒ

–1

ƒ

(

x

) = 2

x

+ 2

y

= 2

x

+ 2

x x

2

y

= 2

y

= + 2 = 2

y

+ 2

x

2 – 2 2 Rewrite the equation using

y

.

Interchange

x

and

y

.

Square both sides.

Solve for

y

.

So,

ƒ

–1 (

x x

2 – 2 ) = .

Inverse Relations and Functions Lesson 7-7 Additional Examples Algebra 2 (continued) c.

Find the domain and range of

ƒ

–1 .

The domain of

ƒ

numbers.

–1 equals the range of

ƒ

, which is the set of nonnegative Since

x

2 – 0,

x

2 – 2 2 –1. Thus the range of

ƒ

–1 greater than or equal to –1.

is the set of numbers Note that the range of

ƒ

–1 is the same as the domain of

ƒ

.

d.

Is

ƒ

–1 a function? Explain.

For each

x

in the domain of

ƒ

–1 , there is only one value of

ƒ

a function.

–1 (

x

). So

ƒ

–1 is

Inverse Relations and Functions Lesson 7-7 Algebra 2

Inverse Relations and Functions Lesson 7-7 Additional Examples Algebra 2

The function

d

= 16

t

2 models the distance

d

in feet that an object falls in

t

seconds. Find the inverse function. Use the inverse to estimate the time it takes an object to fall 50 feet.

d

= 16

t

2

t

2

d

= 16 Solve for

t

. Do not interchange variables.

t

=

d

4 Quantity of time must be positive.

t

= 1 4 50 1.77

The time the object falls is 1.77 seconds.

Inverse Relations and Functions Lesson 7-7 Additional Examples

(

ƒ

° For the function

ƒ

(

x

) =

x

2

ƒ

–1 )( – 86).

+ 5, find (

ƒ

–1 °

ƒ

)(652) and Since

ƒ

is a linear function, so is

ƒ

–1 .

Therefore

ƒ

–1 is a function.

So (

ƒ

–1 °

ƒ

)( 652 ) = 652 and (

ƒ

°

ƒ

–1 )( – 86 ) = – 86.

Algebra 2

Algebra 2

Problems due for tomorrow: Page 404

#29 (Word Problem) #35-43 odd (No need to use check step)

Page 405

#47-57 odd (For all quadratics, just mention that off the bat you know its inverse is NOT a function and stop there.)