11.4 Inverse Relations and Functions

OBJ: Find the inverse of a relation Draw the graph of a function

and its inverse Determine whether the inverse of a function

is a function

DEF: Inverse Relations f(x) and f-1(x)Switch the x value and the y value

P 285

EX A = {(1,2), (2,-3), (5,2)}

Find A-1

A-1 = {(2, 1), (-3, 2), (2, 5 )}

EX1A ={(-3,-2),(-1,2),(3,5)}

FIND: A-1

A-1= {(-2,-3),(2,-1),(5,3)}

Is A-1 a function?

Yes

y

x

5

5

-5-5

NOTE: When the inverse of a given function “f” is a function, f –1 is used to denote it. If a

function is defined by an equation, the equation of the inverse is obtained by interchanging x and y in the original equation. P 285

EX: 2 A function “f” is defined by the equation y = -2/3 x + 4. Find an equation for f –1(x)

x = -2/3 y + 4 (x = -2/3 y + 4)3 3 x = -2 y + 12 3 x– 12 =-2y 3 x – 12 = y or f-1(x) -2

f-1(x) = - 3x + 6 2

y

x

5

5

-5-5

Absolute Value

5) y = |x| D: 8) x = |y| D: 10) y=|x|+6 D:

x | y x | y x | y

(-1, ) R: (1, ) R: (-1, ) R:

(0, ) (0, ) (0, )

(1, ) F ? (1, ) F ? (1 , ) F?y

x

5

5

-5-5

y

x

5

5

-5-5

y

x

5

5

-5-5

Absolute Value

5) y = |x| D: 8) x = |y| D: 10) y=|x|+6 D:

x | yƦ x | y [0, ∞) x | y Ʀ

(-1, 1) R: (1, -1) R: (-1, 7) R:

(0, 0) [0, ∞) (0, 0 ) Ʀ (0, 6) [6, ∞)

(1, 1) F ? (1, 1 ) F ? (1 , 7) F?

Yes No Yesy

x

5

5

-5-5

y

x

5

5

-5-5

y

x

5

5

-5-5

P286 EX: 3 Graph the function defined by y = x and its inverse. Is the

inverse a function?

x x function inverse-2 2 ( -2, 2 ) ( 2, -2 )-1 1 ( -1, 1 ) ( 1, -1 )0 0 ( 0, 0 ) ( 0, 0 )1 1 ( 1, 1 ) ( 1, 1 )2 2 ( 2, 2 ) ( 2, 2 )

y

x

5

5

-5-5

Lines

Ax + By = C

y = mx + b

9) 5x – 2 y = 10

(0, -5) (2, 0)

2y = -5x + 10

y = 5/2 x – 5

Standard Form

Slope-Intercept Form

Standard Form

Cover-up method

Slope-Intercept Formy

x

5

P286

EX:4 Graph the function defined by 4x–2y

=8 and its inverse in the same coordinate

plane. Is its inverse a function?

4x – 2y = 8 Use cover-up (0, ) ( ,0)

(0,-4) (2,0)

4y – 2x = 8 Use cover-up (0, ) ( ,0)

(0,2) (-4,0)

Or get y [f-1(x)] by itself

4y = 2x + 8

f-1(x) or y = ½ x + 2

y

x

5

5

-5-5

NOTE: If a function is defined by an equation in theform y = mx + b, m 0, then its inverse can be writtenin the same form, and the inverse is a function.

P297f ( x ) = 3 x – 5 or y = 3 x – 5 x = 3y – 5 x + 5 = 3y x + 5 = y [f-1(x)]

3 29) f –1 ( 1 ) (1 + 5)/3 = 2 30) f –1 ( 4 )

(4 + 5)/3 = 3

31) f –1 ( -5 ) (-5 + 5)/3= 0

32) f –1 ( 16 ) (16 +5)/3= 7

OR 1 = 3x – 5; 6 = 3x; 2 = x

OR 4 = 3x – 5; 9 = 3x; 3 = x

OR -5 = 3x – 5; 0 = 3x; 0 =x

OR 16 = 3x – 5; 21 = 3x; 7 = x

y

x

5

5

-5-5

Original relation Inverse relation

y 4 2 0 – 2 – 4

x 210– 1– 2

RANGE

FINDING INVERSES OF LINEAR FUNCTIONS

x 4 2 0 – 2 – 4

y 210– 1– 2

An inverse relation maps the output values back to their original input values. This means that the domain of the inverse relation is the range of the original relation and that the range of the inverse relation is the domain of theoriginal relation.

RANGE

DOMAINDOMAIN

FINDING INVERSES OF LINEAR FUNCTIONS

x

y 4 2 0 – 2 – 4

210– 1– 2

Original relation

x 4 2 0 – 2 – 4

y 210– 1– 2

Inverse relation

Graph of originalrelation

Reflection in y = x

Graph of inverserelation

y = xy = x

– 2

4

4

– 2

– 1

2 – 1

20

0

0

0

1

– 2

– 2

1

2

– 4

– 4

2

FINDING INVERSES OF LINEAR FUNCTIONS

To find the inverse of a relation that is given by anequation in x and y, switch the roles of x and y andsolve for y (if possible).

Finding an Inverse Relation

Find an equation for the inverse of the relation y = 2 x – 4.

y = 2 x – 4 Write original relation.

SOLUTION

Divide each side by 2.2x + 2 = y12

The inverse relation is y = x + 2.12

If both the original relation and the inverse relation happen to befunctions, the two functions are called inverse functions.

Switch x and y .x y

Add 4 to each side.4x + 4 = 2 y

x = 2 y – 4

Finding an Inverse Relation

I N V E R S E F U N C T I O N S

Functions f and g are inverses of each other provided:

f (g (x)) = x and g ( f (x)) = x

The function g is denoted by f

– 1, read as “f inverse.”

Given any function, you can always find its inverse relation by switching x and y.

For a linear function f (x ) = mx + b where m 0, the inverse is itself a linear function.

Verifying Inverse Functions

SOLUTION

Show that f (g (x)) = x and g (f (x)) = x.

g (f (x)) = g (2x – 4)

= (2x – 4) + 2

= x – 2 + 2

= x

12

Verify that f (x) = 2 x – 4 and g (x) = x + 2 are inverses.12

f (g (x)) = f x + 2 12( )

= 2 x + 2 – 4

= x + 4 – 4

= x

12( )

Take square roots of each side.

FINDING INVERSES OF NONLINEAR FUNCTIONS

Finding an Inverse Power Function

Find the inverse of the function f (x) = x 2.

SOLUTION

f (x) = x 2

y = x 2

x = y 2

± x = y

Write original function.

Replace original f (x) with y.

Switch x and y.

x 0

Notice that the inverse of g (x) = x 3 is a function, but that

the inverse of f (x) = x 2 is not a function.

If the domain of f (x) = x 2 is restricted, say to only nonnegative

numbers, then the inverse of f is a function.

FINDING INVERSES OF NONLINEAR FUNCTIONS

The graphs of the power functions f (x) = x 2

and g (x) = x

3 are shown along with their reflections in the line y = x.

g (x) = x 3

g (x ) = x 3

f (x) = x 2

f (x ) = x 2

inverse of g (x) = x 3

g –1(x ) = x3

inverse of f (x) = x 2

x = y 2

On the other hand, the graph of g (x) = x

3 cannot be intersected twice with a horizontal line and its inverse is a function.

Notice that the graph of f (x) = x 2

can be intersected twice with a horizontal line and that its inverse is not a function.

FINDING INVERSES OF NONLINEAR FUNCTIONS

H O R I Z O N T A L L I N E T E S T

If no horizontal line intersects the graph of a function f more than once, then the inverse of f is itself a function.

Modeling with an Inverse Function

ASTRONOMY Near the end of a star’s life the star will eject gas, forming a planetary nebula. The Ring Nebula is an example of a planetary nebula.

The volume V (in cubic kilometers) of this nebula can be modeled by V = (9.01 X 10

26 ) t 3 where t is the age (in years) of the nebula. Write the inverse function that gives the age of the nebula as a function of its volume.

Modeling with an Inverse Function

SOLUTION

Write original function.

Isolate power.

Take cube root of each side.

V = (9.01 X 1026 ) t 3

V

9.01 X 10 26 = t

3

Simplify.(1.04 X 10– 9 ) V = t3

Volume V can be modeled by V = (9.01 X 10 26 ) t

3

Write the inverse function that gives the age of the nebula as a function of its volume.

V

9.01 X 10 26

3 = t

Determine the approximate age of the Ring Nebula given

that its volume is about 1.5 X 10 38 cubic kilometers.

To find the age of the nebula, substitute 1.5 X 10 38 for V.

Write inverse function.

Substitute for V.

5500 Use calculator.

The Ring Nebula is about 5500 years old.

Modeling with an Inverse Function

SOLUTION

= (1.04 X 10–

9 ) 1.5 X 1038 3

t = (1.04 X 10–

9 ) V3