3.4 Use Inverse Functions3.4 Use Inverse Functions

p. 190p. 190

What is an inverse relation?What is an inverse relation?

What do you switch to find an inverse relation?What do you switch to find an inverse relation?

What notation is used for an inverse function?What notation is used for an inverse function?

How is it read?How is it read?

What test can you use to verify the inverse of a What test can you use to verify the inverse of a function is a function?function is a function?

ReviewReview

• Relation – a mapping of input values (x-values) onto output values (y-values).

• Here are 3 ways to show the same relation.

y = x2 x y

-2 4

-1 1

0 0

1 1

Equation

Table of values

Graph

• Inverse relation – just think: switch the x & y-values.

x = y2

xy

x y

4 -2

5 -1

0 0

1 1

** the inverse of an

equation: switch the x

& y and solve for y. ** the

inverse of a table:

switch the x & y.

** the inverse of a graph: the reflection of the original graph

in the line y = x.

Ex: Find an inverse of y = -3x+6.• Steps: -switch x & y

-solve for y

y = -3x+6

x = -3y+6

x-6 = -3y

yx

3

6

23

1

xy

Find an equation for the inverse of the relation y = 3x – 5.

Write original relation.y = 3x – 5

Switch x and y.x = 3y – 5

Add 5 to each side.x + 5 = 3y

Solve for y. This is the inverse relation.

13 x +5

3 = y

Inverse Functions

• Given 2 functions, f(x) & g(x), if f(g(x))=x AND g(f(x))=x, then f(x) & g(x) are inverses of each other.

Symbols: f -1(x) means “f inverse of x”

Ex: Verify that f(x)=−3x+6 and g(x)=−1/3x+2 are inverses.

• Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses.

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

= x −6+6

= x

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

= x−2+2

= x

** Because f(g(x))=x and g(f(x))=x, ** Because f(g(x))=x and g(f(x))=x, they are inversesthey are inverses..

To find the inverse of a function:To find the inverse of a function:

1. Change the f(x) to a y.

2. Switch the x & y values.

3. Solve the new equation for y.

STEP 1Show: that f(f –1(x)) = x.

f (f –1(x)) = f 31 x + 5

3

= x + 5 – 5

= x

SOLUTION

31 x + 5

3= 3 – 5

STEP 2Show: that f –1(f(x)) = x.

= 13

53(3x – 5) +

= x – 53

53+

= x

f –1(f(x)) = f –1(3x – 5)

Verify that f(x) = 3x – 5 and f –1(x) =13

x +53

are inverse functions.

38

Elastic bands can be used in exercising to provide a range of resistance. A band’s resistance R (in pounds) can be modeled by R = L – 5 where L is the total

length of the stretched band (in inches).

Fitness

Use the inverse function to find the length at which the band provides 19 pounds of resistance.

•

• Find the inverse of the model.

STEP 1Find: the inverse function.

Write original model.R = L – 538

Add 5 to each side.R + 5 = 38 L

83

403R + = L Multiply each side by

83 .

SOLUTION

Evaluate: the inverse function when R = 19.

403L = 8

3 R + 83= (19) +40

3403

152 3= + 192

3= = 64

ANSWER

The band provides 19 pounds of resistance when it is stretched to 64 inches.

Find the inverse of f(x) = x2, x ≥ 0. Then graph f and f –1.

Replace f (x) with y.y = x2

Switch x and y. x = y2

Take square roots of each side.

± =x y

SOLUTION

Write original function. f(x) = x2

The domain of f is restricted to nonnegative values of x. So, the range of f –1 must also be restricted to nonnegative values, and therefore the inverse is f –1(x) = x. (If the domain was restricted to x ≤ 0, you would choose f –1(x) = – x.)

Vertical Line Test

A vertical line test for functions can be used to see if the relation is a function.

•A relation is a function if and only if no vertical line intersects the graph of the relation at more than one point.

Ex: (a)Find the inverse of f(x)=x5.

1. y = x5

2. x = y5

3. 5 55 yx

yx 5

5 xy

(b) Is f -1(x) a function?

(hint: look at the graph!

Does it pass the vertical line test?)

Yes , f -1(x) is a function.

Horizontal Line TestHorizontal Line Test

• Used to determine whether a function’s inverseinverse will be a function by seeing if the original function passes the horizontal horizontal line testline test.

• If the original function passespasses the horizontal line test, then its inverse is a inverse is a functionfunction.

• If the original function does not passdoes not pass the horizontal line test, then its inverse is not inverse is not a functiona function.

Ex: g(x)=2x3

Inverse is a function!

y=2x3

x=2y3

3

2y

x

yx

3

2

3

2

xy

OR, if you fix the tent in the basement…

2

43 xy

Consider the function f (x) = 2x3 + 1. Determine whether the inverse of f is a function. Then find the inverse.

Graph the function f. Notice that no horizontal line intersects the graph more than once. So, the inverse of f is itself a function. To find an equation for f –1, complete the following steps:

SOLUTION

Find the inverse of a cubic funtionWrite original function.f (x) = 2x3 + 1

Replace f (x) with y.y = 2x3 + 1

Switch x and y.x = 2y3 + 1

Subtract 1 from each side.x – 1 = 2y3

Divide each side by 2.x – 12 = y3

Take cube root of each side.3 x – 12 = y

The inverse of f is f –1(x) = 3 x – 12

.

Find the inverse of the function. Then graph the function and its inverse.

5. f(x) = x6, x ≥ 0

ANSWER f –1(x) = 6√ x

Ex: Graph the function f(x)=x2 and determine whether its inverse is a

function.

Graph does not pass the horizontal line test, therefore the inverse is not a function.

Ex: f(x)=2x2-4 Determine whether f -1(x) is a function, then find the inverse equation.

2

2

4y

x

f -1(x) is not a function.

y = 2x2-4

x = 2y2-4

x+4 = 2y2

2

4x

y

22

1 xyOR, if you fix the

tent in the basement…

• What is an inverse relation?What is an inverse relation?It maps the output values back to the original input It maps the output values back to the original input

values (domain or inverse is range or original).values (domain or inverse is range or original).• What do you switch to find an inverse relation?What do you switch to find an inverse relation?x and yx and y• What notation is used for an inverse function?What notation is used for an inverse function?f f -1-1 (x) (x)• How is it read?How is it read?f inverse of x• What test can you use to verify the inverse of a What test can you use to verify the inverse of a

function is a function?function is a function?Horizontal line testHorizontal line test

Assignment

Page194, 3-39 every 3rd problem, 46, 47