7.8 Inverse Functions and Relations Horizontal line Test

7.8 Inverse Functions and Relations

Horizontal line Test

Look at the functions f(x) and g(x)

f(x) = 2x + 4 g(x) = ½x – 2

(x, f(x)) (x,g(x))

(-1,2) (2,-1)

(0,4) (4,0)

(1,6) (6,1)

(2,8) (8,2)

What do you see?

Look at the functions f(x) and g(x)

f(x) = 2x + 4 g(x) = ½x – 2

(x, f(x)) (x, g(x))

(-1,2) (2,-1)

(0,4) (4,0)

(1,6) (6,1)

(2,8) (8,2)

The two functions are inverses of each other

Definition of Inverse Functions

A function and its inverse function can be described as the "DO" and the "UNDO" functions.  A function takes a starting value, performs some operation on this value, and creates an output answer.  The inverse function takes the output answer, performs some operation on it, and arrives back at the original function's starting value. http://www.regentsprep.org/Regents/math/algtrig/ATP8/inverselesson.htm

How to find an inverse function

Since the input and output switch places.

x and y will switch places.

Function Inverse

y = 4x +12 x = 4y + 12

4y = x- 12

y = ¼x – 3

Find the inverse of y = 5x - 20

Switch x and y 205 yx

Find the inverse of y = 5x - 20

Switch x and y

yx

yx

520

205

Find the inverse of y = 5x - 20

Switch x and y

45

1

45

1

520

205

xy

yx

yx

yx

Graph the function and it inverse 205 xy 4

5

1 xy

The graphs the function and its inverse reflect over a line

y=x

205 xy 45

1 xy

To check if two functions are inverse we use compositions

Let

If both compositions equal x, then the functions are inverses

64

3 xxf 8

3

4)( xxg

xfg

xgf

To check if two functions are inverse we use compositions

If both compositions equal x, then the functions are inverses

64

3 xxf

83

4)( xxg

x

x

xxfg

x

x

xxgf

88

864

3

3

4))((

66

683

4

4

3))((

Inverses can be written as

If both compositions equal x, then the functions are inverses

64

3 xy

83

4 xy

x

x

xxfg

x

x

xxgf

88

864

3

3

4))((

66

683

4

4

3))((

yandy

Horizontal Line test

If a Horizontal line can pass through a graph of a function only touching it at one point, then the graph has a inverse.

Yes No

Homework

Page 393 –

# 15, 21, 24,

27, 30, 33,

36

Homework

Page 393 –

# 18, 20, 23,

26, 29, 32,

35