9.1 1

Section 9.1 Composite and Inverse Functions

Composite Functions (f◦g)(x)=f(g(x)) Inverses and 1-to-1 Functions Finding Formulas for Inverses Graphing Functions and Their Inverses Inverse Functions and Composition

9.1 2

Two Functions:Concept and Notation for Composition

9.1 3

Women’s Shoe Sizes

9.1 4

Is Composition Commutative?

226151)15(

15)5(3)5(

226)5)((

78)26(3)26(

2625151)5(

78)5)((

2

5

g

f

fg

f

g

gf

226)25(91)5)((

91

)3(1))((

))((

78)25(33)5)((

33

)1(3))((

))((

2

2

2

2

fg

x

xxfg

xfg

gf

x

xxgf

xgf

9.1 5

Inverses and One-to-One Functions

9.1 6

Does an Inverse Function Exist?Tests for One-To-One Functions

9.1 7

Thinking about Inverse Functions Do all Linear Functions have Inverse Functions? All except Horizontal and Vertical Lines What about Quadratic Functions (Parabolas)?

No: y=4 fails HLT

9.1 8

Inverse Function Notation: f -1(x)

2)(

2

2

2

1

xxf

yforsolvexy

yandxswitchyx

xy

2

3)(

2

3

32

32

1

xxf

xy

yx

xy

9.1 9

Graphing Functions & Their Inverses

9.1 10

Consider g(x) = x3 + 2 and g -1(x) Is g(x) one-to-one?

9.1 11

Inverse Functions and Composition

9.1 12

What Next? Exponential Functions Present

Section 9.2