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
ffg
fg
gf
226)25(91)5)((91
)3(1))((
))((
78)25(33)5)((33
)1(3))((
))((
2
2
2
2
fgx
xxfg
xfg
gfx
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)(
222
1
xxf
yforsolvexyyandxswitchyx
xy
23)(
233232
1
xxf
xy
yxxy
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