©1999 by design science, inc. pre-calculus lesson 2: algebra of functions

©1999 by Design Science, Inc.

Pre-CalculusLesson 2: Algebra of

Functions

©1999 by Design Science, Inc.2

Basic function operationsSum

Difference

Product

Quotient , 0

f xff g x x g x

g g x

f g x f x g x+ +

– –f g x f x g x

f g x f x g x x x


Composition of functionsComposition of functions is the successive

application of the functions in a specific order.

Given two functions f and g, the composite function is defined by and is read “f of g of x.”f g f g x f g x


g x

A different way to look at it…

FunctionMachine

x f g x

FunctionMachine

gf

Evaluate and :


Example f g x g f x

3f x x

22 1g x x

Evaluate and :


2

2

2 1 3

2 4x

xf g x

Example f g x g f x

3f x x

22 1g x x

Evaluate and :


2

2

2 1 3

2 4x

xf g x

2

2

2

2 1

2 6 9 1

2 12 18 1

3g

x x

x

f x

x

x

Example f g x g f x

3f x x

22 1g x x

Evaluate and :


2

2

2 1 3

2 4x

xf g x

2

2

2

2 1

2 6 9 1

2 12 18 1

3g

x x

x

f x

x

x

Example f g x g f x

3f x x

22 1g x x

You can see that function composition is not commutative!


THE DIFFERENCE QUOTIENT


THE DIFFERENCE QUOTIENT

h

xfhxf


Example: Given f(x)=3x-7, find

h

xfhxf



h

xfhxf

h

xhx 7373



h

xfhxf

h

xhx 7373

h

xhx 73733


h

xhx 73733


h

xhx 73733

h

h3


h

xhx 73733

h

h3

3


Example: f(x) = 3x2 - 4x - 5

http://www.pstcc.edu/facstaff/jahrens/math1730/worksheets/diffquot.pdf

h

xfhxf


Example: f(x) = 3x2 - 4x - 5


h

xfhxf

h

xxhxhx 543543 22


Example: f(x) = 3x2 - 4x - 5


h

xfhxf

h

xxhxhx 543543 22

h

xxhxhxhx 54354423 222


Example: f(x) = 3x2 - 4x - 5


h

xfhxf

h

xxhxhx 543543 22

h

xxhxhxhx 54354423 222

h

xxhxhxhx 543544363 222


h

hhxh 436 2


h

hhxh 436 2

h

hxh 436


h

hhxh 436 2

h

hxh 436

436 hx

Practice:For the function f(x)=2x2 + 3, find:

a)f(2)

b)f(t)

c)f(x+h)

d)


h

xfhxf

©1999 by design science, inc. pre-calculus lesson 2: algebra of functions

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