Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.

Chapter 3Chapter 3

FunctionsFunctionsand Their Graphsand Their Graphs

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.Chapter 3Chapter 3OverviewOverview

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.Chapter 3Chapter 3ObjectivesObjectives

Find the domain and range of a function. Sketch the graphs of common functions. Sketch graphs of general functions

employing translations of common functions.

Perform composition of functions. Find the inverse of a function. Model applications with functions using

variation.

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.Section 3.1Section 3.1FunctionsFunctions

Skills Objectives Determine whether a

relation is a function. Determine whether an

equation represents a function.

Use function notation. Find the value of a

function. Determine the domain

and range of a function.

Conceptual Objectives Think of function notation

as a placeholder or mapping.

Understand that all functions are relations but not all relations are functions.

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.

FunctionFunction

A function is a correspondence between two sets where each element in the first set corresponds exactly to one element in the second set.

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.

Vertical Line TestVertical Line Test

Given a graph of an equation, if any vertical line that can be drawn intersects the graph at no more than one point, the equation defines y as a function of x. This test is called the vertical line test.

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.

Common MistakeCommon Mistake

.3 that given 1 Evaluate 2 xxxfxf

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.

Domain of a FunctionDomain of a Function

5. and 5 except numbers real all to restricted is domain The

,55,55, notation. interval in domain Write

5 ns.restrictio domain the State

525 or25 equation. nrestrictio the Solve

025 . of values the on nsrestrictioany Determine

25

3 equation. orginial the Write

:Solution

25

3 function. given the of domain the State

2

2

2

2

x

xx

xx

xF(x)

xxF

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.Section 3.2 Section 3.2 Graphs of Functions; Piecewise-Defined Functions; Graphs of Functions; Piecewise-Defined Functions;

Increasing and Decreasing Functions; Average Rate of Increasing and Decreasing Functions; Average Rate of Change Change

Skills Objectives Classify functions as even, odd,

or neither. Determine whether functions are

increasing, decreasing, or constant.

Calculate the average rate of change of a function.

Evaluate the difference quotient for a function.

Graph piecewise-defined functions.

Conceptual Objectives Identify common functions. Develop and graph piecewise-

defined functions: Identify and graph points of

discontinuity. State the domain and range.

Understand that even functions have graphs that are symmetric about the y-axis.

Understand that odd functions have graphs that are symmetric about the origin.

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.

Your Turn!Your Turn!

1

112

1

xx

x

xx

xf

Click mouse to continueClick mouse to continue

Graph the piecewise-defined function, and state the intervals where the function is increasing, decreasing, or constant, along with the domain and range.

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.

Your Turn!Your Turn!

1

112

1

xx

x

xx

xf

Graph the piecewise-defined function, and state the intervals where the function is increasing, decreasing, or constant, along with the domain and range.

1, :Range

,11, :Domain

1,1 :Constant

1, :Decreasing

,1 :Increasing

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.Section 3.3 Section 3.3 Graphing Techniques: TransformationsGraphing Techniques: Transformations

Skills Objectives Sketch the graph of a function

using horizontal and vertical shifting of common functions.

Sketch the graph of a function by reflecting a common function about the x-axis or y-axis.

Sketch the graph of a function by stretching or compressing a common function.

Sketch the graph of a function using a sequence of transformations.

Conceptual Objectives Identify the common

functions by their graphs. Apply multiple

transformations of common functions to obtain graphs of functions.

Understand that domain and range are also transformed.

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.Vertical andVertical and Horizontal Shifts Horizontal Shifts

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.

Reflection About the AxesReflection About the Axes

The graph of –f(x) is obtained by reflecting the function f (x) about the x-axis.

The graph of f(-x) is obtained by rotating the function f(x) about the y-axis.

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.

Your Turn!Your Turn!

Click mouse to continueClick mouse to continue

xf

xxf

of range anddomain theState

21function graph the toreflection and shifts Use

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.

Your Turn!Your Turn!

xf

xxf

of range anddomain theState

21function graph the toreflection and shifts Use

2,- :Range

1, :Domain

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.Vertical Stretching and Vertical Vertical Stretching and Vertical Compressing of GraphsCompressing of Graphs

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.Horizontal Stretching and Horizontal Horizontal Stretching and Horizontal Compressing of GraphsCompressing of Graphs

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.Section 3.4 Section 3.4 OperationsOperations onon FunctionsFunctions

andand Composition of FunctionsComposition of FunctionsSkills Objectives

Add, subtract, multiply, and divide functions.

Evaluate composite functions.

Determine domain of functions resulting from operations and composition of functions.

Conceptual Objectives Understand domain

restrictions when dividing functions.

Realize that the domain of a composition of functions excludes the values that are not in the domain of the inside function.

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.

Composition of FunctionsComposition of Functions

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.

91

9744 .4 Evalue

41 . into 41 Subsitute

4151 . function, inner the of value the Find

1 quantity. desired the Write

2

2

gf

ff

fgffg

gg

gf

Evaluating aEvaluating a Composite Function Composite Function

1 Evaluate

5 and 7 functions the Given 22

gf

xxgxxf

Solution:

One way of evaluating these composite functions is to calculate the two individual composites in terms of x: f(g(x)) and g(f(x)). Once those functions are known, the values can be substituted for x and evaluated. Another way of proceeding is as follows:

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.Section 3.5 Section 3.5 One-to-One Functions One-to-One Functions and Inverse Functionsand Inverse Functions

Skills Objectives Determine whether a

function is a one-to-one function.

Verify that two functions are inverses of one another.

Graph the inverse function given the graph of the function.

Find the inverse of a function.

Conceptual Objectives Visualize the relationships

between the domain and range of a function and the domain and range of its inverse.

Understand why functions and their inverses are symmetric about y = x.

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.

Horizontal Line TestHorizontal Line Test

Co

lleg

e A

lgeb

ra, T

hir

d E

dit

ion

by

Cyn

thia

Y.

Yo

un

g, ©

201

2 Jo

hn

Wile

y an

d S

on

s. A

ll ri

gh

ts r

eser

ved

.

Inverse FunctionsInverse Functions