NAME DATE PERIOD 1-5 Study Guide and · PDF fileLesson 1-5 NAME DATE PERIOD ... Lesson 1-6...

Less

on

1-5

NAME DATE PERIOD

Copyright

© G

lencoe/M

cG

raw

-Hill

, a

div

isio

n o

f T

he

McG

raw

-Hill

Co

mp

an

ies,

Inc.

PDF Pass

Chapter 1 27 Glencoe Precalculus

1-5

Parent Functions A parent function is the simplest of the functions in a family.

Parent Function Form Notes

constant function f(x) = c graph is a horizontal line

identity function f(x) = x points on graph have coordinates (a, a)

quadratic function f(x) = x2 graph is U-shaped

cubic function f(x) = x3 graph is symmetric about the origin

square root function f(x) = √ �

x graph is in first quadrant

reciprocal function f(x) = 1 − x graph has two branches

absolute value function f(x) = | x | graph is V-shaped

greatest integer function f(x) = �x�

defined as the greatest integer less than or equal to x; type of step function

Describe the following characteristics of the graph of the parent function f(x) = x3 : domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing.

The graph confirms that D = {x | x ∈ �} and R = {y | y ∈ �}.

The graph intersects the origin, so the x-intercept is 0 and the y-intercept is 0.

It is symmetric about the origin and it is an odd function:

f(-x) = -f(x).

The graph is continuous because it can be traced without lifting the pencil off the paper.

As x decreases, y approaches negative infinity, and as x increases, y approaches positive infinity.

lim x → -∞

f(x) = -∞ and lim x → ∞

f(x) = ∞

The graph is always increasing, so it is increasing for (-∞, ∞).

Exercise

Describe the following characteristics of the graph of the parent function f(x) = x2 : domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing.

Study Guide and InterventionParent Functions and Transformations

Example

y

x

f (x ) = x 3

027-042_PCCRMC01_893802.indd 27027-042_PCCRMC01_893802.indd 27 10/5/09 10:28:21 PM10/5/09 10:28:21 PM

Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pan

ies, In

c.

NAME DATE PERIOD

PDF Pass


1-5

Transformations of Parent Functions Parent functions can be transformed to create other members in a family of graphs.

Translations

g(x) = f(x) + k is the graph of f(x) translated…

…k units up when k > 0.

…k units down when k < 0.

g(x) = f(x - h) is the graph of f(x) translated…

…h units right when h > 0.

…h units left when h < 0.

Reflections

g(x) = -f(x) is the graph of f(x)… …reflected in the x-axis.

g(x) = f(-x) is the graph of f(x)… …reflected in the y-axis.

Dilations

g(x) = a � f(x) is the graph of f(x)…

…expanded vertically if a > 1.

…compressed vertically if 0 < a < 1.

g(x) = f(ax) is the graph of f(x)…

…compressed horizontally if a > 1.

…expanded horizontally if 0 < a < 1.

Identify the parent function f(x) of g(x) = √��-x - 1, and describe how the graphs of g(x) and f(x) are related. Then graph f(x) and g(x) on the same axes.

The graph of g(x) is the graph of the square root function f(x) = √�x reflected in the y-axis and then translated one unit down.

ExercisesIdentify the parent function f(x) of g(x), and describe how the graphs of g(x) and f(x) are related. Then graph f(x) and g(x) on the same axes.

1. g(x) = 0.5 ⎪x + 4⎥ 2. g(x) = 2x2- 4

Study Guide and Intervention (continued)

Parent Functions and Transformations

Example

y

x

g(x) = √-x - 1f(x) = √x

y

x4 8−4−8

4

8

−8

−4

y

x84−4−8

4

8

−8

−4


Copyright

© G

lencoe/M

cG

raw

-Hill

, a

div

isio

n o

f T

he

McG

raw

-Hill

Co

mp

an

ies,

Inc.

Less

on

1-5

NAME DATE PERIOD

Pdf Pass


1-5

1. Use the graph of f(x) = √ �

x to graph g(x) = √ �� x + 3 + 1.

y

x

2. Use the graph of f(x) = ⎪x⎥ to graph g(x) = -|2x|.

y

x

3. Describe how the graph of f(x) = x2 and g(x) are related. Then write an equation for g(x).

4. Identify the parent function f(x) of g(x) = 2|x + 2| - 3. Describe how the graphs of g(x) and f(x) are related. Then graph f(x) and g(x) on the same axes.

5. Graph f(x) =

y

x

6. Use the graph of f(x) = x3 to graph g(x) = ⎪(x + 1)3

⎥ .

y

x

PracticeParent Functions and Transformations

y

x

⎧

⎨

⎩

y

x2 4 6 8−4−6−8

2468

−8−6−4

-1 if x ≤ -31 + x if -2 < x ≤ 2. x� if 4 ≤ x ≤ 6


Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pa

nie

s, In

c.

NAME DATE PERIOD

Pdf Pass


1-5

1. AREA The width w of a rectangular plot of land with fixed area A is modeled by the function w(�) = A

−

�

, where � is the length.

a. If the area is 1000 square feet, describe the transformations of the parent function f(x) = 1

−x used to graph w(x).

b. Describe a function of the length that could be used to find a minimum perimeter for a given area

c. Is the function you found in part b a transformation of f(x)? Explain.

d. Find the minimum perimeter for an area of 1000 square feet.

2. GOLF The path of the flight of a golf ball

can be modeled by h(x) = -

1−

10x2

+ 2x, where h(x) is the distance above the ground in yards and x is the horizontal distance from the tee in yards.

a. Describe the transformation of the parent function f(x) = x2 used to graph h(x).

b. Suppose the same shot was made from a tee located 10 yards behind the original tee. Rewrite h(x) to reflect this change.

3. TAXES Graph the tax rates for the different incomes by using a step function.

4. HORIZON The function f(x) = √��1.5x can be used to approximate the distance to the apparent horizon, or how far a person can see on a clear day, where f(x) is the distance in miles and x is the person’s elevation in feet.

a. How does the graph of f(x) compare to the graph of its parent function?

b. The function g(x) = 1.2 √�x is also

used to approximate the distance to the horizon. How does the graph of g(x) compare to the graph of its parent function?

Word Problem PracticeParent Functions and Transformations

Tax

Rate

(%)

20

10

30

40

50

Taxable Income(thousands)

30 60 90 120 150 180 210 240 270 300

Source: Information Please Almanac

Income Tax Rates for a Couple

Filing Jointly

Limits of Taxable

Income ($)

Tax Rate

(%)

0 to 41,200 15

41,201 to 99,600 28

99,601 to 151,750 31

151,751 to 271,050 36

271,051 and up 39.6


Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pa

nie

s, In

c.

NAME DATE PERIOD

Pdf Pass


Operations with Functions Two functions can be added, subtracted, multiplied, or divided to form a new function. For the new function, the domain consists of the intersection of the domains of the two functions, excluding values that make a denominator equal to zero.

Given f(x) = x2 - x - 6 and g(x) = x + 2, find each function and its domain. a. (f + g)(x)

(f + g)x = f(x) + g(x) = x2 - x - 6 + x + 2

= x2 - 4

The domains of f and g are both (-∞, ∞), so the domain of (f + g) is (-∞, ∞).

b. ( f − g ) (x)

( f − g ) x =

f(x) −

g(x)

= x2 - x - 6 −

x + 2

= (x - 3)(x + 2) −

x + 2 = x - 3

The domains of f and g are both (-∞, ∞), but x = -2 yields a zero in

the denominator of ( f − g ) . So, the domain

is {x | x ≠ -2, x ∈ �}.

Given f(x) = x2 - 3 and g(x) = 1 −

x , find each function and its domain.

a. (f - g)(x)

(f - g)x = f(x) - g(x)= x2 - 3 - 1 −

x

The domain of f is (-∞, ∞) and the domain of g is (−∞, 0) ∪ (0, ∞), so the domain of (f - g) is (−∞, 0) ∪ (0, ∞).

b. (f � g)(x)

(f � g)x = f(x) � g(x)

= (x2 - 3) 1 −

x

= x - 3 −

x

The domain of f is (-∞, ∞) and the domain of g is (−∞, 0) ∪ (0, ∞), so the domain of (f - g) is (−∞, 0) ∪ (0, ∞).

Exercises

Find (f + g)(x), (f - g)(x), (f � g)(x), and ( f −

g ) (x) for each f(x) and g(x).

State the domain of each new function.

1. f(x) = x2 - 1, g(x) = 2 −

x 2. f(x) = x2

+ 4x − 7, g(x) = √

x

Study Guide and InterventionFunction Operations and Composition of Functions

Example 1

1-6

Example 2


Copyright

© G

lencoe/M

cG

raw

-Hill

, a

div

isio

n o

f T

he

McG

raw

-Hill

Co

mp

an

ies,

Inc.

Less

on

1-6

NAME DATE PERIOD

Pdf Pass


Compositions of Functions In a function composition, the result of one function is used to evaluate a second function.

Given functions f and g, the composite function f ◦ g can be described by the equation [f ◦ g](x) = f[g(x)]. The domain of f ◦ g includes all x-values in the domain of g for which g(x) is in the domain of f.

Given f(x) = 3x2 + 2x - 1 and g(x) = 4x + 2, find [f ◦ g](x) and [g ◦ f](x).

[f ◦ g](x) = f[g(x)] Defi nition of composite functions

= f(4x + 2) Replace g(x) with 4x + 2.

= 3(4x + 2)2 + 2(4x + 2) - 1 Substitute 4x + 2 for x in f(x).

= 3(16x2 + 16x + 4) + 8x + 4 - 1 Simplify.

= 48x2 + 56x + 15

[g ◦ f](x) = g(f(x)) Defi nition of composite functions

= g(3x2 + 2x - 1) Replace f(x) with 3x2 + 2x - 1.

= 4(3x2 + 2x - 1) + 2 Substitute 3x2 + 2x - 1 for x in g(x).

= 12x2 + 8x - 2 Simplify.

Exercises

For each pair of functions, find [f ◦ g](x), [g ◦ f](x), and [f ◦ g](4).

1. f(x) = 2x + 1, g(x) = x2 - 2x - 4 2. f(x) = 3x2 − 4, g(x) = 1 − x

3. f(x) = x3, g(x) = 5x 4. f(x) = 4x − 2, g(x) = √ �� x + 3

5. f(x) = 3x - 5, g(x) = x2 + 1 6. f(x) = 1 −

x - 1 , g(x) = x2 - 1

7. f(x) = 2x - 3, g(x) = 1 −

x - 2 8. f(x) = x - 8, g(x) = x + 4

Study Guide and Intervention (continued)

Function Operations and Composition of Functions

Example

1-6


Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pan

ies, In

c.

NAME DATE PERIOD

PDF Pass


Find (f + g)(x), (f - g)(x), (f · g)(x), and ( f − g ) (x) for each f(x) and

g(x). State the domain of each new function.

1. f(x) = 2x2 + 8 and g(x) = 5x - 6 2. f(x) = x3 and g(x) = √ �� x + 1

For each pair of functions, find [f ◦ g](x), [g ◦ f](x), and [f ◦ g](3).

3. f(x) = x + 5 and g(x) = x - 3 4. f(x) = 2x3 - 3x2 + 1 and g(x) = 3x

5. f(x) = 2x2 - 5x + 1 and g(x) = 2x - 3 6. f(x) = 3x2 - 2x + 5 and g(x) = 2x - 1

Find f ◦ g.

7. f(x) = √ �� x - 2 8. f(x) = 1 −

x - 8

g(x) = 3x g(x) = x2 + 5

Find two functions f and g such that h(x) = [f ◦ g](x). Neither function may be the identity function f(x) = x.

9. h(x) = √ �� 2x - 6 -1 10. h(x) = 1 −

3x +3

11. RESTAURANT A group of three restaurant patrons order the same meal and drink and leave an 18% tip. Determine functions that represent the cost of all of the meals before tip, the actual tip, and the composition of the two functions that gives the cost for all of the meals including tip.

PracticeFunction Operations and Composition of Functions

1-6


Copyright

© G

lencoe/M

cG

raw

-Hill

, a

div

isio

n o

f T

he

McG

raw

-Hill

Co

mp

an

ies,

Inc.

Less

on

1-6

NAME DATE PERIOD

PDF 2nd PDF 2nd


1. MARCHING BAND Band members form a circle of radius r when the music starts. They march outward as they play. The function f(t) = 2.5t gives the radius ofthe circle in feet after t seconds.Using g(r) = πr2 for the area of the circle, write a composite function that gives the area of the circle after t seconds.Then find the area, to the nearest tenth, after 4 seconds.

2. CANDLES A hobbyist makes and sells candles at a local market. The function c(h) = 4h gives the number of candles she has made after h hours. The function f(c) = 12 + 0.25c gives the cost of making c candles.

a. Write the composite function that gives the cost of candle making after h hours.

b. A sale reduces the cost of making c candles by 10%. Write the sale function s(x) and the composite function that gives the cost of candle making after h hours if materials are purchased during the sale.

3. SCIENCE The function t(x) = √ � 2x

−

28 + 6.25

gives the temperature in degrees Celsius of the liquid in a beaker after x seconds. Decompose the function into two separate functions, s(x) and r(x), so that s(r(x)) = t(x).

4. TRAVEL Two travelers are budgeting money for the same trip. The first traveler’s budget (in dollars) can be represented by f(x) = 45x + 350. The second traveler’s budget (in dollars) can be represented by g(x) = 60x + 475, x is the number of nights.

a. Find (f + g)(x) and the relevant domain.

b. What does the composite function in part a represent?

c. Find (f + g)(7) and explain what the value represents.

d. Repeat parts a–c for (g - f)(x).

5. POPULATION The function p(x) = 2x2

- 12x + 18 predicts the population of elk in a forest for the years 2010 through 2015 where x is the number of years since 2000. Decompose the function into two separate functions, a(x) and b(x), so that [a ◦ b](x) = p(x) and a(x) is a quadratic function and b(x) is a linear function.

Word Problem PracticeFunction Operations and Composition of Functions

1-6

027-042_PCCRMC01_893802.indd 35027-042_PCCRMC01_893802.indd 35 10/16/09 10:37:15 AM10/16/09 10:37:15 AM

Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pa

nie

s, In

c.

Chapter 1 A12 Glencoe Precalculus

Answers (Lesson 1-4 and Lesson 1-5)

Pdf Pass

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

1

26

Gle

ncoe

Pre

calc

ulus

Find

ing

an A

vera

ge R

ate

of C

hang

e G

iven

a fu

ncti

on, y

ou c

an

draw

tw

o po

ints

on

the

func

tion

, con

nect

the

poi

nts

wit

h a

line,

and

the

n fin

d th

e sl

ope

of t

hat

line,

giv

ing

you

the

aver

age

rate

of c

hang

e fo

r th

at

inte

rval

.

F

or t

he f

unct

ion

f (x)

= x

3 + 4

x2 - 6

x -

5,

find

the

ave

rage

rat

e of

cha

nge

for

the

inte

rval

[-2,

0].

Step

1:

Add

a G

RA

PH

S &

GE

OM

ET

RY

pag

e. E

nter

the

func

tion

rul

e in

the

func

tion

ent

ry li

ne.

Pres

s /

+ G

to

hide

the

func

tion

ent

ry li

ne.

Step

2:

Pres

s b

and

cho

ose

PO

INT

S &

LIN

ES

>

PO

INT

ON

. Pla

ce t

wo

poin

ts a

nyw

here

on

the

grap

h.

Dou

ble-

clic

k on

eac

h x-

coor

dina

te, c

hang

ing

one

to -

2an

d th

e ot

her

to 0

. The

y-c

oord

inat

es w

ill u

pdat

e. Y

oum

ay n

eed

to a

djus

t yo

ur v

iew

ing

win

dow

to

see

the

poin

ts.

Step

3:

Pres

s b

and

cho

ose

PO

INT

S &

LIN

ES

> L

INE

. Con

nect

the

poin

ts o

n th

e gr

aph.

Step

4:

Pres

s b

and

sel

ect

ME

AS

UR

EM

EN

T >

SL

OP

E. C

hoos

e th

elin

e. T

he s

lope

is -

10, s

o th

e av

erag

e ra

te o

f cha

nge

for

the

inte

rval

[-2,

0] is

-10

.

Exer

cise

s

Use

the

met

hod

show

n ab

ove

to f

ind

the

aver

age

rate

of

chan

ge o

fea

ch f

unct

ion

on t

he g

iven

inte

rval

.

1. f

(x)

= x

3 + 4

x2 - 6

x -

5; [

-4,

-2]

2.

f(x)

= x

2 + 7

x -

11;

[-8,

-6]

-

17

−

2

-

7

3. f

(x)

=x2

+ 7

x-

11;

[4, 5

]

4. f

(x)

=x4

-x3

+ 7

x; [-

2, -

1]

1

6

-

15

5. f

(x)

= x

4 - x

3 + 7

x; [0

, 2]

6. f

(x)

= x

4 - 3

x3 - x2 -

6; [

-2,

-1]

1

1

-

33

7. f

(x)

=x4

- 3

x3-

x2

- 6

; [1,

2]

8. f

(x)

=x2

- 1

; [1,

5]

-9

6

TI-N

spir

e Ac

tivity

Exam

ple

1-4

005_

026_

PC

CR

MC

01_8

9380

2.in

dd26

3/22

/09

5:51

:32

PM

Lesson 1-5

NA

ME

DA

TE

PE

RIO

D


Ch

ap

ter

1

27

Gle

ncoe

Pre

calc

ulus

1-5

Pare

nt F

unct

ions

A p

aren

t fu

ncti

on is

the

sim

ples

t of

the

func

tion

s in

a fa

mily

.

Par

ent

Fun

ctio

nF

orm

Not

es

cons

tant

func

tion

f(x) =

cgr

aph

is a

hor

izon

tal l

ine

iden

tity

func

tion

f(x) =

xpo

ints

on

grap

h ha

ve c

oord

inat

es (a

, a)

quad

rati

c fu

ncti

onf(x

) = x

2gr

aph

is U

-sha

ped

cubi

c fu

ncti

onf(x

) = x

3gr

aph

is s

ymm

etri

c ab

out

the

orig

in

squa

re r

oot

func

tion

f(x) =

√ �

x gr

aph

is in

firs

t qu

adra

nt

reci

proc

al fu

ncti

onf(x

) = 1 −

x gr

aph

has

two

bran

ches

abso

lute

val

ue fu

ncti

onf(x

) = |

x |gr

aph

is V

-sha

ped

grea

test

inte

ger

func

tion

f(x) =

�x�

defin

ed a

s th

e gr

eate

st in

tege

r le

ss t

han

or e

qual

to

x; t

ype

of s

tep

func

tion

D

escr

ibe

the

foll

owin

g ch

arac

teri

stic

s of

the

gra

ph o

f th

e pa

rent

fun

ctio

n f(

x) =

x3 : d

omai

n, r

ange

, int

erce

pts,

sym

met

ry,

cont

inui

ty, e

nd b

ehav

ior,

and

inte

rval

s on

whi

ch t

he g

raph

is

incr

easi

ng/d

ecre

asin

g.

The

grap

h co

nfir

ms

that

D =

{x | x

∈ �

} and

R =

{y | y

∈ �

}.

The

grap

h in

ters

ects

the

ori

gin,

so

the

x-in

terc

ept

is 0

and

th

e y-

inte

rcep

t is

0.

It is

sym

met

ric

abou

t th

e or

igin

and

it is

an

odd

func

tion

:

f(x) =

-f(x

).

The

grap

h is

con

tinu

ous

beca

use

it c

an b

e tr

aced

wit

hout

lif

ting

the

pen

cil o

ff th

e pa

per.

As

x de

crea

ses,

y a

ppro

ache

s ne

gati

ve in

finit

y, a

nd a

s x

incr

ease

s, y

app

roac

hes

posi

tive

infin

ity.

lim

x

→ -

∞

f(x) =

-∞

and

lim

x

→ ∞

f(x) =

∞

The

grap

h is

alw

ays

incr

easi

ng, s

o it

is in

crea

sing

for

(-∞

, ∞).

Exer

cise

Des

crib

e th

e fo

llow

ing

char

acte

rist

ics

of t

he g

raph

of

the

pare

nt f

unct

ion

f(x)

= x

2 : d

omai

n, r

ange

, int

erce

pts,

sym

met

ry, c

onti

nuit

y, e

nd b

ehav

ior,

an

d in

terv

als

on w

hich

the

gra

ph is

incr

easi

ng/d

ecre

asin

g.D

= {

x | x

∈ �

}, R

= {

y | y

≥ 0

, y

∈ �

}; x

-in

t: 0

; y-

int:

0;

sy

mm

etr

ic w

ith

re

sp

ec

t

to y

-ax

is;

ev

en

fu

nc

tio

n;

co

nti

nu

ou

s;

li

m

x

→ -

∞ f(

x) =

∞ a

nd

lim

x

→ ∞

f(x)

= ∞

;

de

cre

as

ing

fo

r (-

∞,

0)

an

d i

nc

rea

sin

g f

or

(0,

∞)

Stud

y Gu

ide

and

Inte

rven

tion

Pare

nt

Fu

ncti

on

s an

d T

ran

sfo

rmati

on

s

Exam

ple

y

x

f(x)

=x3

027-

042_

PC

CR

MC

01_8

9380

2.in

dd27

3/22

/09

5:51

:42

PM

A01_A20_PCCRMC01_893802.indd 12A01_A20_PCCRMC01_893802.indd 12 3/23/09 11:12:17 PM3/23/09 11:12:17 PM

Copyright

© G

lencoe/M

cG

raw

-Hill

, a

div

isio

n o

f T

he

McG

raw

-Hill

Co

mp

an

ies,

Inc.


An

swer

s

Answers (Lesson 1-5)

Pdf 3rd


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

1

28

Gle

ncoe

Pre

calc

ulus

1-5

Tran

sfor

mat

ions

of

Pare

nt F

unct

ions

Par

ent

func

tion

s ca

n be

tra

nsfo

rmed

to

crea

te o

ther

mem

bers

in a

fam

ily o

f gra

phs.

Tra

nsla

tion

s

g(x)

= f(

x) +

k is

the

gra

ph

of f(

x) t

rans

late

d……

k un

its

up w

hen

k >

0.

…k

unit

s do

wn

whe

n k

< 0

.

g(x)

= f(

x -

h) i

s th

e gr

aph

of f(

x) t

rans

late

d……

h un

its

righ

t w

hen

h >

0.

…h

unit

s le

ft w

hen

h <

0.

Ref

lect

ions

g(x)

= -

f(x) i

s th

e gr

aph

of f(

x)…

…re

flect

ed in

the

x-a

xis.

g(x)

= f(

-x )

is t

he g

raph

of

f(x)

…

…re

flect

ed in

the

y-a

xis.

Dil

atio

ns

g(x)

= a

� f(x

) is

the

grap

h of

f(x)

…

…ex

pand

ed v

erti

cally

if a

> 1

.

…co

mpr

esse

d ve

rtic

ally

if 0

< a

< 1

.

g(x)

= f(

ax) i

s th

e gr

aph

of f(

x)…

…co

mpr

esse

d ho

rizo

ntal

ly if

a >

1.

…ex

pand

ed h

oriz

onta

lly if

0 <

a <

1.

Id

enti

fy t

he p

aren

t fu

ncti

on f(

x) o

f g(x

) =√

��

-x

- 1

, and

des

crib

e ho

w

the

grap

hs o

f g(x

) and

f(x)

are

rel

ated

. The

n gr

aph

f(x)

and

g(x

) on

the

sam

e ax

es.

The

grap

h of

g(x

) is

the

grap

h of

the

squ

are

root

fu

ncti

on f(

x) =

√� x

refle

cted

in t

he y

-axi

s an

d th

en t

rans

late

d on

e un

it d

own.

Exer

cise

sId

enti

fy t

he p

aren

t fu

ncti

on f

(x)

of g

(x),

and

desc

ribe

how

the

gra

phs

of g

(x)

and

f(x)

are

rel

ated

. The

n gr

aph

f(x)

and

g(x

) on

the

sam

e ax

es.

1.

g(x)

= 0

.5 ⎪ x

+ 4

⎥

2. g

(x) =

2x

2-

4

Stud

y Gu

ide

and

Inte

rven

tion

(con

tinu

ed)

Pare

nt

Fu

ncti

on

s an

d T

ran

sfo

rmati

on

s

Exam

ple

y

x

g(x)

= √

-x

-1

f(x)=

√x

y

x4

8−

4−

8

48

−8

−4

Th

e g

rap

h o

f g

(x)

is t

he g

rap

h o

f th

e

ab

so

lute

valu

e f

un

cti

on

f(x

) =

|x|

co

mp

ressed

vert

ically a

nd

tra

nsla

ted

4 u

nit

s l

eft

.

Th

e g

rap

h o

f g

(x)

is t

he g

rap

h o

f th

e

sq

uare

fu

ncti

on

f(x

) =

x2

exp

an

ded

vert

ically a

nd

tra

nsla

ted

4 u

nit

s d

ow

n.

y

x8

4−

4−

8

48

−8

−4

027-

042_

PC

CR

MC

01_8

9380

2.in

dd28

9/30

/09

2:04

:21

PM


Lesson 1-5

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

1

29

Gle

ncoe

Pre

calc

ulus

1-5

1. U

se t

he g

raph

of f

(x) =

√ �

x to

gra

ph

g(x)

= √

��

�

x +

3 +

1.

y

x

f(x)

g(x)

2.

Use

the

gra

ph o

f f(x

) = ⎪

x⎥ t

o gr

aph

g(x)

= -

|2x|

.

y

x

f(x)

g(x)

3. D

escr

ibe

how

the

gra

ph o

f f(x

) = x

2 and

g(x

) are

re

late

d. T

hen

wri

te a

n eq

uati

on fo

r g(

x).

g

(x)

is f

(x)

refl

ec

ted

in

th

e x

-ax

is,

tra

ns

late

d 1

un

it r

igh

t a

nd

1 u

nit

up

.

g

(x)

= -

(x -

1)2

+ 1

4. I

dent

ify t

he p

aren

t fu

ncti

on f(

x) o

f g(x

) = 2

|x +

2|

- 3

. D

escr

ibe

how

the

gra

phs

of g

(x) a

nd f(

x) a

re r

elat

ed.

Then

gra

ph f(

x) a

nd g

(x) o

n th

e sa

me

axes

.

T

he

gra

ph

of

g(x

) is

th

e g

rap

h o

f f(

x) =

| x

| s

tre

tch

ed

ve

rtic

all

y a

nd

tra

ns

late

d 2

un

its

le

ft

an

d 3

un

its

do

wn

.

5. G

raph

f(x)

=

y

x

6.

Use

the

gra

ph o

f f(x

) = x

3 to

grap

h g(

x) =

⎪(x

+ 1

)3 ⎥ .

y

x

g(x)

Prac

tice

Pare

nt

Fu

ncti

on

s an

d T

ran

sfo

rmati

on

s

y

x

g(x)

⎧

⎨

⎩

y

x2

46

8−

4−

6−

8

2468

−8

−6

−4

f(x)

g(x)

-

1 if

x ≤

-3

1 +

x if

-2

< x

≤ 2

. �x�

if

4 ≤

x ≤

6

027-

042_

PC

CR

MC

01_8

9380

2.in

dd29

10/2

3/09

4:24

:52

PM


Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pa

nie

s, In

c.



Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

1

30

Gle

ncoe

Pre

calc

ulus

1-5

1. A

REA

The

wid

th w

of a

rec

tang

ular

plo

t of

land

wit

h fix

ed a

rea

A is

mod

eled

by

the

func

tion

w(�

) =A −

�

, whe

re� is

the

le

ngth

.

a. I

f the

are

a is

100

0 sq

uare

feet

, de

scri

be t

he t

rans

form

atio

ns o

f the

pa

rent

func

tion

f(x)

=1 − x

used

to

grap

h w

(x).

f(x)

is

ex

pa

nd

ed

ve

rtic

all

y.

b. D

escr

ibe

a fu

ncti

on o

f the

leng

th t

hat

coul

d be

use

d to

find

a m

inim

um

peri

met

er fo

r a

give

n ar

ea

P(ℓ

)=

2ℓ

+2

(A − ℓ)

c. I

s th

e fu

ncti

on y

ou fo

und

in p

art

ba

tran

sfor

mat

ion

of f(

x)?

Exp

lain

.N

o;

sa

mp

le a

ns

we

r: t

he

y a

re t

wo

d

iffe

ren

t k

ind

s o

f ra

tio

na

l fu

nc

tio

ns

.

d. F

ind

the

min

imum

per

imet

er fo

r an

ar

ea o

f 100

0 sq

uare

feet

.1

26

.5 f

t

2. G

OLF

The

path

of t

he fl

ight

of a

gol

f bal

l

can

be m

odel

ed b

y h(

x) =

-

1− 10

x2+

2x,

whe

re h

(x) i

s th

e di

stan

ce a

bove

the

gr

ound

in y

ards

and

x is

the

hor

izon

tal

dist

ance

from

the

tee

in y

ards

.

a. D

escr

ibe

the

tran

sfor

mat

ion

of t

he

pare

nt fu

ncti

on f(

x) =

x2 use

d to

gra

ph

h(x)

.

h(x

) is

th

e g

rap

h o

f f(

x)

tra

ns

late

d 1

0 u

nit

s r

igh

t,

co

mp

res

se

d v

ert

ica

lly

, re

fl ec

ted

in

th

e x

-ax

is,

an

d t

he

n

tra

ns

late

d 1

0 u

nit

s u

p.

b. S

uppo

se t

he s

ame

shot

was

mad

e fr

om a

tee

loca

ted

10 y

ards

beh

ind

the

orig

inal

tee

. Rew

rite

h(x

) to

refle

ct

this

cha

nge.

h(x

) =

- 1 − 1

0x2

+ 1

0

3. T

AX

ES G

raph

the

tax

rat

es fo

r th

e di

ffere

nt in

com

es b

y us

ing

a st

ep

func

tion

.

4. H

ORI

ZON

The

func

tion

f(x)

=√

��

1.5x

ca

n be

use

d to

app

roxi

mat

e th

e di

stan

ce

to t

he a

ppar

ent

hori

zon,

or

how

far

a pe

rson

can

see

on

a cl

ear

day,

whe

re

f(x) i

s th

e di

stan

ce in

mile

s an

d x

is t

he

pers

on’s

elev

atio

n in

feet

.

a. H

ow d

oes

the

grap

h of

f(x)

com

pare

to

the

gra

ph o

f its

par

ent

func

tion

?

It

is

th

e p

are

nt

fun

cti

on

c

om

pre

ss

ed

ho

rizo

nta

lly

.

b. T

he fu

ncti

on g

(x) =

1.2

√�x

is a

lso

used

to

appr

oxim

ate

the

dist

ance

to

the

hori

zon.

How

doe

s th

e gr

aph

of

g(x)

com

pare

to

the

grap

h of

its

pare

nt fu

ncti

on?

It

is

th

e p

are

nt

fun

cti

on

e

xp

an

de

d v

ert

ica

lly

.

Wor

d Pr

oble

m P

ract

ice

Pare

nt

Fu

ncti

on

s an

d T

ran

sfo

rmati

on

s

Tax Rate (%)

20 10304050

Taxa

ble

Inco

me

(thou

sand

s)

3060

9012

015

018

021

024

027

030

0

Sour

ce:I

nfor

mat

ion

Plea

se A

lman

ac

Inc

om

e T

ax

Ra

tes

fo

r a

Co

up

le

Fil

ing

Jo

intl

y

Lim

its

of

Ta

xa

ble

Inc

om

e (

$)

Ta

x R

ate

(%)

0 to

41

,20

0

15

41

,20

1 t

o 9

9,6

00

2

8

99

,60

1 t

o 1

51

,75

0

31

15

1,7

51

to

27

1,0

50

3

6

27

1,0

51

an

d u

p

39

.6

027-

042_

PC

CR

MC

01_8

9380

2.in

dd30

3/22

/09

5:52

:06

PM


Lesson 1-5

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

1

31

Gle

ncoe

Pre

calc

ulus

1-5

Ro

tati

on

sA

rot

atio

n is

a r

igid

tra

nsfo

rmat

ion.

A r

otat

ion

turn

s a

figur

e ab

out

a po

int

a ce

rtai

n nu

mbe

r of

deg

rees

. The

rot

atio

n ca

n be

clo

ckw

ise

or

coun

terc

lock

wis

e. F

or t

his

acti

vity

, ass

ume

all r

otat

ions

are

abo

ut t

he o

rigi

n an

d in

the

cou

nter

cloc

kwis

e di

rect

ion.

To

rota

te a

poi

nt 9

0° a

bout

the

ori

gin,

us

e th

e ru

le (x

, y) →

(-y,

x).

1. R

otat

e po

int

A b

y 90

° u

sing

the

rul

e. G

raph

the

poi

nt.

G

ive

the

coor

dina

tes

of A

'.

(-2

, 3

)

2. R

otat

e po

int

A' b

y 90

°. G

raph

the

poi

nt. G

ive

the

coor

dina

tes

of A

''. T

hen

use

the

resu

lt t

o w

rite

a r

ule

for

rota

ting

(x, y

) by

180°

. (-

3,

-2

); (

x, y

) →

(-

x, -

y)

3. R

otat

e po

int

A'' b

y 90

°. G

raph

the

poi

nt. G

ive

the

coor

dina

tes

of A

'''. T

hen

use

the

resu

lt t

o w

rite

a r

ule

for

rota

ting

(x, y

) by

270°

. (2

, -

3);

(x,

y)

→ (

y, -

x)

To r

otat

e a

func

tion

, you

can

plo

t se

vera

l im

age

poin

ts a

nd t

hen

conn

ect

them

.

Gra

ph e

ach

func

tion

. The

n gr

aph

the

func

tion

aft

er it

is r

otat

ed 9

0°.

4. f

(x) =

x3 -

2

5. f

(x) =

1 −

2 x2 -

1

y

x8−

4−

8

48

−8

y

x4

8−

4−

8

48

−8

−4

Gra

ph e

ach

func

tion

. The

n gr

aph

the

func

tion

aft

er it

is r

otat

ed 2

70°.

6. f

(x) =

⎪x3 +

2x

- 4

⎥

7. f

(x) =

√ �

��

-x

- 4

y

x4

8−

4−

8

4

−8

−4

y

x4

8−

4−

8

48

−8

−4

8. T

he g

raph

of t

he fu

ncti

on f(

x) =

2x

- 3

is r

otat

ed 9

0°. W

hat

func

tion

repr

esen

ts t

he r

otat

ed g

raph

? f(

x) =

- 1

−

2 x

+ 3

−

2

Enri

chm

ent

y

x

'

�

�

027-

042_

PC

CR

MC

01_8

9380

2.in

dd31

3/22

/09

5:52

:11

PM


Copyright

© G

lencoe/M

cG

raw

-Hill

, a

div

isio

n o

f T

he

McG

raw

-Hill

Co

mp

an

ies,

Inc.


An

swer

s


PDF Pass


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

1

32

Gle

ncoe

Pre

calc

ulus

Ope

rati

ons

wit

h Fu

ncti

ons

Two

func

tion

s ca

n be

add

ed, s

ubtr

acte

d,

mul

tipl

ied,

or

divi

ded

to fo

rm a

new

func

tion

. For

the

new

func

tion

, the

do

mai

n co

nsis

ts o

f the

inte

rsec

tion

of t

he d

omai

ns o

f the

tw

o fu

ncti

ons,

ex

clud

ing

valu

es t

hat

mak

e a

deno

min

ator

equ

al t

o ze

ro.

G

iven

f(x

) =

x2 -

x -

6 a

nd g

(x)

= x

+ 2

, fin

d ea

ch

func

tion

and

its

dom

ain.

a. (

f +

g)(

x)

(f

+ g

)x =

f(x)

+ g

(x)

=

x2 -

x -

6 +

x +

2

=

x2 -

4

Th

e do

mai

ns o

f f a

nd g

are

bot

h

(-∞

, ∞),

so t

he d

omai

n of

(f +

g) i

s

(-∞

, ∞).

b.

( f −

g ) (x)

( f −

g ) x =

f(x)

−

g(x)

=

x2 - x

- 6

−

x +

2

=

(x

- 3

)(x +

2)

−

x +

2

= x

- 3

Th

e do

mai

ns o

f f a

nd g

are

bot

h

(-

∞, ∞

), bu

t x

= -

2 yi

elds

a z

ero

in

th

e de

nom

inat

or o

f ( f −

g ) . S

o, t

he d

omai

n

is

{x | x

≠ -

2, x

∈ �

}.

G

iven

f(x

) =

x2 -

3 a

nd g

(x)

= 1 −

x , fin

d ea

ch f

unct

ion

and

its

dom

ain.

a. (

f -

g)(

x)

(f

- g

)x =

f(x)

- g

(x)

= x

2 - 3

- 1 −

x

Th

e do

mai

n of

f is

(-∞

, ∞) a

nd t

he

dom

ain

of g

is (−

∞, 0

) ∪ (0

, ∞),

so t

he

dom

ain

of (f

- g

) is

(−∞

, 0) ∪

(0, ∞

).

b.

(f �

g)(

x)

(f

� g

)x =

f(x)

� g(

x)

= (x

2 - 3

) 1 −

x

= x

- 3 −

x

Th

e do

mai

n of

f is

(-∞

, ∞) a

nd t

he

dom

ain

of g

is (−

∞, 0

) ∪ (0

, ∞),

so t

he

dom

ain

of (f

- g

) is

(−∞

, 0) ∪

(0, ∞

).

Exer

cise

s

Fin

d (f

+ g

)(x)

, (f

- g

)(x)

, (f

� g)

(x),

and

( f −

g ) (x)

for

each

f(x

) an

d g(

x).

Stat

e th

e do

mai

n of

eac

h ne

w f

unct

ion.

1. f

(x) =

x2

- 1

, g(x

) = 2 −

x 2.

f(x

) = x

2 +

4x

− 7

, g(x

) = √

x

x3

- x

+ 2

−

x ;

D =

(-

∞,

0)

∪ (

0,

∞)

x2 +

4x

- 7

+ √

�

x ;

D =

[0

, ∞

)

x3

- x

- 2

−

x ;

D =

(-

∞,

0)

∪ (

0,

∞)

x2 +

4x

- 7

− √

�

x ;

D =

[0

, ∞

)

2

x2 -

2

−

x ;

D =

(-

∞,

0)

∪ (

0,

∞)

x2 √

�

x +

4x

√ �

x

- 7

√ �

x

; D

= [

0,

∞)

x3

- x

−

2

; D

= (

-∞

, 0

) ∪

(0

, ∞

) x2

+ 4

x -

7

−

√ �

x

;

D =

(0

, ∞

)

Stud

y Gu

ide

and

Inte

rven

tion

Fu

ncti

on

Op

era

tio

ns

an

d C

om

po

siti

on

of

Fu

ncti

on

s

Exam

ple

1

1-6

Exam

ple

2

027-

042_

PC

CR

MC

01_8

9380

2.in

dd32

9/30

/09

2:04

:35

PM


Lesson 1-6

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

1

33

Gle

ncoe

Pre

calc

ulus

Com

posi

tion

s of

Fun

ctio

ns I

n a

func

tion

com

posi

tion

, the

res

ult

of

one

func

tion

is u

sed

to e

valu

ate

a se

cond

func

tion

.

Giv

en fu

ncti

ons

f and

g, t

he c

ompo

site

func

tion

f ◦ g

can

be

desc

ribe

d by

the

eq

uati

on [f

◦ g

](x) =

f[g(

x)].

The

dom

ain

of f

◦ g

incl

udes

all

x-va

lues

in t

he

dom

ain

of g

for

whi

ch g

(x) i

s in

the

dom

ain

of f.

G

iven

f(x

) =

3x2 +

2x

- 1

and

g(x

) =

4x

+ 2

, fin

d [f

◦ g

](x)

an

d [g

◦ f

](x)

.

[f ◦ g

](x) =

f[g(

x)]

De

fi niti

on

of

com

po

site

fu

nct

ion

s

=

f(4x

+ 2

) R

ep

lace

g(x

) w

ith 4

x +

2.

=

3(4

x +

2)2 +

2(4

x +

2) -

1

Su

bst

itute

4x

+ 2

fo

r x

in f(

x).

=

3(1

6x2 +

16x

+ 4

) + 8

x +

4 -

1

Sim

plif

y.

=

48x

2 + 5

6x +

15

[g ◦

f](x

) = g

(f(x)

) D

efi n

itio

n o

f co

mp

osi

te f

un

ctio

ns

=

g(3

x2 + 2

x -

1)

Re

pla

ce f

(x)

with

3x2

+ 2

x -

1.

=

4(3

x2 + 2

x -

1) +

2

Su

bst

itute

3x2

+ 2

x -

1 f

or

x in

g(x

).

=

12x

2 + 8

x -

2

Sim

plif

y.

Exer

cise

s

For

eac

h pa

ir o

f fu

ncti

ons,

fin

d [f

◦ g

](x)

, [g

◦ f

](x)

, and

[f

◦ g

](4)

.

1. f

(x) =

2x

+ 1

, g(x

) = x

2 - 2

x -

4

2. f

(x) =

3x2 −

4, g

(x) =

1 −

x

2

x2 -

4x

- 7

; 4

x2 -

5;

9

3

- 4

x2

−

x2

;

1

−

3x2

- 4

; -

61

−

16

3. f

(x) =

x3 ,

g(x)

= 5

x 4.

f(x

) = 4

x −

2, g

(x) =

√

x +

3

1

25

x3;

5x3

; 8

00

0

4

√ �

��

x +

3 -

2;

√ �

��

4x

+ 1

; 4

√ �

7

- 2

5. f

(x) =

3x

- 5

, g(x

) = x

2 + 1

6.

f(x

) =

1 −

x -

1 ,

g(x)

= x

2 - 1

3

x2 -

2;

9x2

- 3

0x

+ 2

6;

46

1

−

x2 -

2 ;

2x

- x

2

−

x2

- 2

x +

1 ;

1

−

14

7. f

(x) =

2x

- 3

, g(x

) =

1 −

x -

2

8. f

(x) =

x

- 8

, g(x

) = x

+ 4

8

- 3

x −

x

- 2

;

1

−

2x

- 5

; -

2

x -

4;

x -

4;

0

Stud

y Gu

ide

and

Inte

rven

tion

(con

tinu

ed)

Fu

ncti

on

Op

era

tio

ns

an

d C

om

po

siti

on

of

Fu

ncti

on

s

Exam

ple

1-6

027-

042_

PC

CR

MC

01_8

9380

2.in

dd33

3/22

/09

5:52

:26

PM


Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pan

ies, In

c.



PDF Pass


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

1

34

Gle

ncoe

Pre

calc

ulus

Fin

d (f

+ g

)(x)

, (f

- g

)(x)

, (f

· g)(

x), a

nd ( f −

g ) (x

) fo

r ea

ch f

(x)

and

g(x)

. Sta

te t

he d

omai

n of

eac

h ne

w f

unct

ion.

1.

f(x) =

2x2 +

8 a

nd g

(x) =

5x

- 6

2.

f(x

) = x

3 and

g(x

) = √

��

�

x +

1

2x2

+ 5

x +

2,

D =

(-

∞,

∞)

x3

+ √

��

�

x +

1 ,

D =

[-

1,

∞)

2x2

- 5

x +

14

, D

= (

-∞

, ∞

)

x3 -

√ �

��

x +

1 ,

D =

[-

1,

∞)

10

x3 -

12

x2 +

40

x -

48

,

x3 √

��

�

x +

1 ,

D =

[-

1,

∞)

D =

(-

∞,

∞)

2x2

+ 8

−

5x

- 6

, D

= {x

|x ≠

6

−

5 ,

x ∈

�}

x3

−

√

��

�

x +

1 ,

D =

(-

1,

∞)

For

eac

h pa

ir o

f fu

ncti

ons,

fin

d [f

◦ g

](x)

, [g

◦ f

](x)

, and

[f

◦ g

](3)

.

3.

f(x) =

x +

5 a

nd g

(x) =

x -

3

4. f

(x) =

2x3 -

3x2 +

1 a

nd g

(x) =

3x

x +

2;

x +

2;

4

54x3

- 2

7x2

+ 1

; 6x3

- 9

x2 +

3;

1216

5.

f(x) =

2x2 -

5x

+ 1

and

g(x

) = 2

x -

3

6. f

(x) =

3x2 -

2x

+ 5

and

g(x

) = 2

x -

1

8x2

- 3

4x

+ 3

4;

4x2

- 1

0x

- 1

; 4

12

x2 -

16

x +

10

; 6

x2 -

4x

+ 9

; 7

0

Fin

d f

◦ g

.

7.

f(x) =

√ �

��

x -

2

8. f

(x) =

1

−

x -

8

g(x)

= 3

x

g(x)

= x

2 + 5

{x| x

≥ 2

−

3 ,

x ∈

�} ;

f ◦

g =

√ �

��

3x

- 2

{ x | x

≠ ±

√ �

3

, x

∈ �

} ; f

◦ g

=

1

−

x2 -

3

Fin

d tw

o fu

ncti

ons

f an

d g

such

tha

t h(

x) =

[f

◦ g

](x)

. Nei

ther

fu

ncti

on m

ay b

e th

e id

enti

ty f

unct

ion

f(x)

= x

.

9.

h(x)

= √

��

�

2x -

6 -

1 10

. h(

x) =

1

−

3x +

3

Sa

mp

le a

ns

we

r: f

(x)

= √

�

x -

1,

S

am

ple

an

sw

er:

f(x

) =

1

−

3x ,

g

(x)

= 2

x -

6

g

(x)

= x

+ 1

11. R

ESTA

URA

NT

A g

roup

of t

hree

res

taur

ant

patr

ons

orde

r th

e sa

me

mea

l an

d dr

ink

and

leav

e an

18%

tip

. Det

erm

ine

func

tion

s th

at r

epre

sent

the

co

st o

f all

of t

he m

eals

bef

ore

tip,

the

act

ual t

ip, a

nd t

he c

ompo

siti

on o

f th

e tw

o fu

ncti

ons

that

giv

es t

he c

ost

for

all o

f the

mea

ls in

clud

ing

tip.

f(x)

= 3

x, w

he

re x

is

th

e c

os

t fo

r o

ne

me

al;

g(x

) =

1.1

8x;

g(f

(x))

= 3

.54

x

Prac

tice

Fu

ncti

on

Op

era

tio

ns

an

d C

om

po

siti

on

of

Fu

ncti

on

s

1-6

027-

042_

PC

CR

MC

01_8

9380

2.in

dd34

9/30

/09

2:04

:49

PM


Lesson 1-6

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

1

35

Gle

ncoe

Pre

calc

ulus

1.

MA

RC

HIN

G B

AN

D B

and

mem

bers

fo

rm a

cir

cle

of r

adiu

s r

whe

n th

e m

usic

st

arts

. The

y m

arch

out

war

d as

the

y pl

ay. T

he fu

ncti

on f(

t) =

2.5

t giv

es t

he

radi

us o

fth

e ci

rcle

in fe

et a

fter

t se

cond

s.U

sing

g(r

) = π

r2 fo

r th

e ar

ea o

f the

ci

rcle

, wri

te a

com

posi

te fu

ncti

on t

hat

give

s th

e ar

ea o

f the

cir

cle

afte

r t

seco

nds.

Then

find

the

are

a, t

o th

e ne

ares

t te

nth,

af

ter

4 se

cond

s.

g

[f(t

)] =

6.2

5π

t2;

31

4.2

ft2

2. C

AN

DLE

S A

hob

byis

t m

akes

and

sel

ls

cand

les

at a

loca

l mar

ket.

The

func

tion

c(

h) =

4h

give

s th

e nu

mbe

r of

can

dles

sh

e ha

s m

ade

afte

r h

hour

s. T

he fu

ncti

on

f(c) =

12

+ 0

.25c

giv

es t

he c

ost

of m

akin

g c

cand

les.

a. W

rite

the

com

posi

te fu

ncti

on t

hat

give

s th

e co

st o

f can

dle

mak

ing

afte

r h

hour

s.

f[

c(h

)] =

12

+ h

b. A

sal

e re

duce

s th

e co

st o

f mak

ing

c ca

ndle

s by

10%

. Wri

te t

he s

ale

func

tion

s(x

) and

the

com

posi

te

func

tion

tha

t gi

ves

the

cost

of c

andl

e m

akin

g af

ter

h ho

urs

if m

ater

ials

are

pu

rcha

sed

duri

ng t

he s

ale.

s(

x) =

0.9

x;

s{f[

c(h

)]}

= 1

0.8

+ 0

.9h

3. S

CIEN

CE T

he fu

ncti

on t(

x) =

√ �

2x

−

28

+ 6

.25

give

s th

e te

mpe

ratu

re in

deg

rees

Cel

sius

of

the

liqu

id in

a b

eake

r af

ter

x se

cond

s.

Dec

ompo

se t

he fu

ncti

on in

to t

wo

sepa

rate

func

tion

s, s

(x) a

nd r

(x),

so t

hat

s(r(

x)) =

t(x)

.

S

am

ple

an

sw

er:

s(x

) =

x

−

28 +

6.2

5;

r(x

) =

√ ��

2

x

4. T

RAV

ELTw

o tr

avel

ers

are

budg

etin

g m

oney

for

the

sam

e tr

ip. T

he fi

rst

trav

eler

’s bu

dget

(in

dolla

rs) c

an b

e re

pres

ente

d by

f(x)

= 4

5x+

350

. The

se

cond

tra

vele

r’s b

udge

t (in

dol

lars

) can

be

rep

rese

nted

by

g(x)

= 6

0x+

475

x is

th

e nu

mbe

r of

nig

hts.

a. F

ind

(f+

g)(x

) and

the

rel

evan

t do

mai

n. (

f+

g)(

x) =

10

5x

+ 8

25

; D

= {

x | x

≥ 0

, x

∈�

}

b. W

hat

does

the

com

posi

te fu

ncti

on in

pa

rt a

rep

rese

nt?

the

co

mb

ine

d

bu

dg

et

of

bo

th t

rav

ele

rs

c.

Find

(f+

g)(7

) and

exp

lain

wha

t th

e va

lue

repr

esen

ts.

$1

56

0;

the

c

om

bin

ed

am

ou

nt

tha

t c

an

be

s

pe

nt

by

th

e t

rav

ele

rs o

n a

7

-nig

ht

trip

d. R

epea

t pa

rts

a–c

for

(g-

f)(x)

.

a

: (g

-f)

(x)

= 1

5x

+ 1

25

;

D

= {

x | x

≥ 0

, x

∈�

}

b

: h

ow

mu

ch

mo

re t

he

se

co

nd

tr

av

ele

r c

an

sp

en

d t

ha

n t

he

fi

rst

c:

$2

30

; h

ow

mu

ch

mo

re t

he

s

ec

on

d t

rav

ele

r c

an

sp

en

d o

n a

7

-nig

ht

trip

5. P

OPU

LATI

ON

The

func

tion

p(

x) =

2x2

- 1

2x+

18

pred

icts

the

po

pula

tion

of e

lk in

a fo

rest

for

the

year

s 20

10 t

hrou

gh 2

015

whe

re x

is t

he

num

ber

of y

ears

sin

ce 2

000.

Dec

ompo

se

the

func

tion

into

tw

o se

para

te fu

ncti

ons,

a(

x) a

nd b

(x),

so t

hat

[a◦

b](x

) =p(

x) a

nd

a(x)

is a

qua

drat

ic fu

ncti

on a

nd b

(x) i

s a

linea

r fu

ncti

on.

S

am

ple

an

sw

er:

a(x

) =

2x2

; b

(x)

=x

- 3

Wor

d Pr

oble

m P

ract

ice

Fu

ncti

on

Op

era

tio

ns

an

d C

om

po

siti

on

of

Fu

ncti

on

s

1-6

027-

042_

PC

CR

MC

01_8

9380

2.in

dd35

10/1

/09

10:1

5:58

PM


NAME DATE PERIOD 1-5 Study Guide and · PDF fileLesson 1-5 NAME DATE PERIOD ... Lesson 1-6...

Documents

Transcript of NAME DATE PERIOD 1-5 Study Guide and · PDF fileLesson 1-5 NAME DATE PERIOD ... Lesson 1-6...