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Less
on
1-5
NAME DATE PERIOD
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© G
lencoe/M
cG
raw
-Hill
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isio
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-Hill
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Inc.
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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
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Chapter 1 28 Glencoe Precalculus
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
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Copyright
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lencoe/M
cG
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-Hill
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isio
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-Hill
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Inc.
Less
on
1-5
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Chapter 1 29 Glencoe Precalculus
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
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-Hill C
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s, In
c.
NAME DATE PERIOD
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Chapter 1 30 Glencoe Precalculus
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
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Chapter 1 32 Glencoe Precalculus
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
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Chapter 1 33 Glencoe Precalculus
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
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Chapter 1 34 Glencoe Precalculus
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
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PDF 2nd PDF 2nd
Chapter 1 35 Glencoe Precalculus
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
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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
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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_
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/09
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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.
Chapter 1 A13 Glencoe Precalculus
An
swer
s
Answers (Lesson 1-5)
Pdf 3rd
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
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/09
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:21
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
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PM
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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 A14 Glencoe Precalculus
Answers (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
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_
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/09
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:06
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
A01_A20_PCCRMC01_893802.indd 14A01_A20_PCCRMC01_893802.indd 14 3/23/09 11:12:26 PM3/23/09 11:12:26 PM
Copyright
© G
lencoe/M
cG
raw
-Hill
, a
div
isio
n o
f T
he
McG
raw
-Hill
Co
mp
an
ies,
Inc.
Chapter 1 A15 Glencoe Precalculus
An
swer
s
Answers (Lesson 1-6)
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
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
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
A01_A20_PCCRMC01_893802.indd 15A01_A20_PCCRMC01_893802.indd 15 10/5/09 10:39:44 PM10/5/09 10:39:44 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.
Chapter 1 A16 Glencoe Precalculus
Answers (Lesson 1-6)
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
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
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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
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