..
Date
Name
Practice B
Foruse withpages 428-435
·
Let"(x)
= 7x1/2 - 2,
g(x)
= _x1/2
+ 4, and h(x)
= -4x1/2
+ 1.
Perform the indicated operation.
1. f(x) + g(x)
2. f(x) + hex)
3.
4. f(x) - g(x)
5. hex)- f(x)
6. g(x) - hex)
hex)+ g(x)
Let ,(x) = 4x2, g(x) = -3x4l3, and hex)= .1/2. Perform the
indicated operation.
7.
10.
f(x). g(x)
8. f(x). hex)
/
h(i)
11. f(x)
fix)
g(x)
Let ,(x)
indicated
"..
.
= 2x + 3, g(x) = x ~ l'
h(x)
12.
and h(x)
operation.
9. h(x). g(x)
=x ~ 5. Perform
13. f(g(x»
14. g(h(x»
15. f(h(x»
"16. g(f(x»
17. h(f(x»
18. g(g(x»
g(x)
the
I;
Let ,(x)
= 3x + 2, g(x) = 2x2,
and h(x)
of the operation.
""0
;~ti'
19. f(x)+g(x)
lXL.-\-'}x>rL.-
21. h(x). g(x)
-t,)(L.
23.
h(g(x»
In Exercises
= .J+-4 3 . State
the domain
20. h(x)-f(x)
g()
.
22. jiyx)
,x
~
-..!:!-
24.
2x't.-t2r
25-29, use the following
_3){1.._\l~-lQ.
Xt-}
~'> 'I.t-"1..
f(g(x»
'3( 1K1.)r"2..
1-
= <c x t 1-
information.
Computer Sale You have a coupon for $200 off the price' of a personal computer. When
you~ve at the store, you find thayhe computers are on sale for 20% off. Let x represent
the onmnal price of the computot.'
25.
to describe your cost,j(x), using only the coupon.
26.
,tationto describe your co~t,g(x), with only the 20% discount.
27.
Form the ~).psition
couponjl"rst, thefi~e
28.
F<>Jdithe composi~ of the functionsf and g that represents your cost if you use the
scount first, the;:~~e
coupon.
2...
Would you pay less for th~uter
discount first?
Algebra2
32
Chapter6 ResourceBook
of the functionsf and g that represents your cost if you use the
the 20% discount.
if you used the coupo~ first or took the 20%
>c:
m
c..
E
uo
c:
~
~
c:
S!
..c:
C>
::>
o
,:J:
-o
c:
o
'w
";;
'6
m
Lesson 6.2, continued
~;
5. (2x - 1)-1I2(2x-~)
6. (3x + 2)3/2(~x+ 2)
7. i(4 + x)2/3(3x- 4) 8. k(2x - 3)3/2(2x+ 3)
28.f(g(x» = 0.8x - 200 29. pay less with
discount first
Practice
Level C
1. 2x2 -+- x - xl/2 - 4 2. 6x2 + 2x1l2- 5
3. 4x2 + x -
9. ~x(x2 + 1)-2/3(4x2+ 3)
-
4.2x2
10. ~(3x + 2)-3/2(x - 6)1/2(38- 3x)
Lesson
6.3
Teaching Guide
1. -x2 + 7x - 18 2. x2 - llx + 42
3. - 3xl/2+ 9x2/3 4. x3 + 4x2 - 6x - 9
4x2
~ 8
6 18x11l15 7 .3.4
5 .6.
Y
X
Y
Practice Level A
1. 4xl/2 + 1 2. -x1l2 + 5 3. X1/2+ 2
4. -2x1l2 + 3 5. :-3xl/2 + 1 6. 5xll2 - 4
7. 8x 1I/6 8. - 24x2 9. -12x516 10. 2x7/6
125x3
"
3
11. -2x
16.48
'.
116
3/2
. 11.x
I
+
+
!
..,
Practice Level B
1. 6xl/2 + 2 2. 3xll2 - 1 3. -5x1l2 + 5
4. 8xl/2 - 6 5. -llxll2 + 3 6. 3xll2 + 3
7. -12xID/3 8.4xs/2 9. -3x11l6 10. _ 4x3213
1
1
3x+9
6
.t
J
l
".
11. 4x312 12.
15.x+8
-
3x516 13.
3
16.2x+4
-;+T
14. x + 7
3x+ 3
18.x+4
17.x+4
19. all real numbers 20. aUreal numbers except
x
=-3
21. all real numbers except x
= -3
2
22. all real numbers except x.=
~3
23. all real numbers 24. all real numbers
25.f(x)
27.g(f(x»
=x
- 200 26. g(x)
= 0.8x - 160
= 0.8x
x + 7xl/2 - 6 5.2x2 - 4xll2 + 5
1
X5/6
+ Ii2
12. -- 3
x
2 - xll2
x ; positive real numbers
13.
(
1/2
3
)
14. x-I
2x2
15.
16.
; all real numbers greater than 1
- 7x + 5
9
1
; all real numbers
(2x2 -x )
112;
1
all real numbers less than 0 and greater than 2
2x2 -
17.
21.~
22. 5x+4 23. -2x+ 1
x-4
x
x
24. 4x + 15 25. all real numbers 26. all real
numbers except x = 2 27. all real numbers
except x = 2 28. all real numbers except x = -1
29. all real numbers except x ="j:Y2 30. all real
numbers 31. P(x) = 0.65x - 15,000; $310,000
I
3
6. -4x2 + x - 3xl/2 + 1 7. -3x5/3 - ill
x
_ 7/3_ 1/3
8. xS/2 + xl/2 9. -3x1/6 10. x
x
16
3
12. - x
13.3 14.5" 15. 1
2
2
17.-'5
18.8 19.2x+5
20.x+ 1
'
5x 112+
x-I
3
; all real numbers
18. x 114;positive real numbers 19. True
20. False; Sample answer: letf(x) = x and
g(x)
=x +
1 21. True
22. False; Sample answer: let/ex)
. g(x)
.
23. False; Sample answer: letf(x)
g(x)
= x and
=x + I
= "';-;.'and
=x + 1
.24. False; Sample answer: letf(x)
=x +
I
26. Sample answer:f(x)
= ~, g(x) = x + 1
= x3 + 2, g(x) = ...;-;
27. Sample answer: f(x)
= -y,x
25. Sample answer: f(x)
x+ 1
g(x)
= x-I
28. Letf(x) = 0.7x,g(x) = x - 10,hex) = 0.9x
f(g(h(x))) = 0.63x - 7
f(h(g(x») = 0.63x - 6.3
g(f(h(x») = 0.63x - 10
.g(h(f(x») = 0.63x - 10
h(f(g(x») = 0.63x - 6.3
h(g(f(x») = 0.63x - 9
The store will most likely deduct the $10 coupon
first and then take the 30% and 10% discount in
any order.
Algebra2
Chapter6 Resource
Book
-
A3
.
~Chapter 6,
continued
38. A;
'}2./(-9) = 3(-9) + 2 = -25
-25 - 2
27
h(j(-9» = h(-25) = -S= -5
8- 2
6
= --S = 5.
23. h(8)
g(h(8»
24. g(5)
J6
6 2
36
-25 - 2
= h(-25) = -S-
27
40. Sampleanswer:
= -5
f(x)
= 3(7) + 2 = 23
f(f(7» =f(23) = 3(23) + 2 = 71
-4 - 2
-5 -
27. g(-5)
2
16
= h(-5 )= ---s- = - 25
= _(-5)2 = -25
f(x)
~7
28.f(g(x» = f(a - 7) = 3(a - 7)-1 = 2x
The domain off(g(x» consists of all real numbers except
=
f
29. g(f(x»
because
~ f) = 0
is not in the domain
off
= g(3x-l) = 2(3x-l)- 7 = 6x-1- 7 = ~ - 7
= 0 because 0 is not in the domain off
30. h(f(x»
= h(3x-l) = 3X-~+ 4 = 3X3-1+ j = ~ + j
Thedomainof h(f(x» consistsof all realnumbersexcept
x
= 0 because 0 is not in the domain off
31. g(h(x»
= ~x; 4) = 2(X; 4) - 7
= 2x 3+ 8 _ 7 = 2x + 83 - 21 = 2x -3 13
The domain of g(h(x» consists of all real numbers.
32. h(g(x»
= h(a _
7)
= 2x -
;
+ 4
= 2x; 3
The domain of h(g(x» consists of all real numbers.
33. f(f(x»
= j(3x-l) = 3(3x-I)-I= 3(rlx) = 30x= x
The domain off(f(x» consists of all real numbers except
x = 0, because 0 is not in the domain off
x+4+4
34. h(h(x»=h(x;4)=
33
_x+~+
12=x~16
The domain of h(h(x» consists of all real numbers.
35. g(g(x»
= g(a -7) = 2(a - 7) =4x - 14 - 7 = 4x - 21
3
37. The product 4(x2 - 3) was'not performed correctly.
g(f(x»
= g(x2 -
3)
= 4(x2 -
+ 9)
= la
3) = 4x2
-
12
+
91
Problem Solving
1.1 wO.734
.
= b(w) -
d(w)
1.1wO.734
- O.007w- O.002w
_
-
1.1 wO.734
O.005w
= 220w(0.734 - I)
= 220w-0.266
r(w) = 220w-O.266
r(6.5) = 220(6.5)-0.266 = 134
.
The breathing rate of a mammal that weighs 6.5 grams is
about 134breaths per minute.
r(300)
= 220(300)-0.266 = 48.3
The breathing rate of a mammal that weighs 300 grams is
about 48.3 breaths per minute.
r(70,000) = 220(70,000)-0.266= 11.3
The breathing rate of a mammal that weighs 70,000 grams
is about 11.3 breaths per minute.
= C(50t) = 60(50t) + 750 = 3000t +
C(x(5» = 3000(5) + 750 = 15,750
44. C(x(t»
750
This number represents the cost ($15,750) of5 hours of
production in the factory.
45. Let x represent the regular price.
Function
36. When performingf(4x), 4x should have been -substituted
for x in the functionf
=f(4x) = (4x)2 - 3 = 1&2 -
= f(a
Function for $15 discount:f(x)
7
The domain of g(g(x» consists of all real numbers.
f(g(x»
= Ixl. g(x)= a + 9
hex) = f(g(x»
43. r(w)
Thedomainof g(f(x» consistsof all real numbersexcept
x
+2
hex) = f(g(x» = f(3x2) = --f3x + 7
42. Sample answer:
g(g(-5».= g(-25) = -(-25)2 = -625
x
=x + 2
=f(x + 2) = \Ix
g(x)
41. Sampleanswer:
4
f(x) = x + 7' g(x) = 3x2
6
6
= ~,
hex) =f(g(x»
=---s- = -5 6
h(h(-4»
= x, g(x)= x-I
f(g(x» = g(f(x»
j(x-I) = g(x)
x-I = x-I
25. f(7)
26. h(-4)
= g(7x2) = 3(7x2)-2 = 3(T2x-4) = --L
49x4
39. Sampleanswer:f(x)
= gu )= - (5) = -25
= -52 = -25
h(g(5»
g(f(x»
for 10% discount:
g(x)
=x -
15
= x - O.lx = 0.9x
a. g(f(x» = g(x - 15) = 0.9(x - 15)
g(f(85» = 0.9(85 - 15) = 0.9(70) = 63
The sale price is $63 when.the $15 discount is applied
before the 10% discount.
b. f(g(x»
f(g(85»
=f(0.9x) = 0.9x - 15
= 0.9(85)- 15 = 76.5 -
.
15 = 61.50
The sale price is $61.50 when the I 0% discount is
applied before the $15 discount.
Algebra2
Worked-Out
SolutionKey
--
373
Chapter 6,
6.3 Exerci.e.
continued
(pp. 432-434J
12. f(x)
.
g(x)
Skill Practice
5xl12
+ 112)
=2Ox7116
Domain off all real numbers
2. The sum of.two power functions is sometimes a power
Domain of g: all nonnegative real numbers
Domain off. g: all nonnegative real numbers
.
13. g(x)
Sample answer:
.
j{x) = 5xl12. 4x213
f(x) = 2x1l3,g(x) = 4x-1/3
= 20X<112 + 213)
fix) + g(x) = 2x1l3+ 4x-l13
= 20x7/6
f(x) = 2x113,g(x)
= 4xl13
Domain of g: all nonnegative real numbers
+ g(x) = 2x1/3 + 4xll3 = 6x113
3. f(x) + g(x) = -3xl13 + 4xl12+ 5xl13+ 4xl12
Domain off all real numbers
f(x)
= (-3
+ 5)x1l3 + (4 + 4)x112
= 2x113
+
.
..f(x) = 4x213.4x213
Domain of g f all nonnegative real numbers
14. f(x)
=
&x112
4. g(x) + f(x)
= 5xl13 + 4xl12 -
= (5 -
3)x113
=
+
2x113
3xl13
+
Domain off all real numbers
4xl12
Domain off. f all real numbers
+ (4 + 4)x112
&x112
15. . g(x)
.g(x) =
=
Domain: all nonnegative real nUmbers
5. f(x) + f(x) = "'-3xl/3 + 4xl12 + (-
= (-3
3xl/3
+
- 3)x113+ (4 + 4)x112
&x112
Domain: all nonnegative real numbers
= 5xl13 + 4xl12 + 5xl13 + 4xl12
= (5 + 5)x1l3 + (4
= lOxl13 + &x112
+ 4xl12 - (- 3xl13 + 4xl12)
3)x113+ (4 - 4)x112
.
Domain: all nonnegative real numbers
9. f(x)
- f(x) = - 3xl13
= (-3
+ 4xl12 - (- 3xl13 + 4xl12)
+ 3)x113+ (4 - 4)x112
16. g(x)
4x2f3
= Sxl12 =
g: all nonnegative real numbers
4X<213
- 112)
5
4xl/6
=5
Domain off all real numbers
Domain of£: all positive real numbers
g
17
g(x)
-
Sxl12
. f(x) -
4x2f3
SX<112 213)
-
4
5x-I/6
---- 4 -
5
4x\l6
Domain of g: all nonnegative.real numbers
Dom,ainof]: all positive real numbers
f(x)
4x2f3
18. f(x)
= 4x213 = I
Domain off all real numbers
Domain or§: all real numbers except x
=0
Domain: all nonnegative real numbers
10. g(x) - g(x) =,5xl13 + 4xl12- (5xl13+ 4xl12)
= (5 -
.
Domain of g
f(x)
.
Domain off all real numbers
Domain:all nonnegativerealnumbers
=&x1/3
5xl12
+ 112)
Domain of g: all nonnegative real numbers
= _&xI13
- f(x) = 5xl/3
= (5 +
.
Domain of g: all nonnegative real numbers
+ 4)x112
Domain:all nonnegativerealnumbers
7. f(x) - g(x) = -3xl13 + 4xl12- (5xl13+ 4xl12)
= (-3 - 5)x1l3+ (4 - 4)x112
8. g(x)
5xl12
25X<112
= 25x
4xl12)
= -6xI13 +
6. g(x) + g(x)
16x<2I3 + 213)
= 16x413
Domain: all nonnegative real numbers
5)x1/3 + (4
-
g(x)
19.
g( x)
=0
5xl12
= Sxm= 1
Domain of g: all nonnegative real numbers
4)x112
=0
Domain:all nonnegativerealnumbers
11. B;
f(x) + g(x) = -7x213- I + 2?3 + 6
~(-7+2)x2l3-1
+6
= -5x213+ 5
372
.
1. Thefunctionh(x) = g(f(x)) is caIled the composition of
the functiong withthe function!
function.
.
= 4x213
= 20X<2I3
Domain of~: all positive real numbers
g
20. g(-3)
= -(-3i = -9
f(g(-3)) = f(-9) = 3(-9) + 2 = -25
21. f(2) = 3(2) + 2= 8
g(f(2)) = g(8) = -82 = -64
Algebra2
Worked-Out
SolutionKey
--