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CHAPTER 1 FUNCTIONS
1.1 FUNCTIONS AND THEIR GRAPHS
1. domain œ (_ß _); range œ [1ß _) 2. domain œ [0ß _); range œ (_ß 1]
3. domain œ Ò2ß _); y in range and y œ È5x 10 ! Ê y can be any positive real number Ê range œ Ò!ß _).
4. domain œ (_ß 0Ó Ò3, _); y in range and y œ Èx2 3x ! Ê y can be any positive real number Ê range œ Ò!ß _).
4 4
5. domain œ (_ß 3Ñ Ð3, _); y in range and y œ 3t, now if t 3 Ê 3 t ! Ê 3t !, or if t 3
4
Ê3 t!Ê 3t ! Ê y can be any nonzero real number Ê range œ Ð_ß 0Ñ Ð!ß _).
2 2
6. domain œ (_ß %Ñ Ð4, 4Ñ Ð4, _); y in range and y œ t2 16 , now if t % Ê t2 16 ! Ê t2 16 !, or if
2 # 2 2 2
% t 4 Ê 16 Ÿ t 16 ! Ê "' Ÿ t2 16 !, or if t % Ê t 16 ! Ê t2 16 ! Ê y can be any
nonzero real number Ê range œ Ð_ß 18 Ó Ð!ß _).
7. (a) Not the graph of a function of x since it fails the vertical line test.
(b) Is the graph of a function of x since any vertical line intersects the graph at most once.
8. (a) Not the graph of a function of x since it fails the vertical line test.
(b) Not the graph of a function of x since it fails the vertical line test.
# È3 È3 È3
9. base œ x; (height)# ˆ #x ‰ œ x# Ê height œ # x; area is a(x) œ "
# (base)(height) œ "
# (x) Š # x‹ œ 4 x# ;
perimeter is p(x) œ x x x œ 3x.
"
10. s œ side length Ê s# s# œ d# Ê s œ d
È2 ; and area is a œ s# Ê a œ # d#
11. Let D œ diagonal length of a face of the cube and j œ the length of an edge. Then j# D# œ d# and
$Î#
6d# #
d$
D# œ 2j# Ê 3j# œ d# Ê j œ d
È3 . The surface area is 6j# œ 3 œ 2d# and the volume is j$ œ Š d3 ‹ œ 3È 3
.
Èx
12. The coordinates of P are ˆxß Èx‰ so the slope of the line joining P to the origin is m œ x œ "
Èx (x 0). Thus,
ˆx, Èx‰ œ ˆ m"# , "‰
m .
13. 2x 4y œ 5 Ê y œ "# x 54 ; L œ ÈÐx 0Ñ2 Ðy 0Ñ2 œ Éx2 Ð "# x 54 Ñ2 œ Éx2 4" x2 54 x 25
16
20x 25 È20x2 20x 25
œ É 54 x2 54 x œ É 20x
25 2
16 16 œ 4
14. y œ Èx 3 Ê y2 3 œ x; L œ ÈÐx 4Ñ2 Ðy 0Ñ2 œ ÈÐy2 3 4Ñ2 y2 œ ÈÐy2 1Ñ2 y2
œ Èy4 2y2 1 y2 œ Èy4 y2 1
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2 Chapter 1 Functions
15. The domain is a_ß _b. 16. The domain is a_ß _b.
17. The domain is a_ß _b. 18. The domain is Ð_ß !Ó.
19. The domain is a_ß !b a!ß _b. 20. The domain is a_ß !b a!ß _b.
21. The domain is a_ß 5b Ð5ß 3Ó Ò3, 5Ñ a5, _b 22. The range is Ò2, 3Ñ.
23. Neither graph passes the vertical line test
(a) (b)
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Section 1.1 Functions and Their Graphs 3
24. Neither graph passes the vertical line test
(a) (b)
Ú xyœ" Þ Ú yœ1x Þ
kx yk œ 1 Í Û or ß Í Û or ß
Ü x y œ " à Ü y œ " x à
25. x 0 1 2 26. x 0 1 2
y 0 1 0 y 1 0 0
"
4 x2 , x Ÿ 1 , x0
27. Faxb œ œ 28. Gaxb œ œ x
x2 2x, x 1 x, 0 Ÿ x
29. (a) Line through a!ß !b and a"ß "b: y œ x; Line through a"ß "b and a#ß !b: y œ x 2
x, 0 Ÿ x Ÿ 1
f(x) œ œ
x 2, 1 x Ÿ 2
Ú
Ý 2, ! Ÿ x "
Ý
!ß " Ÿ x #
(b) f(x) œ Û
Ý
Ý 2ß # Ÿ x $
Ü !ß $ Ÿ x Ÿ %
30. (a) Line through a!ß 2b and a#ß !b: y œ x 2
Line through a2ß "b and a&ß !b: m œ !& "
# œ
"
$ œ "$ , so y œ "$ ax 2b " œ "$ x &
$
x #, 0 x Ÿ #
f(x) œ œ "
$ x &$ , # x Ÿ &
$ !
(b) Line through a"ß !b and a!ß $b: m œ ! Ð"Ñ œ $, so y œ $x $
" $ %
Line through a!ß $b and a#ß "b: m œ #! œ # œ #, so y œ #x $
$x $, " x Ÿ !
f(x) œ œ
#x $, ! x Ÿ #
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4 Chapter 1 Functions
31. (a) Line through a"ß "b and a!ß !b: y œ x
Line through a!ß "b and a"ß "b: y œ "
Line through a"ß "b and a$ß !b: m œ !"
$" œ
"
# œ "# , so y œ "# ax "b " œ "# x $
#
Ú x " Ÿ x !
f(x) œ Û " !xŸ"
Ü "# x $# "x$
(b) Line through a2ß 1b and a0ß 0b: y œ 12 x Ú 1
2 Ÿ x Ÿ 0
2x
Line through a0ß 2b and a1ß 0b: y œ 2x 2 faxb œ Û 2x 2 0xŸ1
Ü 1 1xŸ3
Line through a1ß 1b and a3ß 1b: y œ 1
32. (a) Line through ˆ T# ß !‰ and aTß "b: m œ "!
TaTÎ#b œ T# , so y œ T# ˆx T# ‰ 0 œ T# x "
!, 0 Ÿ x Ÿ T#
faxb œ # T
T x ", # x Ÿ T
Ú
Ý A, ! Ÿ x T#
Ý
Ý T
Aß # Ÿx T
(b) faxb œ Û
Ý
Ý Aß T Ÿ x $#T
Ý $T
Ü Aß # Ÿ x Ÿ #T
33. (a) ÚxÛ œ 0 for x − [0ß 1) (b) ÜxÝ œ 0 for x − (1ß 0]
34. ÚxÛ œ ÜxÝ only when x is an integer.
35. For any real number x, n Ÿ x Ÿ n ", where n is an integer. Now: n Ÿ x Ÿ n " Ê Ðn "Ñ Ÿ x Ÿ n. By
definition: ÜxÝ œ n and ÚxÛ œ n Ê ÚxÛ œ n. So ÜxÝ œ ÚxÛ for all x − d .
36. To find f(x) you delete the decimal or
fractional portion of x, leaving only
the integer part.
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