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# ISMT12 C01 D

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```Section 1.4 Graphing with Calculators and Computers 29
25. 0 03 0 03 by 1 25 1 25 26. 0 1 0 1 by 3 3Ò\ufffd Þ ß Þ Ó Ò\ufffd Þ ß Þ Ó Ò\ufffd Þ ß Þ Ó Ò\ufffd ß Ó

27. 300 300 by 1 25 1 25 28. 50 50 by 0 1 0 1Ò\ufffd ß Ó Ò\ufffd Þ ß Þ Ó Ò\ufffd ß Ó Ò\ufffd Þ ß Þ Ó

29. 0 25 0 25 by 0 3 0 3 30. 0 15 0 15 by 0 02 0 05Ò\ufffd Þ ß Þ Ó Ò\ufffd Þ ß Þ Ó Ò\ufffd Þ ß Þ Ó Ò\ufffd Þ ß Þ Ó

31. x x y y y x x .# # #\ufffd # \u153 % \ufffd % \ufffd Ê \u153 # \u201e \ufffd \ufffd # \ufffd )È
The lower half is produced by graphing
y x x .\u153 # \ufffd \ufffd \ufffd # \ufffd )È #

32. y x y x . The upper branch# # #\ufffd &quot;' \u153 &quot; Ê \u153 \u201e &quot; \ufffd &quot;'È
is produced by graphing y x .\u153 &quot; \ufffd &quot;'È #

30 Chapter 1 Functions
33. 34.
35. 36.
37. 38 Þ
39. 40.
CHAPTER 1 PRACTICE EXERCISES
1. The area is A r and the circumference is C r. Thus, r A .\u153 \u153 # \u153 Ê \u153 \u1531 1 1# # # %
#C C C
1 1 1
\u2c6 \u2030 #
2. The surface area is S r r . The volume is V r r . Substitution into the formula for\u153 % Ê \u153 \u153 Ê \u1531 1# \$% \$ %
&quot;Î# % \$\u2c6 \u2030 ÉS V
1 1
\$
surface area gives S r .\u153 % \u153 %1 1# \$%
#Î\$\u2c6 \u2030V
1
Chapter 1 Practice Exercises 31
3. The coordinates of a point on the parabola are x x . The angle of inclination joining this point to the origin satisfiesa bß # )
the equation tan x. Thus the point has coordinates x x tan tan .) ) )\u153 \u153 ß \u153 ßx
x
# a b a b# #
4. tan h tan ft.) )\u153 \u153 Ê \u153 &!!rise h
run &!!
5. 6.
Symmetric about the origin. Symmetric about the y-axis.
7. 8.
Neither Symmetric about the y-axis.
9. y x x x y x . Even.a b a b a b\ufffd \u153 \ufffd \ufffd &quot; \u153 \ufffd &quot; \u153# #
10. y x x x x x x x y x . Odd.a b a b a b a b a b\ufffd \u153 \ufffd \ufffd \ufffd \ufffd \ufffd \u153 \ufffd \ufffd \ufffd \u153 \ufffd& \$ & \$
11. y x cos x cos x y x . Even.a b a b a b\ufffd \u153 &quot; \ufffd \ufffd \u153 &quot; \ufffd \u153
12. y x sec x tan x sec x tan x y x . Odd.a b a b a b a b\ufffd \u153 \ufffd \ufffd \u153 \u153 \u153 \ufffd \u153 \ufffdsin x
cos x cos x
sin xa b
a b
\ufffd
\ufffd
\ufffd
# #
13. y x y x . Odd.a b a b\ufffd \u153 \u153 \u153 \ufffd \u153 \ufffda ba b a b\ufffd \ufffd&quot;\ufffd \ufffd# \ufffd \ufffd&quot; \ufffd&quot;\ufffd \ufffd# \ufffd#xx x x xx x x x%\$ % %\$ \$
14. y x x sin x x sin x x sin x y x . Odd.a b a b a b a b a b a b\ufffd \u153 \ufffd \ufffd \ufffd \u153 \ufffd \ufffd \u153 \ufffd \ufffd \u153 \ufffd
15. y x x cos x x cos x. Neither even nor odd.a b a b\ufffd \u153 \ufffd \ufffd \ufffd \u153 \ufffd \ufffd
16. y x x cos x x cos x y x . Odd.a b a b a b a b\ufffd \u153 \ufffd \ufffd \u153 \ufffd \u153 \ufffd
17. Since f and g are odd f x f x and g x g x .Ê \ufffd \u153 \ufffd \ufffd \u153 \ufffda b a b a b a b
(a) f g x f x g x f x g x f x g x f g x f g is evena ba b a b a b a b a b a b a b a ba b\u2020 \ufffd \u153 \ufffd \ufffd \u153 Ò\ufffd ÓÒ\ufffd Ó \u153 \u153 \u2020 Ê \u2020
(b) f x f x f x f x f x f x f x f x f x f x f x f is odd.3 3 3a b a b a b a b a b a b a b a b a b a b a b\ufffd \u153 \ufffd \ufffd \ufffd \u153 Ò\ufffd ÓÒ\ufffd ÓÒ\ufffd Ó \u153 \ufffd \u2020 \u2020 \u153 \ufffd Ê
(c) f sin x f sin x f sin x f sin x is odd.a b a b a b a ba b a b a b a b\ufffd \u153 \ufffd \u153 \ufffd Ê
(d) g sec x g sec x g sec x is even.a b a b a ba b a b a b\ufffd \u153 Ê
(e) g x g x g x g is evenl \ufffd l \u153 l\ufffd l \u153 l l Ê l l Þa b a b a b
32 Chapter 1 Functions
18. Let f a x f a x and define g x f x a . Then g x f x a f a x f a x f x a g xa b a b a b a b a b a b a b a b a b a ba b\ufffd \u153 \ufffd \u153 \ufffd \ufffd \u153 \ufffd \ufffd \u153 \ufffd \u153 \ufffd \u153 \ufffd \u153
g x f x a is even.Ê \u153 \ufffda b a b
19. (a) The function is defined for all values of x, so the domain is .a b\ufffd_ß _
(b) Since x attains all nonnegative values, the range is .l l Ò\ufffd#ß _Ñ
20. (a) Since the square root requires x , the domain is .&quot; \ufffd   ! Ð\ufffd_ß &quot;Ó
(b) Since x attains all nonnegative values, the range is .È&quot; \ufffd Ò\ufffd#ß _Ñ
21. (a) Since the square root requires x , the domain is .&quot;' \ufffd   ! Ò\ufffd%ß %Ó#
(b) For values of x in the domain, x , so x . The range is .! \u178 &quot;' \ufffd \u178 &quot;' ! \u178 &quot;' \ufffd \u178 % Ò!ß %Ó# #È
22. (a) The function is defined for all values of x, so the domain is .a b\ufffd_ß _
(b) Since attains all positive values, the range is .\$ &quot;ß _#\ufffdx a b
23. (a) The function is defined for all values of x, so the domain is .a b\ufffd_ß _
(b) Since e attains all positive values, the range is .# \ufffd\$ß _\ufffdx a b
24. (a) The function is equivalent to y tan x, so we require x for odd integers k. The domain is given by x for\u153 # # Á Ák k1 1# %
odd integers k.
(b) Since the tangent function attains all values, the range is .a b\ufffd_ß _
25. (a) The function is defined for all values of x, so the domain is .a b\ufffd_ß _
(b) The sine function attains values from to , so sin x and hence sin x . The\ufffd&quot; &quot; \ufffd# \u178 # \$ \ufffd \u178 # \ufffd\$ \u178 # \$ \ufffd \ufffd &quot; \u178 &quot;a b a b1 1
range is 3 1 .Ò\ufffd ß Ó
26. (a) The function is defined for all values of x, so the domain is .a b\ufffd_ß _
(b) The function is equivalent to y x , which attains all nonnegative values. The range is .\u153 Ò!ß _ÑÈ& #
27. (a) The logarithm requires x , so the domain is .\ufffd \$ \ufffd ! \$ß _a b
(b) The logarithm attains all real values, so the range is .a b\ufffd_ß _
28. (a) The function is defined for all values of x, so the domain is .a b\ufffd_ß _
(b) The cube root attains all real values, so the range is .a b\ufffd_ß _
29. (a) Increasing because volume increases as radius increases
(b) Neither, since the greatest integer function is composed of horizontal (constant) line segments
(c) Decreasing because as the height increases, the atmospheric pressure decreases.
(d) Increasing because the kinetic (motion) energy increases as the particles velocity increases.
30. (a) Increasing on 2, (b) Increasing on 1, Ò _Ñ Ò\ufffd _Ñ
(c) Increasing on , (d) Increasing on , a b\ufffd_ _ Ò _Ñ&quot;#
31. (a) The function is defined for x , so the domain is .\ufffd% \u178 \u178 % Ò\ufffd%ß %Ó
(b) The function is equivalent to y x , x , which attains values from to for x in the domain. The\u153 l l \ufffd% \u178 \u178 % ! #È
range is .Ò!ß #Ó
Chapter 1 Practice Exercises 33
32. (a) The function is defined for x , so the domain is .\ufffd# \u178 \u178 # Ò\ufffd#ß #Ó
(b) The range is .Ò\ufffd&quot;ß &quot;Ó
33. First piece: Line through and . m y x xa b a b!ß &quot; &quot;ß ! \u153 \u153 \u153 \ufffd&quot; Ê \u153 \ufffd \ufffd &quot; \u153 &quot; \ufffd!\ufffd&quot; \ufffd&quot;&quot;\ufffd! &quot;
Second piece: Line through and . m y x x xa b a b a b&quot;ß &quot; #ß ! \u153 \u153 \u153 \ufffd&quot; Ê \u153 \ufffd \ufffd &quot; \ufffd &quot; \u153 \ufffd \ufffd # \u153 # \ufffd!\ufffd&quot; \ufffd&quot;#\ufffd&quot; &quot;
f x x, x
x, x
a b \u153\u153 &quot; \ufffd ! \u178 \ufffd &quot;# \ufffd &quot; \u178 \u178 #
34. First piece: Line through and 2 5 . m y xa b a b!ß ! ß \u153 \u153 Ê \u1535 5 52 2 2\ufffd!\ufffd!
Second piece: Line through 2 5 and 4 . m y x 2 5 x 10 10a b a b a bß ß ! \u153 \u153 \u153 \ufffd Ê \u153 \ufffd \ufffd \ufffd \u153 \ufffd \ufffd \u153 \ufffd!\ufffd \ufffd\ufffd 5 5 5 5 5 5x4 2 2 2 2 2 2
f x (Note: x 2 can be included on either piece.) x, x 2
10 , 2 x 4
a b \ufffd\u153 \u153! \u178 \ufffd\ufffd \u178 \u178
5
2
5x
2
35. (a) f g f g f fa ba b a b a ba b \u160 \u2039\u2030 \ufffd&quot; \u153 \ufffd&quot; \u153 \u153 &quot; \u153 \u153 &quot;&quot; &quot;
\ufffd&quot;\ufffd# &quot;È
(b) g f g f g or a ba b a ba b \u2c6 \u2030 É\u2030 # \u153 # \u153 \u153 \u153&quot; &quot; &quot; #
\ufffd# #Þ& &2 É È&quot;
#
(c) f f x f f x f x, xa ba b a ba b \u2c6 \u2030\u2030 \u153 \u153 \u153 \u153 Á !&quot; &quot;&quot;Îx x
(d) g g x g g x ga ba b a ba b \u160 \u2039\u2030 \u153 \u153 \u153 \u153&quot; &quot;
\ufffd# \ufffd#
\ufffd#
&quot;\ufffd# \ufffd#
È É É
È
Èx
x
x
&quot;
\ufffd#
%
Èx
36. (a) f g f g f fa ba b a b a ba b \u2c6 \u2030È\u2030 \ufffd&quot; \u153 \ufffd&quot; \u153 \ufffd&quot; \ufffd &quot; \u153 ! \u153 # \ufffd ! \u153 #\$
(b) g f f g g ga ba b a b a b a ba b È\u2030 # \u153 # \u153 # \ufffd # \u153 ! \u153 ! \ufffd &quot; \u153 &quot;\$
(c) f f x f f x f x x xa ba b a b a b a ba b\u2030 \u153 \u153 # \ufffd \u153 # \ufffd # \ufffd \u153
(d) g g x g g x g x xa ba b a ba b \u2c6 \u2030È ÈÉ\u2030 \u153 \u153 \ufffd &quot; \u153 \ufffd &quot; \ufffd &quot;\$ \$\$
37. (a) f g x f g x f x x x, x .a ba b a ba b \u2c6 \u2030 \u2c6 \u2030È È\u2030 \u153 \u153 \ufffd # \u153 # \ufffd \ufffd # \u153 \ufffd   \ufffd##
g f x f g x g x x xa ba b a b a b a ba b È È\u2030 \u153 \u153 # \ufffd \u153 # \ufffd \ufffd # \u153 % \ufffd# # #
(b) Domain of f g: (c) Range of f g: \u2030 Ò\ufffd#ß _ÑÞ \u2030 Ð\ufffd_ß #ÓÞ
Domain of g f: Range of g f: \u2030 Ò\ufffd#ß #ÓÞ \u2030 Ò!ß #ÓÞ
38. (a) f g x f g x f x x x.a ba b a ba b \u160 \u2039È È ÈÉ\u2030 \u153 \u153 &quot; \ufffd \u153 &quot; \ufffd \u153 &quot; \ufffd%
g f x f g x g x xa ba b a ba b \u2c6 \u2030È ÈÉ\u2030 \u153 \u153 \u153 &quot; \ufffd
(b) Domain of f g: (c) Range of f g: \u2030 Ð\ufffd_ß &quot;ÓÞ \u2030 Ò!ß _ÑÞ
Domain of g f: Range of g f: \u2030 Ò!ß &quot;ÓÞ \u2030 Ò!ß &quot;ÓÞ
39. y f x y f f x\u153 \u153 \u2030a b a ba b