我是亲狂， 我为自己代言！ 浙一贴吧队 收妹子 基友！！！！！_穿越火线吧_百度贴吧
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本吧签到人数：0可签7级以上的吧50个
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我是亲狂， 我为自己代言！ 浙一贴吧队 收妹子 基友！！！！！
帮蝈蝈艾特几个 妹子来
很多妹子居然在跟我们汉...
我来领几个回战队～～
发张在打游戏的图片吗？...
看着求好友的。我现在竟...
基沙不是人妖吗。
来张福利给大家
湖南妹子哦
过百 给地址 联系方式
呀，浙一路过
. 大家好。我孙子患有精神病， 昨天晚上不幸走失。 它不管遇到什么都会说一些话，第一句是“你只看到我的XXX&。 最后一句是“我为XX代言”。如果遇到这样 的人必是我孙子请 速与本人联系（拨打10 086通后按0即可）或直接送进精神病院进 行电疗。必有重谢 ！！
楼上全撒比，楼下更撒比
我就不说这一楼 全是妹子了
穿越火线 枪战王者全新版本-生化统领,策略对战,惊喜不断~更新领200钻哦穿越火线 枪战王者策略对抗,变身大BOSS!CF正版全新推车玩法,斗智斗勇!
YY558267 哦 有浙一的妹子 和大神吗
贴吧需要支援
我去要不，小白笑脸
大家好。我孙子患有精神病， 昨天晚上不幸走失。 它不管遇到什么都会说一些话，第一句是“你只看到我的XXX&。 最后一句是“我为XX代言”。如果遇到这样 的人必是我孙子请 速与本人联系（拨打10 086通后按0即可）或直接送进精神病院进 行电疗。必有重谢 ！！！
(献花) 我去要不，小白笑脸
浙一路过。。。
晚上八点以后都有时间
男小彩笔一只，请问要不要？
我和女朋友交往了有4年了，她肚子里也有了我的孩子，8个月了。我们非常穷，经常没东西吃，就在刚才我实在忍不住，和女朋友商量把她肚子里的小孩拿出来红烧。她怎么样都不肯，没办法，我只能用棍子把她打晕。于是我模范着电视上那样用力一棍子砸在她脖子上面，可她居然没有晕过去，反而被我打的大哭起来。我没办法，又砸了她几下，她才晕了过去。于是我用手伸进她下体里掏东西，可是怎么也也掏不出来，她的洞貌似太窄了。于是我一咬牙，使劲把洞掰大，然后伸进去用力一扯。拿出来一看居然是个“猴头”这玩意儿没肉，不好吃。于是又把手伸进去，摸索了半天才找到我儿子的身体，拉出来一看，血肉模糊的。本来我想洗一洗红烧，可是肚子实在太饿，我就直接一口啃在我儿子的手臂部位。味道没有我想象中那么好，不过也凑合。吃完我儿子的上半身，突然想起来，我女朋友还饿着呢，她是我最爱的人，我当然要给她留一点。于是把她弄醒，她看起来有点憔悴，一醒来就抓着我问孩子呢？我抓过那剩下的下半身递给她，我以为她会很开心，没想到她抓着大喊让我还她孩子。我微微思考了一下，笑着解下裤带…
一直想加问了好久没人跟我说战队名~
这个妹子也是哦
我想知道有几个人。。。每天晚上或白天人多么
那么多妹纸浙1贴吧好像在野战兵服都见过不过你们有多少人
此人为一个会玩cf的软妹子可以让她来
贴吧热议榜
使用签名档&&
保存至快速回贴有种想法 ，建个战队，全是新手＿＿然后一起练成大手！_穿越火线吧_百度贴吧
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&签到排名：今日本吧第个签到，本吧因你更精彩，明天继续来努力！
本吧签到人数：0可签7级以上的吧50个
本月漏签0次！成为超级会员，赠送8张补签卡连续签到：天&&累计签到：天超级会员单次开通12个月以上，赠送连续签到卡3张
关注：5,357,442贴子：
有种想法 ，建个战队，全是新手＿＿然后一起练成大手！
我要建立一个是由穿吧吧友组成的一只队伍，我们都是菜鸟，什么改名，什么全装，我们全都不需要，我们只需要一颗注重情谊的心， 清一色的小白，清一色的菜鸟，我们不崇拜大手，因为我们只希望在一起娱乐，. 一起成长，一起进步！来了都是兄弟！不论技术，只看人品！！
随后会放制作过程
筒子们，请容我平复下此...
万圣夜（英语：Hallowee...
前排熟悉吧友
此贴为吧友爆照评分贴，...
不知道能否镇的住
今年听到不少吧友们表示...
大家好~我是电影二组的...
此帖里星乔君教你怎么做...
186cm 大一 阳光开朗 喜...
97年双子不杂 处一个可...
第一次做，有什么建议和...
我要建立一个是由穿吧吧友组成的一只队伍，我们都是菜鸟，什么改名，什么全装，我们全都不需要，我们只需要一颗注重情谊的心， 清一色的小白，清一色的菜鸟，我们不崇拜大手，因为我们只希望在一起娱乐，. 一起成长，一起进步！来了都是兄弟！不论技术，只看人品！！
有意者留扣扣哈！来了就是兄弟！
有意者留扣扣哈！来了就是兄弟！
有意者留扣扣哈！来了就是兄弟！
我会成功的信不信
有意者留扣扣哈！来了就是兄弟！
有意者留扣扣哈！来了就是兄弟！
穿越火线 枪战王者全新版本-生化统领,策略对战,惊喜不断~更新领200钻哦穿越火线 枪战王者策略对抗,变身大BOSS!CF正版全新推车玩法,斗智斗勇!
楼主哪个区的啊
欢迎所有网通的菜鸟的战友来吧！
我是北方大区的，欢迎所有网通的菜鸟的战友来吧！
咱们一起努力
一起成大手！
我们是北方大区的，欢迎所有网通的菜鸟的战友来吧！
我要建立一个是由穿吧吧友组成的一只队伍，我们都是菜鸟，什么改名，什么全装，我们全都不需要，我们只需要一颗注重情谊的心， 清一色的小白，清一色的菜鸟，我们不崇拜大手，因为我们只希望在一起娱乐，. 一起成长，一起进步！来了都是兄弟！不论技术，只看人品！！
噢，那只能说明咱们没缘份了
顶一个，，，
清一色的小白，清一色的菜鸟，我们不崇拜大手，因为我们只希望在一起娱乐，. 一起成长，一起进步！来了都是兄弟！不论技术，只看人品！！
有意者留扣扣哈！来了就是兄弟！
哪天真了，战队都成立了
天真才是我们的秉性
我说会成功
是因为我有一帮支持我的兄弟
河南一区的
兄弟们咱们顶起！
河南一区的也能玩北大啊
贴吧热议榜
使用签名档&&
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" }{HYPERLNK 17 "definition of an inver
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The data consists of the time required
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_________________
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how fast is the sports car
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If we denote the speed at time t by V(t) we write V(8.0)=80.
ther hand we can also ask the question when has the sports car reached
the speed of 80 mph?
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of course, 8.0 seconds,
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{PARA 0 "" 0 "" {TEXT -1 256 "One thing to note about this is that whe
n a function has an inverse it simply amounts to interchanging the rol
es of the dependent and independent variables.
Thus if the point (a,b
) is on the graph of the curve v=V(t) then (b,a) is a point on the gra
ph of " }{XPPEDIT 18 0 "t=(V^(-1))(v)" "6#/%\"tG-)%\"VG,$\"\"\"!\"\"6#
%\"vG" }{TEXT -1 1 "=" }{XPPEDIT 18 0 "T(v)" "6#-%\"TG6#%\"vG" }{TEXT
Another thing that you need to consider is that time alway
s increases,
but eventually the sports car will slow down and even st
say, that the driver takes his foot off of the accelerator
after he reaches 110 mph.
Then maybe at a time of 20 seconds after h
e started he has slowed to 100 mph.
In that case we could no longer h
ave a unique time t for a given value of the speed in speeds between 1
00 mph and 110 mph since now there would be at least two possible time
s at which the car is traveling 100 mph." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "invdef"
{TEXT 258 11 "Definition:" }{TEXT -1 36 "
If the function f has an in
verse, " }{XPPEDIT 18 0 "f^(-1)" "6#)%\"fG,$\"\"\"!\"\"" }{TEXT -1 29
", then the following is true:" }}{PARA 0 "" 0 "" {TEXT -1 27 "
" }{XPPEDIT 18 0 "f^(-1)" "6#)%\"fG,$\"\"\"!\"\""
}{TEXT -1 23 "(x)= w
f(w)=x ." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
216 "If a function is defined by a formula, it is sometimes possible t
o find a formula for the inverse function. In Section 1.1
we defined \+
the linear function for converting from degrees Fahrenheit to degrees \+
centigrade," }}{PARA 0 "" 0 "" {TEXT -1 48 "
C = f(F)= " }{XPPEDIT 18 0 "(5/9)" "6#*&\"\"&\"\"\"\"\"*!
\"\"" }{TEXT -1 10 " (F - 32)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 21 "To find the inverse, " }{XPPEDIT 18 0 "f^
(-1)" "6#)%\"fG,$\"\"\"!\"\"" }{TEXT -1 48 ",
we solve for F in terms
C. Thus we obtain" }}{PARA 0 "" 0 "" {TEXT -1 42 "
F = " }{XPPEDIT 18 0 "(9/5" "6#*&\"\"*\"\"\"\"
\"&!\"\"" }{TEXT -1 6 "C +32." }}{PARA 0 "" 0 "" {TEXT -1 15 "It follo
ws that" }}{PARA 0 "" 0 "" {TEXT -1 37 "
" }{XPPEDIT 18 0 "f^(-1)" "6#)%\"fG,$\"\"\"!\"\"" }{TEXT -1 5 "
(C) =" }{XPPEDIT 18 0 " (9/ 5)" "6#*&\"\"*\"\"\"\"\"&!\"\"" }{TEXT -1
6 "C +32." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
122 "We see that the inverse function in this case is the formula for \+
converting from degrees centigrade to degrees Fahrenheit." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 203 "Generally,
it is conventional to use x as the
independent variable and y as the dependent variable.
a formula \+
y=f(x) is known sometimes we can solve for x in terms of y to arrive a
t a formula for
" }{XPPEDIT 18 0 "f^(-1)" "6#)%\"fG,$\"\"\"!\"\"" }
{TEXT -1 88 "(y).
After doing this we usually replace y by
ain an expression of the form
" }{XPPEDIT 18 0 "f^(-1)" "6#)%\"fG,$\"
\"\"!\"\"" }{TEXT -1 75 "(x).
Perhaps the following example will sh
ed some light on this process." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 15 "E
xample 1.6.1: " }{TEXT -1 32 "
Find the inverse function for " }
{XPPEDIT 18 0 "f(x)=x^3" "6#/-%\"fG6#%\"xG*$F'\"\"$" }{TEXT -1 1 "." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 260 9 "Solution:" }{TEXT -1 22 "
equation " }}{PARA 0 "" 0 "" {TEXT -1 49 "
" }{XPPEDIT 18 0 "y=x^3" "6#/%\"yG*$%\"xG\"\"$"
}}{PARA 0 "" 0 "" {TEXT -1 30 "for x in terms of
y to obtain" }}
{PARA 0 "" 0 "" {TEXT -1 54 "
x = " }{XPPEDIT 18 0 "f^(-1" "6#)%\"fG,$\"\"\"!\"\"" }{TEXT
-1 6 "(y) = " }{XPPEDIT 18 0 "y^(1/3)" "6#)%\"yG*&\"\"\"\"\"\"\"\"$!\"
\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 33 "Now write the inver
se function as" }}{PARA 0 "" 0 "" {TEXT -1 54 "
y = " }{XPPEDIT 18 0 "f^(-1)" "6#)%\"fG,$\"
\"\"!\"\"" }{TEXT -1 6 "(x) = " }{XPPEDIT 18 0 "x^(1/3)" "6#)%\"xG*&\"
\"\"\"\"\"\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 60 "W
e can use Maple V to make a plot of the three functions
{XPPEDIT 18 0 "x^3" "6#*$%\"xG\"\"$" }{TEXT -1 14 ",
y=x, and y=" }
{XPPEDIT 18 0 "x^(1/3)" "6#)%\"xG*&\"\"\"\"\"\"\"\"$!\"\"" }{TEXT -1
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "root"
{MPLTEXT 1 0 47 "plot1 := plot(\{root(x,3),x^3\},x=-2..8,y=-2..8):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot2 := plots[textplot](\{[3,8,`y=
x^3`],[6,8,`y=x`],[7.5,3,`y=x^(1/3)`]\}):" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 59 "plot3 := plot(x,x=-2..8,y=-2..8,style=POINT,symbol=ci
rcle):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plots[display](\{plot1,pl
ot2,plot3\});" }}{PARA 13 "" 1 "" {GLPLOT2D 262 262 262 {PLOTDATA 2 "6
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F*FjtF*-%%TEXTG6$7$$\"\"'F*Fdt%$y=xG-Fagl6$7$$\"\"$F*Fdt%&y=x^3G-Fagl6
$7$$\"#vF\\uFjgl%*y=x^(1/3)G-%+AXESLABELSG6$Q\"x6\"Q\"yFghl-%%VIEWG6$;
F(FdtF\\il" 1 2 0 1 0 2 9 1 4 2 1... }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Th
is plot displays a property about the graphs of
{XPPEDIT 18 0 "f^(-1)" "6#)%\"fG,$\"\"\"!\"\"" }{TEXT -1 21 "(x):
e graph of
" }{XPPEDIT 18 0 "f^(-1) " "6#)%\"fG,$\"\"\"!\"\"" }{TEXT
-1 60 " is the reflection of the graph of
f through the line
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 261 37 "When Does A Function Have An Inverse
?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "As \+
mentioned in the example involving the acceleration test we know that \+
some functions do not have an inverse.
" }{TEXT 278 8 "Monotone" }
{TEXT -1 42 " functions have inverses.
in other words:" }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "suffcond" {TEXT 277 122 "If
a function which is increasing (or decreasing) on an interval, [a,b], \+
then there is a unique inverse function
" }{XPPEDIT 262 0 "f^(-1)" "
6#)%\"fG,$\"\"\"!\"\"" }{TEXT 263 56 " defined on the interval
,f(b)] (or ([f(b),f(a)])" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 42 "Recall that any function of the form f(x)=" }{XPPEDIT 18 0 "x^n
" "6#)%\"xG%\"nG" }{TEXT -1 122 " with n odd is increasing, thus funct
ions of this type always have an inverse function which we can find ex
plicitly to be " }}{PARA 0 "" 0 "" {TEXT -1 40 "
" }{XPPEDIT 18 0 "f^(-1)" "6#)%\"fG,$\"\"\"!\"\"" }
{TEXT -1 6 "(x)=
" }{XPPEDIT 18 0 "n^th" "6#)%\"nG%#thG" }{TEXT 265
10 " root of x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 49 "In case
the exponent n is even the function f(x)=" }{XPPEDIT 18 0 "x^n" "6#)%
\"xG%\"nG" }{TEXT -1 19 " is decreasing on (" }{XPPEDIT 18 0 "-infinit
y" "6#,$%)infinityG!\"\"" }{TEXT -1 25 ",0] and increasing on [0," }
{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 49 ").
This means \+
that a function of the form
f(x)=" }{XPPEDIT 18 0 "x^n" "6#)%\"xG%\"n
G" }{TEXT -1 52 ",
which has domain the entire real line (
" }{XPPEDIT 18 0 "-infinity" "6#,$%)infinityG!\"\"" }{TEXT -1 1 "," }
{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 3 ")
" }{TEXT 264
8 "does not" }{TEXT -1 64 " have an inverse.
On the other hand, if we
consider a function " }{XPPEDIT 18 0 "f[1](x)=x^n" "6#/-&%\"fG6#\"\"
\"6#%\"xG)F*%\"nG" }{TEXT -1 65 " ,
as a function with doma
in the negative half-line
(" }{XPPEDIT 18 0 "-infinity" "6#,$%)infini
tyG!\"\"" }{TEXT -1 76 ",0]
then it is decreasing throughout its doma
in and has an inverse function" }}{PARA 0 "" 0 "" {TEXT -1 38 "
" }{XPPEDIT 18 0 "f[1]^(-1)" "6#)&%\"fG
6#\"\"\",$\"\"\"!\"\"" }{TEXT -1 6 "(x) =-" }{XPPEDIT 18 0 "n^th" "6#)
%\"nG%#thG" }{TEXT 266 11 " root of
x" }{TEXT -1 17 "
with domain [0
," }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 2 ")." }}{PARA
0 "" 0 "" {TEXT -1 24 "Similarly, the function " }}{PARA 0 "" 0 ""
{TEXT -1 37 "
" }{XPPEDIT 18 0 "f[
2]^(-1)" "6#)&%\"fG6#\"\"#,$\"\"\"!\"\"" }{TEXT -1 5 "(x)= " }
{XPPEDIT 18 0 "n^th" "6#)%\"nG%#thG" }{TEXT 267 11 " root of
{TEXT -1 17 "
with domain [0," }{XPPEDIT 18 0 "infinity" "6#%)infinit
yG" }{TEXT -1 3 ")
" }}{PARA 0 "" 0 "" {TEXT -1 42 "is an inverse of \+
the increasing function
" }{XPPEDIT 18 0 "f[2]" "6#&%\"fG6#\"\"#" }
{TEXT -1 4 "(x)=" }{XPPEDIT 18 0 "x^n" "6#)%\"xG%\"nG" }{TEXT -1 27 ",
with domain [0," }{XPPEDIT 18 0 "infinity" "6#%)infinityG"
}{TEXT -1 197 ").
Thus there are two functions, each defined on a ha
lf-line, associated with the original
function. Both of these functio
ns have an inverse. The next example illustrates this for the case
2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 13 "Example1.6.2:" }{TEXT -1 65 " \+
Find the inverses of two functions associated with the function" }}
{PARA 0 "" 0 "" {TEXT -1 50 "
y=f(x)=" }{XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"#" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 ""
}}{PARA 0 "" 0 "" {TEXT 269 10 "Solution: " }{TEXT -1 41 " Note that i
f we solve the equation
y = " }{XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"#" }
{TEXT -1 44 "
x we have the nonunique result
x = +" }{XPPEDIT
18 0 "sqrt(y)" "6#-%%sqrtG6#%\"yG" }{TEXT -1 5 " or -" }{XPPEDIT 18 0
"sqrt(y)" "6#-%%sqrtG6#%\"yG" }{TEXT -1 101 ".
If we solve for x in \+
terms of y and replace
x we would have an \"inverse\" of the \+
form y=+" }{XPPEDIT 18 0 "sqrt(x) " "6#-%%sqrtG6#%\"xG" }{TEXT -1 5 " \+
or -" }{XPPEDIT 18 0 "sqrt(x)" "6#-%%sqrtG6#%\"xG" }{TEXT -1 221 ".
This is not a function since each value of x gives rise to two values \+
But we were expecting two different inverse functions each of w
hich is as inverse to one of the two associated functions: one with ra
nge [0," }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 29 "), an
d the other with range (" }{XPPEDIT 18 0 "-infinity" "6#,$%)infinityG!
\"\"" }{TEXT -1 50 ",0]. This is obtained from our formula by taking
" }}{PARA 0 "" 0 "" {TEXT -1 29 "
{XPPEDIT 18 0 "g[1](x)= sqrt(x)" "6#/-&%\"gG6#\"\"\"6#%\"xG-%%sqrtG6#F
*" }{TEXT -1 15 "
in [0," }{XPPEDIT 18 0 "infinity" "6#%)infin
ityG" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 ""
0 "" {TEXT -1 29 "
" }{XPPEDIT 18 0 "g[2](
x)=-sqrt(x)" "6#/-&%\"gG6#\"\"#6#%\"xG,$-%%sqrtG6#F*!\"\"" }{TEXT -1
13 " for x in [0," }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT
-1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
23 "We illustrate this with" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot1 := plot(\{-sqrt(x),sqr
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ot2 := plots[textplot](\{[1.25,0.5,`y=x^2`],[2.5,1.2,\n
`y=sqrt(x)`],[2.5,-1.2,`y=-sqrt(x)`]\}):" }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 59 "plot3 := plot(x,x=-4..4,y=-2..2,style=POINT,symbol=CI
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"InverseTrig" {TEXT -1 32 "Inverse Trigonometric
Functions" }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The sine functi
on" }}{PARA 0 "" 0 "" {TEXT -1 51 "
f(x) = sin x " }}{PARA 0 "" 0 "" {TEXT -1 370 "oscillates between
varies over the extent of any interval of length 2 Pi
This means that the sine function does not have a unique inverse fu
nction. Rather it has (infinitely) many inverses to (associated) funct
ions defined over intervals in which the sine function is either incre
asing or decreasing. The sine function is increasing on the interval [
" }{XPPEDIT 18 0 "-Pi/2" "6#,$*&%#PiG\"\"\"\"\"#!\"\"F(" }{TEXT -1 1 "
," }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 107 "] \+
with range [-1,1].
In this case Maple V recognizes the inverse to th
e sine function over the interval [" }{XPPEDIT 18 0 "-Pi/2" "6#,$*&%#P
iG\"\"\"\"\"#!\"\"F(" }{TEXT -1 1 "," }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG
\"\"\"\"\"#!\"\"" }{TEXT -1 9 "] as the " }{TEXT 271 6 "arcsin" }
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "solve(y=sin(x),x);" }}{PARA 11 ""
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Let us evaluate this func
x=-1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "arcsin" {MPLTEXT 1 0 11 "arcsin(-1);" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#,$%#PiG#!\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1
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{PARA 0 "" 0 "" {TEXT 272 15 "Example 1.6.3: " }{TEXT -1 73 " A somewh
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{PARA 0 "" 0 "" {TEXT -1 37 "
f(x) = sin x
{XPPEDIT 18 0 "Pi/2 <=x" "6#1*&%#PiG\"\"\"\"\"#!\"\"%\"xG" }{TEXT -1
" }{XPPEDIT 18 0 "x <=
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'\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 139 "which is a decreasing fun
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In this case the inverse function is de
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{XPPEDIT 18 0 "f^(-1)" "6#)%\"fG,$\"\"\"!\"\"" }{TEXT -1 6 "(x) = " }
{XPPEDIT 18 0 "Pi " "6#%#PiG" }{TEXT -1 17 "-
arcsin(x),
{XPPEDIT 18 0 "-1 <=
x" "6#1,$\"\"\"!\"\"%\"xG" }{TEXT -1 7 "
" }{XPPEDIT 18 0 "x " 0 "" {MPLTEXT 1 0 40 "plot1 := pl
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:`ZJaRjqh]gt?#F*Fg`l7$$\":`Iv.2#QAsuoHBF*Fj`l7$$\":SKt\"GJAGV0O\\CF*F]
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f\"[lB4GF*Ffal7$$\":>\"eJ!)=3)*f,eKHF*Fial7$$\":k!)*GR!zbZ!yMWIF*F\\bl
7$$\":=P+O-W)>i$Rl;$F*F_bl7$$\":=nCm%H79&47:G$F*Fbbl7$$\":\\?s#=5O&*GNF*Fhbl7$$\"::qU)['z\"*GM)>UOF*F[cl7$$\":'p
!)zh@u(Goq1w$F*F^cl7$$\":Re$*[2$>&=wD=)QF*Facl7$$\":1j6K7l-chx>+%F*Fdc
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;5_VF*F]dl7$$\":M#yyj)ye9$HxsWF*F`dl7$$\":gi:$p=&oXxq#)e%F*Fcdl7$FfzFf
zFjz-%&STYLEG6#%&POINTG-%%TEXTG6%7$$\"\"&F`[l$\"#NF`[l%2y~=~Pi~-~arcsi
n~xG%+ALIGNRIGHTG-F[el6%7$$\"\"$Fa[lF^el%*y~=~sin~xGFcel-F$6$7S7$$F`[l
Fa[l$\":lRpd)*o%Q!)*)Q7ZF*7$$!:nmmmmmmm\"p0k&*F2$\":z>z:``UMVGgT%F*7$$
!:LLLLLLL3N!)z^
=(QF*7$$!:+++++++vy$pZiF2$\":OgbokC9)*yGk\"QF*7$$!:LLLLLLLLyaE\"eF2$\"
:Pp92z2N,nv=w$F*7$$!:>s%HaF2$\":InH%eO\"oeONbr$F*7$$!:++++++++
N*4)*\\F2$\":T%4]KZ6$\\(>(\\m$F*7$$!:++++++++Db\\c%F2$\":y?G))Q7Ev-Xch
$F*7$$!:++++++++lSv9%F2$\":*G-Yd(fJ>;l#pNF*7$$!:mmmmmmmT?)[oPF2$\":\"R
u85=_p$f%)z_$F*7$$!:LLLLLLLL=exJ$F2$\":F)QOYIIK]!y(zMF*7$$!:LLLLLLLLtI
f$HF2$\":GKG+A!=J^gdRMF*7$$!:*************\\PYx\"\\#F2$\":M!z21R/***[B
MR$F*7$$!:LLLLLLLLB@')4#F2$\":mGHfu7QUfEIN$F*7$$!:*************\\P'psm
\"F2$\":#*GO%\\6%Hb)=54LF*7$$!:*************\\74_c7F2$\":oN%oF94AexdnK
F*7$$!9LLLLLLL$3x%z#)F2$\":]#e)GI#435B[CKF*7$$!9LLLLLLL3s$QM%F2$\":)**
H*=2vsypW]=$F*7$$!7zr)*F2$\":\\@4h!)z;H$)zD9$F*7$$\"8LLLLLL$ez
w5VF*$\":FxbM%yP'=hr%)4$F*7$$\"8++++++]PQ#\\\")F*$\":mw_0]O61!)4+1$F*7
$$\"9LLLLLLLe\"*[H7F*$\":WW\")Q8HnK&=L=IF*7$$\"9++++++++F*$\":@E<C
')*=Q9i/vHF*7$$\"9+++++++D0xw?F*$\":2CNp*=\\^gg:[`2*GF*7$$\"9+++++++vgHKHF*$\":FEt@Ay4dX*)R%GF*7$$\"9nm
mmmmmmZvOLF*$\":Z(Gc&[VPI#GR,GF*7$$\"9+++++++]2goPF*$\":,SAQ\"H@3j&)=b
FF*7$$\"9LLLLLL$eR<*fTF*$\":fV$Ra()4kT&fDr#F*7$$\"9+++++++])Hxe%F*$\":
K\\UPP0N5')y\\m#F*7$$\"9H!o-*\\F*$\":Rhn,tE;^EEF*7$$\"9++++
++]7k.6aF*$\":gzZV(zeFKP%)pDF*7$$\"9WTAeF*$\":!yaRkOM#*G(4,_#F
*7$$\"9++++++]i!*3`iF*$\":\\3kTZhN7MlgY#F*7$$\"9LLLLLLLL*zym'F*$\":npW
o>R1(3Aq6CF*7$$\"9LLLLLLL3N1#4(F*$\":d](GN8**fC3A`BF*7$$\"9HYt7
vF*$\":^o-'yxgJsJg\"H#F*7$$\"9++++++++xG**yF*$\":))>%4y***fi#)**3B#F*7
$$\"9nmmmmmmT6KU$)F*$\":Sa]&>(oQh\"R&[:#F*7$$\"9LLLLLLLLbdQ()F*$\":b%G
^y?F*7$$\"9++++++]i`1h\"*F*$\":gj]KhnHs[LL)>F*7$$\"9++++++]P?Wl
&*F*$\":]UMy0IBb@#om=F*7$$\"\"\"Fa[l$\":A8B>m*[zEjzq:F*Fjz-%+AXESLABEL
SG6$Q\"x6\"%!G-%%VIEWG6$;F^flF_fl%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2
1... }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 273 14 "Example 1.6.4:" }{TEXT -1 44 "
the inverse to the function f(x)=cos(" }{XPPEDIT 18 0 "x^2" "6#*$%\"x
G\"\"#" }{TEXT -1 19 "+1)-2
[2,2.2]." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
274 9 "Solution:" }{TEXT -1 75 "
First define the function and plot i
on the specified interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := x -> cos(x^2+1)-2;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&ar
rowGF(,&-%$cosG6#,&*$)9$\"\"#\"\"\"\"\"\"F6F6F6!\"#F6F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(f(x),x=2..2.2);" }}
{PARA 13 "" 1 "" {GLPLOT2D 225 225 225 {PLOTDATA 2 "6%-%'CURVESG6$7S7$
$\"\"#\"\"!$!:hL`NPxOX\"yL;<!#C7$$\":LLLLLLL3VfV+#F-$!:'=9@MH;l/Ak*p\"
F-7$$\":nmmmmm\"H[D:3?F-$!:Ig9voCs$z.;&o\"F-7$$\":LLLLLL$e0$=C,#F-$!:4
a7sjF;m!*G*o;F-7$$\":LLLLLL$3RBr;?F-$!:gK*p*G?Ojt`El\"F-7$$\":nmmmmm\"
zjf)4-#F-$!:Wu`#Q.$**)H]_O;F-7$$\":LLLLLLe4;[\\-#F-$!:<)etP-u'[a\"F-7
$$\":+++++++l+>+0#F-$!:)G&oQec8kS2#H:F-7$$\":+++++++vW]V0#F-$!:9>=%\\q
t(=C)f8:F-7$$\":+++++++NfC&e?F-$!:50JLay[]fq')\\\"F-7$$\":LLLLLLez6:B1
#F-$!:ajY,j9q<(o@&[\"F-7$$\":nmmmmmm\"=C#o1#F-$!:*)G=!y@EpH6Np9F-7$$\"
:nmmmmmmEpS12#F-$!:#R\"4,r`(*[nFgX\"F-7$$\":++++++DOD#3v?F-$!:?sg&z#e79F-7$$\":++++++v3zMu3#F-$!:cD!4)))*))ROb!))R\"F-7$$
\":H_?<4#F-$!:k#\\!**zy<u?!f%Q\"F-7$$\":nmmmmm\"zihl&4#F-$!:B4^
6;$yH,Zor8F-7$$\":LLLLLL$3#G,**4#F-$!:M\\(R@e7_U\"HzN\"F-7$$\":LLLLLLe
zw5V5#F-$!:a)Qh%yp%)o*y#QM\"F-7$$\":++++++v$Q#\\\"3@F-$!:hrYq*\\Y\"3f7
i*oKkGx_cI\"F-7$$\":++++++]_qn27#F-$!:Cb)4![3g#)>')HH\"F-7$$\":+
+++++Dcp@[7#F-$!:#36JCl`$\\\\>4G\"F-7$$\":++++++]2'HKH@F-$!:'z:VhP%Ql'
\\un7F-7$$\":nmmmmmmwanL8#F-$!:5nt<&3ZL\\G6c7F-7$$\":+++++++v+'oP@F-$!
:6<T#ovx[1O\"RC\"F-7$$\":LLLLLLeR1jgIB\"F-7$$\":++++
+++&)Hxe9#F-$!:*o(RRw!>6_'>9A\"F-7$$\":nmmmmm\"H!o-*\\@F-$!:#Hw@&fz'3
\"*\\o57F-7$$\":++++++DTO5T:#F-$!:4ikUEYsau&p*>\"F-7$$\":nmmmmmmT9C#e@
F-$!:\")>J6@&R)RV&=*=\"F-7$$\":++++++D1*3`i@F-$!:1j^n$eOS()[Vy6F-7$$\"
:LLLLLLL$*#F-$!:i)4;h]_O]-Lo6F-7$$\":LLLLLL$3N1#4<#F-$!:aLu!)GMw9'
\\De6F-7$$\":nmmmmm\"HYt7v@F-$!:AQ#)Q1Us&ye_[6F-7$$\":+++++++q(G**y@F-
$!:*Gk,W<(\\Ty?)R6F-7$$\":9@BM=#F-$!:&e4Hf,!H5XD,8\"F-7$$\":LLL
LLLL`v&Q(=#F-$!:MWdc590\")38#F-$!:Xy0K=K?0.=I6
\"F-7$$\":++++++v.Uac>#F-$!:;FcvL8=C9l\\5\"F-7$$\"#A!\"\"$!:8cG!4TaSs'
4m4\"F--%'COLOURG6&%$RGBG$\"#5FfzF*F*-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEW
G6$;F(Fdz%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1...000000
0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
125 "We observe that f,
is increasing on
[2,2.2], so it has an inve
rse. We can solve for the inverse either by hand or use the " }{TEXT
275 5 "solve" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 0 ""
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(y=f(x),x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$*$-%%sqrtG6#,&!\"\"\"\"\"-%'arccosG6#,
&%\"yGF)\"\"#F)F)\"\"\",$F#F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 168 "Since x is positive we require the p
ositive square root.
as in the previous example,
that the principal value of
is over the interval [0," }
{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 214 "] and thus may not be in t
he desired inverse.
when y is in the interval [-1.7,-1.1] t
arccos(y+2)-1 < 0
and no real root exists. Thus we must replace \+
arccos by the inverse of
" }{XPPEDIT 18 0 "(-1.
7)^2" "6#*$,$$\"# " 0 "" {MPLTEXT 1 0 36 "finv := x->sqrt(2*Pi-arccos(x+2)-1)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%finvGR6#%\"xG6\"6$%)operatorG%
&arrowGF(-%%sqrtG6#,(%#PiG\"\"#-%'arccosG6#,&9$\"\"\"F1F7!\"\"F8F7F(F(
F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 78 "The following Maple V segment is partial check
by evaluating a
t a few points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "evalf(finv(evalf(f(2.2))));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\":+++++++++++?#!#C" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 14 "f(finv(-1.7));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!:
+++++++++++q\"!#C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Some of The Maple V Commands Used
in This Section" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 134 "Examples of Maple V commands, or options, or
variables \+
used in this section can be obtained by clicking on the following book
" }{HYPERLNK 17 "root" 1 "" "root" }{TEXT -1 4 " ,
{HYPERLNK 17 "ln" 1 "" "ln" }{TEXT -1 3 " , " }{HYPERLNK 17 "arcsin"
1 "" "arcsin" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Exercises 1.6" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 18 "1.
Show that if
" }}{PARA 14 "" 0 "" {TEXT -1 38 "
x)=" }{XPPEDIT 18 0 "(x+1)/(2x - 3)" "6#*&,&%\"xG\"\"\"\"\"\"F&F&,&*&
\"\"#F&F%F&F&\"\"$!\"\"F," }}{PARA 14 "" 0 "" {TEXT -1 5 "then " }}
{PARA 14 "" 0 "" {TEXT -1 35 "
{XPPEDIT 18 0 "f^(-1)" "6#)%\"fG,$\"\"\"!\"\"" }{TEXT -1 6 "(y) = " }
{XPPEDIT 18 0 "(3y +1)/( 2y - 1)" "6#*&,&*&\"\"$\"\"\"%\"yGF'F'\"\"\"F
'F',&*&\"\"#F'F(F'F'\"\"\"!\"\"F." }{TEXT -1 1 "." }}{PARA 14 "" 0 ""
{TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 29 "2.
Show that the funct
" }}{PARA 14 "" 0 "" {TEXT -1 30 "
" }{XPPEDIT 18 0 "f(x)= (1-x^p)^(1/p)
" "6#/-%\"fG6#%\"xG),&\"\"\"\"
\"\")F'%\"pG!\"\"*&\"\"\"F+F-F." }}{PARA 14 "" 0 "" {TEXT -1 46 "is it
s own inverse.
Verify this numerically. " }}{PARA 14 "" 0 "" {TEXT
-1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 95 "3. For each of the following f
unctions find the inverse on the specified interval if it exists." }}
{PARA 14 "" 0 "" {TEXT -1 41 "
" }{XPPEDIT 18 0 "1+(x-1)^2 " "6#,&\"\"\"\"\"\"*$,&%\"xGF%\"\"\"!\"\"
\"\"#F%" }{TEXT -1 17 " for x in [-5,-2]" }}{PARA 14 "" 0 "" {TEXT -1
0 "" }}{PARA 14 "" 0 "" {TEXT -1 40 "
f(x) = " }{XPPEDIT 18 0 "sqrt(x^2+2x+2) " "6#-%%sqrtG6#,(*$%\"xG\"\"#
\"\"\"*&\"\"#F*F(F*F*\"\"#F*" }{TEXT -1 18 "
for x in [-3,3]" }}
{PARA 14 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 41 "
f(x) = " }{XPPEDIT 18 0 "sqrt(x^2+2x+2)" "6
#-%%sqrtG6#,(*$%\"xG\"\"#\"\"\"*&\"\"#F*F(F*F*\"\"#F*" }{TEXT -1 17 " \+
for x in [-3,-1]" }}{PARA 14 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 ""
{TEXT -1 67 "
f(x) = cos 3x for x in
[-Pi/3,0]" }}{PARA 14 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT
Find the inverse of the function
" }{XPPEDIT 18 0 "f(x) = \+
ln (x^2+1) -3 " "6#/-%\"fG6#%\"xG,&-%#lnG6#,&*$F'\"\"#\"\"\"\"\"\"F/F/
\"\"$!\"\"" }{TEXT -1 34 "
for x in [0,4.5],
if it exists." }}{PARA
14 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 38 "5.
Find the \+
inverse of the function
" }{XPPEDIT 18 0 "f(x) = 4+cos(x^4+1)" "6#/-%
\"fG6#%\"xG,&\"\"%\"\"\"-%$cosG6#,&*$F'\"\"%F*\"\"\"F*F*" }{TEXT -1
x in [1.6,1.7],
if it exists." }}}}}}{MARK "1" 0 }
{VIEWOPTS 1 1 0 1 1 1803 }
