What is the domain of the function $f(x)=sqrt[3]{x^3-x}$?












5












$begingroup$


Let $f$ be: $f(x) = sqrt[3]{x^3 -x}$, an exercise book asked for the domain of definition. Isn't it over $mathbb R$. The book solution stated $Df = [-1,0] cup [1, +infty[$
I don t get it. Can you explain?






calculus







share|cite|improve this question











$endgroup$












  • $begingroup$
    It is $$x^3-xgeq 0$$
    $endgroup$
    – Dr. Sonnhard Graubner
    17 hours ago










  • $begingroup$
    I think it's better $D(f)=mathbb R$, but if $g(x)=(x^3-x)^{frac{1}{3}}$ so $D(g)={xinmathbb R|x^3-x>0}.$ All these a definition only.
    $endgroup$
    – Michael Rozenberg
    17 hours ago












  • $begingroup$
    @Dr.SonnhardGraubner can you explain why?
    $endgroup$
    – J.Moh
    17 hours ago










  • $begingroup$
    $$g(0)=0$$ dear Michael.
    $endgroup$
    – Dr. Sonnhard Graubner
    17 hours ago










  • $begingroup$
    Are you quite sure it wasn't $sqrt{x^3-x}$? Because the domain of $sqrt[3]{x^3 -x}$ is $Bbb R$.
    $endgroup$
    – TonyK
    9 hours ago
















5












$begingroup$


Let $f$ be: $f(x) = sqrt[3]{x^3 -x}$, an exercise book asked for the domain of definition. Isn't it over $mathbb R$. The book solution stated $Df = [-1,0] cup [1, +infty[$
I don t get it. Can you explain?






calculus







share|cite|improve this question











$endgroup$












  • $begingroup$
    It is $$x^3-xgeq 0$$
    $endgroup$
    – Dr. Sonnhard Graubner
    17 hours ago










  • $begingroup$
    I think it's better $D(f)=mathbb R$, but if $g(x)=(x^3-x)^{frac{1}{3}}$ so $D(g)={xinmathbb R|x^3-x>0}.$ All these a definition only.
    $endgroup$
    – Michael Rozenberg
    17 hours ago












  • $begingroup$
    @Dr.SonnhardGraubner can you explain why?
    $endgroup$
    – J.Moh
    17 hours ago










  • $begingroup$
    $$g(0)=0$$ dear Michael.
    $endgroup$
    – Dr. Sonnhard Graubner
    17 hours ago










  • $begingroup$
    Are you quite sure it wasn't $sqrt{x^3-x}$? Because the domain of $sqrt[3]{x^3 -x}$ is $Bbb R$.
    $endgroup$
    – TonyK
    9 hours ago














5












5








5





$begingroup$


Let $f$ be: $f(x) = sqrt[3]{x^3 -x}$, an exercise book asked for the domain of definition. Isn't it over $mathbb R$. The book solution stated $Df = [-1,0] cup [1, +infty[$
I don t get it. Can you explain?






calculus







share|cite|improve this question











$endgroup$




Let $f$ be: $f(x) = sqrt[3]{x^3 -x}$, an exercise book asked for the domain of definition. Isn't it over $mathbb R$. The book solution stated $Df = [-1,0] cup [1, +infty[$
I don t get it. Can you explain?






calculus




calculus






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 9 hours ago









Asaf Karagila♦

308k33441775




308k33441775










asked 17 hours ago









J.MohJ.Moh

855




855












  • $begingroup$
    It is $$x^3-xgeq 0$$
    $endgroup$
    – Dr. Sonnhard Graubner
    17 hours ago










  • $begingroup$
    I think it's better $D(f)=mathbb R$, but if $g(x)=(x^3-x)^{frac{1}{3}}$ so $D(g)={xinmathbb R|x^3-x>0}.$ All these a definition only.
    $endgroup$
    – Michael Rozenberg
    17 hours ago












  • $begingroup$
    @Dr.SonnhardGraubner can you explain why?
    $endgroup$
    – J.Moh
    17 hours ago










  • $begingroup$
    $$g(0)=0$$ dear Michael.
    $endgroup$
    – Dr. Sonnhard Graubner
    17 hours ago










  • $begingroup$
    Are you quite sure it wasn't $sqrt{x^3-x}$? Because the domain of $sqrt[3]{x^3 -x}$ is $Bbb R$.
    $endgroup$
    – TonyK
    9 hours ago


















  • $begingroup$
    It is $$x^3-xgeq 0$$
    $endgroup$
    – Dr. Sonnhard Graubner
    17 hours ago










  • $begingroup$
    I think it's better $D(f)=mathbb R$, but if $g(x)=(x^3-x)^{frac{1}{3}}$ so $D(g)={xinmathbb R|x^3-x>0}.$ All these a definition only.
    $endgroup$
    – Michael Rozenberg
    17 hours ago












  • $begingroup$
    @Dr.SonnhardGraubner can you explain why?
    $endgroup$
    – J.Moh
    17 hours ago










  • $begingroup$
    $$g(0)=0$$ dear Michael.
    $endgroup$
    – Dr. Sonnhard Graubner
    17 hours ago










  • $begingroup$
    Are you quite sure it wasn't $sqrt{x^3-x}$? Because the domain of $sqrt[3]{x^3 -x}$ is $Bbb R$.
    $endgroup$
    – TonyK
    9 hours ago
















$begingroup$
It is $$x^3-xgeq 0$$
$endgroup$
– Dr. Sonnhard Graubner
17 hours ago




$begingroup$
It is $$x^3-xgeq 0$$
$endgroup$
– Dr. Sonnhard Graubner
17 hours ago












$begingroup$
I think it's better $D(f)=mathbb R$, but if $g(x)=(x^3-x)^{frac{1}{3}}$ so $D(g)={xinmathbb R|x^3-x>0}.$ All these a definition only.
$endgroup$
– Michael Rozenberg
17 hours ago






$begingroup$
I think it's better $D(f)=mathbb R$, but if $g(x)=(x^3-x)^{frac{1}{3}}$ so $D(g)={xinmathbb R|x^3-x>0}.$ All these a definition only.
$endgroup$
– Michael Rozenberg
17 hours ago














$begingroup$
@Dr.SonnhardGraubner can you explain why?
$endgroup$
– J.Moh
17 hours ago




$begingroup$
@Dr.SonnhardGraubner can you explain why?
$endgroup$
– J.Moh
17 hours ago












$begingroup$
$$g(0)=0$$ dear Michael.
$endgroup$
– Dr. Sonnhard Graubner
17 hours ago




$begingroup$
$$g(0)=0$$ dear Michael.
$endgroup$
– Dr. Sonnhard Graubner
17 hours ago












$begingroup$
Are you quite sure it wasn't $sqrt{x^3-x}$? Because the domain of $sqrt[3]{x^3 -x}$ is $Bbb R$.
$endgroup$
– TonyK
9 hours ago




$begingroup$
Are you quite sure it wasn't $sqrt{x^3-x}$? Because the domain of $sqrt[3]{x^3 -x}$ is $Bbb R$.
$endgroup$
– TonyK
9 hours ago










1 Answer
1






active

oldest

votes


















12












$begingroup$

If your book reaches the domain $[-1,0]cup[1,+infty)$, it must be because the book only considers $sqrt[3]{phantom{X}}$ to be defined when the argument is a non-negative real.



Books (and people) differ in how they consider $sqrt[N]{phantom X}$ to be defined.



Some people find it okay to define odd roots on the entire real line -- after all, $xmapsto x^N$ is a bijection on $mathbb R$ when $N$ is positive odd, and every such bijection has a perfectly fine inverse.



Other people prefer to restrict these functions to non-negative reals, no matter what $N$ is -- partially to avoid creating a (confusing?) distinction between odd and even $N$, partially for more subtle reasons that unfortunately are not apparent when one first learns about roots.



(For even subtler reasons, one might even want to reserve the root notation to arguments that are strictly positive, such that $sqrt 0$ is considered undefined. It is somewhat rare to take that position consistently, though).



You'll just have to live with the fact that such questions cannot be answered without knowing which convention for the root sign is to be used. (Arguably it is bad form to let a find-the-domain-of-this-expression exercise depend on such choices, but that's purely the textbook's fault, of course).






share|cite|improve this answer











$endgroup$













  • $begingroup$
    That s why I love computer scientists, they answer as if they re writing code ;) Thanks Henning! Perfect!
    $endgroup$
    – J.Moh
    17 hours ago








  • 1




    $begingroup$
    @J.Moh: If every mathematics student learns programming, we would hardly see any of the silly mistakes arising from imprecision. I agree with Henning's last sentence and even say that such kind of questions are terrible because they encourage imprecision. Moreover, I personally think that we should define $sqrt[n]{x}$ for all real $x$ and odd natural number $n$, because $(mathbb{R} x ↦ x^n)$ is a bijection from $mathbb{R}$ to $mathbb{R}$, so its inverse exists. Similarly for non-negative real $x$ and even natural number $n$.
    $endgroup$
    – user21820
    11 hours ago












  • $begingroup$
    @J.Moh: By the way, if you are satisfied with this answer, you can click the tick to accept it.
    $endgroup$
    – user21820
    11 hours ago






  • 1




    $begingroup$
    I agree! I found refuge in math and programming since I could sense for the first time what honesty was.
    $endgroup$
    – J.Moh
    11 hours ago










  • $begingroup$
    @user21820 I did
    $endgroup$
    – J.Moh
    11 hours ago












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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









12












$begingroup$

If your book reaches the domain $[-1,0]cup[1,+infty)$, it must be because the book only considers $sqrt[3]{phantom{X}}$ to be defined when the argument is a non-negative real.



Books (and people) differ in how they consider $sqrt[N]{phantom X}$ to be defined.



Some people find it okay to define odd roots on the entire real line -- after all, $xmapsto x^N$ is a bijection on $mathbb R$ when $N$ is positive odd, and every such bijection has a perfectly fine inverse.



Other people prefer to restrict these functions to non-negative reals, no matter what $N$ is -- partially to avoid creating a (confusing?) distinction between odd and even $N$, partially for more subtle reasons that unfortunately are not apparent when one first learns about roots.



(For even subtler reasons, one might even want to reserve the root notation to arguments that are strictly positive, such that $sqrt 0$ is considered undefined. It is somewhat rare to take that position consistently, though).



You'll just have to live with the fact that such questions cannot be answered without knowing which convention for the root sign is to be used. (Arguably it is bad form to let a find-the-domain-of-this-expression exercise depend on such choices, but that's purely the textbook's fault, of course).






share|cite|improve this answer











$endgroup$













  • $begingroup$
    That s why I love computer scientists, they answer as if they re writing code ;) Thanks Henning! Perfect!
    $endgroup$
    – J.Moh
    17 hours ago








  • 1




    $begingroup$
    @J.Moh: If every mathematics student learns programming, we would hardly see any of the silly mistakes arising from imprecision. I agree with Henning's last sentence and even say that such kind of questions are terrible because they encourage imprecision. Moreover, I personally think that we should define $sqrt[n]{x}$ for all real $x$ and odd natural number $n$, because $(mathbb{R} x ↦ x^n)$ is a bijection from $mathbb{R}$ to $mathbb{R}$, so its inverse exists. Similarly for non-negative real $x$ and even natural number $n$.
    $endgroup$
    – user21820
    11 hours ago












  • $begingroup$
    @J.Moh: By the way, if you are satisfied with this answer, you can click the tick to accept it.
    $endgroup$
    – user21820
    11 hours ago






  • 1




    $begingroup$
    I agree! I found refuge in math and programming since I could sense for the first time what honesty was.
    $endgroup$
    – J.Moh
    11 hours ago










  • $begingroup$
    @user21820 I did
    $endgroup$
    – J.Moh
    11 hours ago
















12












$begingroup$

If your book reaches the domain $[-1,0]cup[1,+infty)$, it must be because the book only considers $sqrt[3]{phantom{X}}$ to be defined when the argument is a non-negative real.



Books (and people) differ in how they consider $sqrt[N]{phantom X}$ to be defined.



Some people find it okay to define odd roots on the entire real line -- after all, $xmapsto x^N$ is a bijection on $mathbb R$ when $N$ is positive odd, and every such bijection has a perfectly fine inverse.



Other people prefer to restrict these functions to non-negative reals, no matter what $N$ is -- partially to avoid creating a (confusing?) distinction between odd and even $N$, partially for more subtle reasons that unfortunately are not apparent when one first learns about roots.



(For even subtler reasons, one might even want to reserve the root notation to arguments that are strictly positive, such that $sqrt 0$ is considered undefined. It is somewhat rare to take that position consistently, though).



You'll just have to live with the fact that such questions cannot be answered without knowing which convention for the root sign is to be used. (Arguably it is bad form to let a find-the-domain-of-this-expression exercise depend on such choices, but that's purely the textbook's fault, of course).






share|cite|improve this answer











$endgroup$













  • $begingroup$
    That s why I love computer scientists, they answer as if they re writing code ;) Thanks Henning! Perfect!
    $endgroup$
    – J.Moh
    17 hours ago








  • 1




    $begingroup$
    @J.Moh: If every mathematics student learns programming, we would hardly see any of the silly mistakes arising from imprecision. I agree with Henning's last sentence and even say that such kind of questions are terrible because they encourage imprecision. Moreover, I personally think that we should define $sqrt[n]{x}$ for all real $x$ and odd natural number $n$, because $(mathbb{R} x ↦ x^n)$ is a bijection from $mathbb{R}$ to $mathbb{R}$, so its inverse exists. Similarly for non-negative real $x$ and even natural number $n$.
    $endgroup$
    – user21820
    11 hours ago












  • $begingroup$
    @J.Moh: By the way, if you are satisfied with this answer, you can click the tick to accept it.
    $endgroup$
    – user21820
    11 hours ago






  • 1




    $begingroup$
    I agree! I found refuge in math and programming since I could sense for the first time what honesty was.
    $endgroup$
    – J.Moh
    11 hours ago










  • $begingroup$
    @user21820 I did
    $endgroup$
    – J.Moh
    11 hours ago














12












12








12





$begingroup$

If your book reaches the domain $[-1,0]cup[1,+infty)$, it must be because the book only considers $sqrt[3]{phantom{X}}$ to be defined when the argument is a non-negative real.



Books (and people) differ in how they consider $sqrt[N]{phantom X}$ to be defined.



Some people find it okay to define odd roots on the entire real line -- after all, $xmapsto x^N$ is a bijection on $mathbb R$ when $N$ is positive odd, and every such bijection has a perfectly fine inverse.



Other people prefer to restrict these functions to non-negative reals, no matter what $N$ is -- partially to avoid creating a (confusing?) distinction between odd and even $N$, partially for more subtle reasons that unfortunately are not apparent when one first learns about roots.



(For even subtler reasons, one might even want to reserve the root notation to arguments that are strictly positive, such that $sqrt 0$ is considered undefined. It is somewhat rare to take that position consistently, though).



You'll just have to live with the fact that such questions cannot be answered without knowing which convention for the root sign is to be used. (Arguably it is bad form to let a find-the-domain-of-this-expression exercise depend on such choices, but that's purely the textbook's fault, of course).






share|cite|improve this answer











$endgroup$



If your book reaches the domain $[-1,0]cup[1,+infty)$, it must be because the book only considers $sqrt[3]{phantom{X}}$ to be defined when the argument is a non-negative real.



Books (and people) differ in how they consider $sqrt[N]{phantom X}$ to be defined.



Some people find it okay to define odd roots on the entire real line -- after all, $xmapsto x^N$ is a bijection on $mathbb R$ when $N$ is positive odd, and every such bijection has a perfectly fine inverse.



Other people prefer to restrict these functions to non-negative reals, no matter what $N$ is -- partially to avoid creating a (confusing?) distinction between odd and even $N$, partially for more subtle reasons that unfortunately are not apparent when one first learns about roots.



(For even subtler reasons, one might even want to reserve the root notation to arguments that are strictly positive, such that $sqrt 0$ is considered undefined. It is somewhat rare to take that position consistently, though).



You'll just have to live with the fact that such questions cannot be answered without knowing which convention for the root sign is to be used. (Arguably it is bad form to let a find-the-domain-of-this-expression exercise depend on such choices, but that's purely the textbook's fault, of course).







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 17 hours ago

























answered 17 hours ago









Henning MakholmHenning Makholm

243k17312556




243k17312556












  • $begingroup$
    That s why I love computer scientists, they answer as if they re writing code ;) Thanks Henning! Perfect!
    $endgroup$
    – J.Moh
    17 hours ago








  • 1




    $begingroup$
    @J.Moh: If every mathematics student learns programming, we would hardly see any of the silly mistakes arising from imprecision. I agree with Henning's last sentence and even say that such kind of questions are terrible because they encourage imprecision. Moreover, I personally think that we should define $sqrt[n]{x}$ for all real $x$ and odd natural number $n$, because $(mathbb{R} x ↦ x^n)$ is a bijection from $mathbb{R}$ to $mathbb{R}$, so its inverse exists. Similarly for non-negative real $x$ and even natural number $n$.
    $endgroup$
    – user21820
    11 hours ago












  • $begingroup$
    @J.Moh: By the way, if you are satisfied with this answer, you can click the tick to accept it.
    $endgroup$
    – user21820
    11 hours ago






  • 1




    $begingroup$
    I agree! I found refuge in math and programming since I could sense for the first time what honesty was.
    $endgroup$
    – J.Moh
    11 hours ago










  • $begingroup$
    @user21820 I did
    $endgroup$
    – J.Moh
    11 hours ago


















  • $begingroup$
    That s why I love computer scientists, they answer as if they re writing code ;) Thanks Henning! Perfect!
    $endgroup$
    – J.Moh
    17 hours ago








  • 1




    $begingroup$
    @J.Moh: If every mathematics student learns programming, we would hardly see any of the silly mistakes arising from imprecision. I agree with Henning's last sentence and even say that such kind of questions are terrible because they encourage imprecision. Moreover, I personally think that we should define $sqrt[n]{x}$ for all real $x$ and odd natural number $n$, because $(mathbb{R} x ↦ x^n)$ is a bijection from $mathbb{R}$ to $mathbb{R}$, so its inverse exists. Similarly for non-negative real $x$ and even natural number $n$.
    $endgroup$
    – user21820
    11 hours ago












  • $begingroup$
    @J.Moh: By the way, if you are satisfied with this answer, you can click the tick to accept it.
    $endgroup$
    – user21820
    11 hours ago






  • 1




    $begingroup$
    I agree! I found refuge in math and programming since I could sense for the first time what honesty was.
    $endgroup$
    – J.Moh
    11 hours ago










  • $begingroup$
    @user21820 I did
    $endgroup$
    – J.Moh
    11 hours ago
















$begingroup$
That s why I love computer scientists, they answer as if they re writing code ;) Thanks Henning! Perfect!
$endgroup$
– J.Moh
17 hours ago






$begingroup$
That s why I love computer scientists, they answer as if they re writing code ;) Thanks Henning! Perfect!
$endgroup$
– J.Moh
17 hours ago






1




1




$begingroup$
@J.Moh: If every mathematics student learns programming, we would hardly see any of the silly mistakes arising from imprecision. I agree with Henning's last sentence and even say that such kind of questions are terrible because they encourage imprecision. Moreover, I personally think that we should define $sqrt[n]{x}$ for all real $x$ and odd natural number $n$, because $(mathbb{R} x ↦ x^n)$ is a bijection from $mathbb{R}$ to $mathbb{R}$, so its inverse exists. Similarly for non-negative real $x$ and even natural number $n$.
$endgroup$
– user21820
11 hours ago






$begingroup$
@J.Moh: If every mathematics student learns programming, we would hardly see any of the silly mistakes arising from imprecision. I agree with Henning's last sentence and even say that such kind of questions are terrible because they encourage imprecision. Moreover, I personally think that we should define $sqrt[n]{x}$ for all real $x$ and odd natural number $n$, because $(mathbb{R} x ↦ x^n)$ is a bijection from $mathbb{R}$ to $mathbb{R}$, so its inverse exists. Similarly for non-negative real $x$ and even natural number $n$.
$endgroup$
– user21820
11 hours ago














$begingroup$
@J.Moh: By the way, if you are satisfied with this answer, you can click the tick to accept it.
$endgroup$
– user21820
11 hours ago




$begingroup$
@J.Moh: By the way, if you are satisfied with this answer, you can click the tick to accept it.
$endgroup$
– user21820
11 hours ago




1




1




$begingroup$
I agree! I found refuge in math and programming since I could sense for the first time what honesty was.
$endgroup$
– J.Moh
11 hours ago




$begingroup$
I agree! I found refuge in math and programming since I could sense for the first time what honesty was.
$endgroup$
– J.Moh
11 hours ago












$begingroup$
@user21820 I did
$endgroup$
– J.Moh
11 hours ago




$begingroup$
@user21820 I did
$endgroup$
– J.Moh
11 hours ago


















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GameSpot

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body.skin-minerva .mw-parser-output table.infobox caption{text-align:center} GameSpot 戰地風雲：惡名昭彰2在Gamespot的評論 网站类型 新聞 持有者 CBS 创始人 Pete Deemer Vince Broady Jon Epstein 网站 http://www.gamespot.com/ 注册 Optional (free and paid) 推出时间 1996年5月1日 [1] GameSpot （中国大陆：游戏基地），於1996年5月由Pete Deemer和Vince Broady創立，是一個提供新聞、評論、預告片、下載及其他的相關資訊的電子遊戲網站。GameSpot被一間後來被CNET收購的企業ZDNet所收購。根據Alexa，GameSpot.com是200個網路擁擠最嚴重的網站之一。 除了由GameSpot員工創作的內容，網站還允許用戶寫評論、網誌、之後在網路論壇分享。一些在CNET旗下的GameFAQs分享。 2004年， GameSpot被Spike TV的觀眾選上「電子遊戲賞節目」贏得「最傑出遊戲網站。 [2] 其他的遊戲網站還有IGN、1UP.com、GameSpy是它最大的競爭對手。2008年，根據Compete.com的統計，「gamespot.com」吸引了最少6000萬人的點擊率。 [3] GameSpot的主頁鏈結了到最近新聞、評論、預告、和一些有關遊戲機的入口：Wii、任天堂DS、電腦遊戲、Xbox 360、PSP、PlayStation 2、PlayStation 3。它還有一列「最受歡迎遊戲名單」，還有給用戶快速獲得遊戲資訊的搜尋器。GameSpot 還包括一些小範圍的遊戲機：任天堂64、GameCube、Game Boy Color、Game Boy Advance、Xbox、PlayStation、SEGA Saturn、Dreamcast、Neo Geo Pocket Color、N-Gage、手機遊戲。 目录 1 歷史 1.1 國際歷史 1.2 著名的員工 2 評論和分
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connect to host localhost port 22: Connection refused

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.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ margin-bottom:0; } 1 1 I created a Ubuntu Vagrant box and ssh'd into it. Now, created a ssh key and trying to ssh to my localhost but I get the following error. vagrant@vagrant-ubuntu-trusty-64:~$ ssh -vvvv localhost OpenSSH_6.6.1, OpenSSL 1.0.1f 6 Jan 2014 debug1: Reading configuration data /etc/ssh/ssh_config debug1: /etc/ssh/ssh_config line 19: Applying options for * debug2: ssh_connect: needpriv 0 debug1: Connecting to localhost [127.0.0.1] port 22. debug1: connect to address 127.0.0.1 port 22: Connection refused ssh: connect to host localhost port 22: Connection refused When I check the list of ports opened as follows: vagrant@vagrant-ubuntu-trusty-64:~$ s
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Getting a Wifi WPA2 wifi connection

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.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ margin-bottom:0; } 0 I've been trying to get a simple wifi connection ! bu I have to admit I just can't get it done ! here is what I've tried: root@MYONE:/etc# clear root@MYONE:/etc# vi wpa_supplicant.conf ctrl_interface=/var/run/wpa_supplicant ctrl_interface_group=0 update_config=1 network={ ssid="Mi Engine" psk="7407352688e4" } root@MYONE:/etc# iw wlan0 link Not connected. root@MYONE:/etc# ifconfig eth0 Link encap:Ethernet HWaddr 74:FE:48:3A:55:AB inet addr:192.168.178.88 Bcast:192.168.178.255 Mask:255.255.255.0 inet6 addr: fe80::76fe:48ff:fe3a:55ab%4804152/64 Scope:Link UP BRO
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