Ttyjfyk

Direct Experience of Meditation

Is 1 ppb equal to 1 μg/kg?

Complexity of many constant time steps with occasional logarithmic steps

Can a zero nonce be safely used with AES-GCM if the key is random and never used again?

How does modal jazz use chord progressions?

Can I throw a longsword at someone?

Strange behaviour of Check

Why is "Captain Marvel" translated as male in Portugal?

Determine whether f is a function, an injection, a surjection

How are presidential pardons supposed to be used?

How do you clear the ApexPages.getMessages() collection in a test?

Slither Like a Snake

Is there a service that would inform me whenever a new direct route is scheduled from a given airport?

When communicating altitude with a '9' in it, should it be pronounced "nine hundred" or "niner hundred"?

Is it possible to ask for a hotel room without minibar/extra services?

What to do with post with dry rot?

What is the order of Mitzvot in Rambam's Sefer Hamitzvot?

Biased dice probability question

What do you call a plan that's an alternative plan in case your initial plan fails?

What LEGO pieces have "real-world" functionality?

New Order #5: where Fibonacci and Beatty meet at Wythoff

How should I respond to a player wanting to catch a sword between their hands?

What computer would be fastest for Mathematica Home Edition?

I'm having difficulty getting my players to do stuff in a sandbox campaign

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

edited 11 hours ago

Asaf Karagila♦

308k33441775

asked 19 hours ago

J.MohJ.Moh

695

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I think it's better $D(f)=mathbb R$, but if $g(x)=(x^3-x)^frac13$ so $D(g)=xinmathbb R.$ All these a definition only.
  $endgroup$
  – Michael Rozenberg
  19 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  

 | 
show 1 more comment

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

edited 11 hours ago

Asaf Karagila♦

308k33441775

asked 19 hours ago

J.MohJ.Moh

695

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I think it's better $D(f)=mathbb R$, but if $g(x)=(x^3-x)^frac13$ so $D(g)=xinmathbb R.$ All these a definition only.
  $endgroup$
  – Michael Rozenberg
  19 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  

 | 
show 1 more comment

$begingroup$

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

edited 11 hours ago

Asaf Karagila♦

308k33441775

asked 19 hours ago

J.MohJ.Moh

695

$endgroup$

share|cite|improve this question

edited 11 hours ago

Asaf Karagila♦

308k33441775

asked 19 hours ago

J.MohJ.Moh

695

share|cite|improve this question

edited 11 hours ago

Asaf Karagila♦

308k33441775

asked 19 hours ago

J.MohJ.Moh

695

edited 11 hours ago

Asaf Karagila♦

308k33441775

edited 11 hours ago

Asaf Karagila♦

308k33441775

asked 19 hours ago

J.MohJ.Moh

695

asked 19 hours ago

J.MohJ.Moh

695

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 $sqrt[3]phantomX$ 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

edited 19 hours ago

answered 19 hours ago

Henning MakholmHenning Makholm

243k17312556

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That s why I love computer scientists, they answer as if they re writing code ;) Thanks Henning! Perfect!
  $endgroup$
  – J.Moh
  18 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 $(mathbbR x ↦ x^n)$ is a bijection from $mathbbR$ to $mathbbR$, so its inverse exists. Similarly for non-negative real $x$ and even natural number $n$.
  $endgroup$
  – user21820
  13 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
  13 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
  13 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  

 | 
show 2 more comments

Your Answer

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3187198%2fwhat-is-the-domain-of-the-function-fx-sqrt3x3-x%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

12

$begingroup$

If your book reaches the domain $[-1,0]cup[1,+infty)$, it must be because the book only considers $sqrt[3]phantomX$ 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

edited 19 hours ago

answered 19 hours ago

Henning MakholmHenning Makholm

243k17312556

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That s why I love computer scientists, they answer as if they re writing code ;) Thanks Henning! Perfect!
  $endgroup$
  – J.Moh
  18 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 $(mathbbR x ↦ x^n)$ is a bijection from $mathbbR$ to $mathbbR$, so its inverse exists. Similarly for non-negative real $x$ and even natural number $n$.
  $endgroup$
  – user21820
  13 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
  13 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
  13 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  

 | 
show 2 more comments

12

$begingroup$

If your book reaches the domain $[-1,0]cup[1,+infty)$, it must be because the book only considers $sqrt[3]phantomX$ 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

edited 19 hours ago

answered 19 hours ago

Henning MakholmHenning Makholm

243k17312556

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That s why I love computer scientists, they answer as if they re writing code ;) Thanks Henning! Perfect!
  $endgroup$
  – J.Moh
  18 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 $(mathbbR x ↦ x^n)$ is a bijection from $mathbbR$ to $mathbbR$, so its inverse exists. Similarly for non-negative real $x$ and even natural number $n$.
  $endgroup$
  – user21820
  13 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
  13 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
  13 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  

 | 
show 2 more comments

$begingroup$

If your book reaches the domain $[-1,0]cup[1,+infty)$, it must be because the book only considers $sqrt[3]phantomX$ 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

edited 19 hours ago

answered 19 hours ago

Henning MakholmHenning Makholm

243k17312556

$endgroup$

share|cite|improve this answer

edited 19 hours ago

answered 19 hours ago

Henning MakholmHenning Makholm

243k17312556

answered 19 hours ago

Henning MakholmHenning Makholm

243k17312556

answered 19 hours ago

Henning MakholmHenning Makholm

243k17312556