How to find domain of a function in the implicit function theorem?
So i am working with an excercise reguarding the implicit function theorem .
$$F(x, y) = y(e^y + x) − log(x).$$
I know $(x_0,y_0)=(1,0)$ such that $F(x_0, y_0) = 0$.
I want to check that there exist a strictly positive real number $delta > 0$ and a function $y =f(x)$ such that $$y_0 = f(x_0)quad text{ and }quad F(x, f(x)) = 0 qquad forall x ∈ (x_0 − delta, x_0 + delta).$$
Is it fair to say that , in order to solve the question, we have to check if it possible to apply the first part of the Implicit Function Theorem?
Which would imply, as the answer sheet says, that the point $(1, 0)$ belongs to $$DF = {(x, y) in mathbb{R}^2∶ x > 0, y in mathbb{R}}$$
the domain of $F$.
I have dyscalculia and it appears confusing how to solve the domain of this function, where $x$ is greater than $0$ and $y$ exists in real numbers..
Can somebody please show me, in the simplest way possible, how to find the domain so I can proceed with the excercise? Thanks!
implicit-function-theorem
add a comment |
So i am working with an excercise reguarding the implicit function theorem .
$$F(x, y) = y(e^y + x) − log(x).$$
I know $(x_0,y_0)=(1,0)$ such that $F(x_0, y_0) = 0$.
I want to check that there exist a strictly positive real number $delta > 0$ and a function $y =f(x)$ such that $$y_0 = f(x_0)quad text{ and }quad F(x, f(x)) = 0 qquad forall x ∈ (x_0 − delta, x_0 + delta).$$
Is it fair to say that , in order to solve the question, we have to check if it possible to apply the first part of the Implicit Function Theorem?
Which would imply, as the answer sheet says, that the point $(1, 0)$ belongs to $$DF = {(x, y) in mathbb{R}^2∶ x > 0, y in mathbb{R}}$$
the domain of $F$.
I have dyscalculia and it appears confusing how to solve the domain of this function, where $x$ is greater than $0$ and $y$ exists in real numbers..
Can somebody please show me, in the simplest way possible, how to find the domain so I can proceed with the excercise? Thanks!
implicit-function-theorem
add a comment |
So i am working with an excercise reguarding the implicit function theorem .
$$F(x, y) = y(e^y + x) − log(x).$$
I know $(x_0,y_0)=(1,0)$ such that $F(x_0, y_0) = 0$.
I want to check that there exist a strictly positive real number $delta > 0$ and a function $y =f(x)$ such that $$y_0 = f(x_0)quad text{ and }quad F(x, f(x)) = 0 qquad forall x ∈ (x_0 − delta, x_0 + delta).$$
Is it fair to say that , in order to solve the question, we have to check if it possible to apply the first part of the Implicit Function Theorem?
Which would imply, as the answer sheet says, that the point $(1, 0)$ belongs to $$DF = {(x, y) in mathbb{R}^2∶ x > 0, y in mathbb{R}}$$
the domain of $F$.
I have dyscalculia and it appears confusing how to solve the domain of this function, where $x$ is greater than $0$ and $y$ exists in real numbers..
Can somebody please show me, in the simplest way possible, how to find the domain so I can proceed with the excercise? Thanks!
implicit-function-theorem
So i am working with an excercise reguarding the implicit function theorem .
$$F(x, y) = y(e^y + x) − log(x).$$
I know $(x_0,y_0)=(1,0)$ such that $F(x_0, y_0) = 0$.
I want to check that there exist a strictly positive real number $delta > 0$ and a function $y =f(x)$ such that $$y_0 = f(x_0)quad text{ and }quad F(x, f(x)) = 0 qquad forall x ∈ (x_0 − delta, x_0 + delta).$$
Is it fair to say that , in order to solve the question, we have to check if it possible to apply the first part of the Implicit Function Theorem?
Which would imply, as the answer sheet says, that the point $(1, 0)$ belongs to $$DF = {(x, y) in mathbb{R}^2∶ x > 0, y in mathbb{R}}$$
the domain of $F$.
I have dyscalculia and it appears confusing how to solve the domain of this function, where $x$ is greater than $0$ and $y$ exists in real numbers..
Can somebody please show me, in the simplest way possible, how to find the domain so I can proceed with the excercise? Thanks!
implicit-function-theorem
implicit-function-theorem
edited Dec 8 at 10:15
smcc
4,297517
4,297517
asked Dec 8 at 9:19
BM97
688
688
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
The formula defining $F$ only makes sense for $x>0$ because pf the second term, $log(x)$. The largest possible domain of the logarithm function is all positive real numbers, so we need $x>0$. There is no restriction on $y$ because $y(e^y+x)$ makes sense for any real number $y$ (the largest possible domain of the exponential function is all real numbers).
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
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
});
}
});
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3030864%2fhow-to-find-domain-of-a-function-in-the-implicit-function-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
The formula defining $F$ only makes sense for $x>0$ because pf the second term, $log(x)$. The largest possible domain of the logarithm function is all positive real numbers, so we need $x>0$. There is no restriction on $y$ because $y(e^y+x)$ makes sense for any real number $y$ (the largest possible domain of the exponential function is all real numbers).
add a comment |
The formula defining $F$ only makes sense for $x>0$ because pf the second term, $log(x)$. The largest possible domain of the logarithm function is all positive real numbers, so we need $x>0$. There is no restriction on $y$ because $y(e^y+x)$ makes sense for any real number $y$ (the largest possible domain of the exponential function is all real numbers).
add a comment |
The formula defining $F$ only makes sense for $x>0$ because pf the second term, $log(x)$. The largest possible domain of the logarithm function is all positive real numbers, so we need $x>0$. There is no restriction on $y$ because $y(e^y+x)$ makes sense for any real number $y$ (the largest possible domain of the exponential function is all real numbers).
The formula defining $F$ only makes sense for $x>0$ because pf the second term, $log(x)$. The largest possible domain of the logarithm function is all positive real numbers, so we need $x>0$. There is no restriction on $y$ because $y(e^y+x)$ makes sense for any real number $y$ (the largest possible domain of the exponential function is all real numbers).
answered Dec 8 at 10:18
smcc
4,297517
4,297517
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3030864%2fhow-to-find-domain-of-a-function-in-the-implicit-function-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown