Closed-form solution for $f(x)/x=y$ using $f^{-1}$
$begingroup$
I'm programming a piece of math that requires solving an equation of a form $f(x)/x=y$. Now I already have $f^{-1}(z)$ coded (efficiently, and not by me) so I'd prefer using this implementation instead of, say, coding bisection for $f(x)/x$. Is there a way to do that?
If it were $f(x)=y$, I would just use $x=f^{-1}(y)$. But with $f(x)/x=y$ I doubt it's even possible to have a closed-form solution with $f$ and $f^{-1}$ but I can't come up with a reasoning for that.
functions closed-form applications inverse-function
asked Jan 21 at 17:47
GlinkaGlinka
1,576828
$endgroup$
add a comment |
$begingroup$
I'm programming a piece of math that requires solving an equation of a form $f(x)/x=y$. Now I already have $f^{-1}(z)$ coded (efficiently, and not by me) so I'd prefer using this implementation instead of, say, coding bisection for $f(x)/x$. Is there a way to do that?
If it were $f(x)=y$, I would just use $x=f^{-1}(y)$. But with $f(x)/x=y$ I doubt it's even possible to have a closed-form solution with $f$ and $f^{-1}$ but I can't come up with a reasoning for that.
functions closed-form applications inverse-function
asked Jan 21 at 17:47
GlinkaGlinka
1,576828
$endgroup$
$begingroup$
What is formula for $f(x)$ ?
$endgroup$
– coffeemath
Jan 21 at 17:49
1
$begingroup$
@coffeemath, it's very large and clumsy and is constructed in several steps. It's not the first time I faced this problem, previously with different expressions for $f(x)$, so I'm asking a question in general, for arbitrary $f(x)$.
$endgroup$
– Glinka
Jan 21 at 17:55
add a comment |
$begingroup$
I'm programming a piece of math that requires solving an equation of a form $f(x)/x=y$. Now I already have $f^{-1}(z)$ coded (efficiently, and not by me) so I'd prefer using this implementation instead of, say, coding bisection for $f(x)/x$. Is there a way to do that?
If it were $f(x)=y$, I would just use $x=f^{-1}(y)$. But with $f(x)/x=y$ I doubt it's even possible to have a closed-form solution with $f$ and $f^{-1}$ but I can't come up with a reasoning for that.
functions closed-form applications inverse-function
asked Jan 21 at 17:47
GlinkaGlinka
1,576828
$endgroup$
I'm programming a piece of math that requires solving an equation of a form $f(x)/x=y$. Now I already have $f^{-1}(z)$ coded (efficiently, and not by me) so I'd prefer using this implementation instead of, say, coding bisection for $f(x)/x$. Is there a way to do that?
If it were $f(x)=y$, I would just use $x=f^{-1}(y)$. But with $f(x)/x=y$ I doubt it's even possible to have a closed-form solution with $f$ and $f^{-1}$ but I can't come up with a reasoning for that.
functions closed-form applications inverse-function
functions closed-form applications inverse-function
asked Jan 21 at 17:47
GlinkaGlinka
1,576828
asked Jan 21 at 17:47
GlinkaGlinka
1,576828
asked Jan 21 at 17:47
GlinkaGlinka
1,576828
asked Jan 21 at 17:47
GlinkaGlinka
1,576828
asked Jan 21 at 17:47
GlinkaGlinka
1,576828
1,576828
$begingroup$
What is formula for $f(x)$ ?
$endgroup$
– coffeemath
Jan 21 at 17:49
1
$begingroup$
@coffeemath, it's very large and clumsy and is constructed in several steps. It's not the first time I faced this problem, previously with different expressions for $f(x)$, so I'm asking a question in general, for arbitrary $f(x)$.
$endgroup$
– Glinka
Jan 21 at 17:55
add a comment |
$begingroup$
What is formula for $f(x)$ ?
$endgroup$
– coffeemath
Jan 21 at 17:49
1
$begingroup$
@coffeemath, it's very large and clumsy and is constructed in several steps. It's not the first time I faced this problem, previously with different expressions for $f(x)$, so I'm asking a question in general, for arbitrary $f(x)$.
$endgroup$
– Glinka
Jan 21 at 17:55
$begingroup$
What is formula for $f(x)$ ?
$endgroup$
– coffeemath
Jan 21 at 17:49
$begingroup$
What is formula for $f(x)$ ?
$endgroup$
– coffeemath
Jan 21 at 17:49
1
1
$begingroup$
@coffeemath, it's very large and clumsy and is constructed in several steps. It's not the first time I faced this problem, previously with different expressions for $f(x)$, so I'm asking a question in general, for arbitrary $f(x)$.
$endgroup$
– Glinka
Jan 21 at 17:55
$begingroup$
@coffeemath, it's very large and clumsy and is constructed in several steps. It's not the first time I faced this problem, previously with different expressions for $f(x)$, so I'm asking a question in general, for arbitrary $f(x)$.
$endgroup$
– Glinka
Jan 21 at 17:55
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Not possible in general. For example, if $f(x) = e^x$ then $f^{-1}(x) = log x$. But the solution $x$ of
$$
frac{e^x}{x} = y
$$
is not elementary:
$$
x = -Wleft(frac{-1}{y}right)
$$
where W is the Lambert W function.
Knowing how to compute $e^x$ and $log x$ does not tell you how to compute this solution.
edited Jan 21 at 18:09
NickD
1,1311512
answered Jan 21 at 18:02
GEdgarGEdgar
62.6k267171
$endgroup$
$begingroup$
Thanks, that answers it
$endgroup$
– Glinka
Jan 21 at 18:11
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%2f3082166%2fclosed-form-solution-for-fx-x-y-using-f-1%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
$begingroup$
Not possible in general. For example, if $f(x) = e^x$ then $f^{-1}(x) = log x$. But the solution $x$ of
$$
frac{e^x}{x} = y
$$
is not elementary:
$$
x = -Wleft(frac{-1}{y}right)
$$
where W is the Lambert W function.
Knowing how to compute $e^x$ and $log x$ does not tell you how to compute this solution.
edited Jan 21 at 18:09
NickD
1,1311512
answered Jan 21 at 18:02
GEdgarGEdgar
62.6k267171
$endgroup$
$begingroup$
Thanks, that answers it
$endgroup$
– Glinka
Jan 21 at 18:11
add a comment |
$begingroup$
Not possible in general. For example, if $f(x) = e^x$ then $f^{-1}(x) = log x$. But the solution $x$ of
$$
frac{e^x}{x} = y
$$
is not elementary:
$$
x = -Wleft(frac{-1}{y}right)
$$
where W is the Lambert W function.
Knowing how to compute $e^x$ and $log x$ does not tell you how to compute this solution.
edited Jan 21 at 18:09
NickD
1,1311512
answered Jan 21 at 18:02
GEdgarGEdgar
62.6k267171
$endgroup$
$begingroup$
Thanks, that answers it
$endgroup$
– Glinka
Jan 21 at 18:11
add a comment |
$begingroup$
Not possible in general. For example, if $f(x) = e^x$ then $f^{-1}(x) = log x$. But the solution $x$ of
$$
frac{e^x}{x} = y
$$
is not elementary:
$$
x = -Wleft(frac{-1}{y}right)
$$
where W is the Lambert W function.
Knowing how to compute $e^x$ and $log x$ does not tell you how to compute this solution.
edited Jan 21 at 18:09
NickD
1,1311512
answered Jan 21 at 18:02
GEdgarGEdgar
62.6k267171
$endgroup$
Not possible in general. For example, if $f(x) = e^x$ then $f^{-1}(x) = log x$. But the solution $x$ of
$$
frac{e^x}{x} = y
$$
is not elementary:
$$
x = -Wleft(frac{-1}{y}right)
$$
where W is the Lambert W function.
Knowing how to compute $e^x$ and $log x$ does not tell you how to compute this solution.
edited Jan 21 at 18:09
NickD
1,1311512
answered Jan 21 at 18:02
GEdgarGEdgar
62.6k267171
edited Jan 21 at 18:09
NickD
1,1311512
edited Jan 21 at 18:09
NickD
1,1311512
edited Jan 21 at 18:09
NickD
1,1311512
1,1311512
answered Jan 21 at 18:02
GEdgarGEdgar
62.6k267171
answered Jan 21 at 18:02
GEdgarGEdgar
62.6k267171
answered Jan 21 at 18:02
GEdgarGEdgar
62.6k267171
62.6k267171
$begingroup$
Thanks, that answers it
$endgroup$
– Glinka
Jan 21 at 18:11
add a comment |
$begingroup$
Thanks, that answers it
$endgroup$
– Glinka
Jan 21 at 18:11
$begingroup$
Thanks, that answers it
$endgroup$
– Glinka
Jan 21 at 18:11
$begingroup$
Thanks, that answers it
$endgroup$
– Glinka
Jan 21 at 18:11
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.
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%2f3082166%2fclosed-form-solution-for-fx-x-y-using-f-1%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
$begingroup$
What is formula for $f(x)$ ?
$endgroup$
– coffeemath
Jan 21 at 17:49
1
$begingroup$
@coffeemath, it's very large and clumsy and is constructed in several steps. It's not the first time I faced this problem, previously with different expressions for $f(x)$, so I'm asking a question in general, for arbitrary $f(x)$.
$endgroup$
– Glinka
Jan 21 at 17:55