Find all $f:mathbb Rtomathbb R$ such that $forall x,yinmathbb R$ the given equality holds:...
$begingroup$
Find all $f:mathbb Rto mathbb R$ such that $forall x,yinmathbb R$ the given equality holds: $$xf(y)+yf(x)=(x+y)f(x)f(y)$$
My try:
whenever $y=0$, we have $$xcdot f(0)cdot(1-f(x))=0$$
So the solutions can be either generated by: $$f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$$
where $c$ generate an infinite amount of solutions, or it can be any function satisfying $f(0)=0$.
However, I can't seem to find a way to check such a solution on the initial equation (we have to check it since we've simply eliminated the impossible solutions and we've only found that the solutions can be somehow generated using the way I've shown).
Note that this generates the trivial solutions $f(x)=0$ and $f(x)=1$ as well.
Is my reasoning correct?
If so, how can I check whether all the generated functions satisfy the initial equation?
Off-topic, but do $f^2(x)$ and $f(x)^2$ denote the same concept? My teacher uses the former while my book uses the latter (for squaring $f(x)$). Which one is correct, or are both of them correct?
functions functional-equations
asked Jun 3 '14 at 18:16
user26486user26486
9,28221948
$endgroup$
add a comment |
$begingroup$
Find all $f:mathbb Rto mathbb R$ such that $forall x,yinmathbb R$ the given equality holds: $$xf(y)+yf(x)=(x+y)f(x)f(y)$$
My try:
whenever $y=0$, we have $$xcdot f(0)cdot(1-f(x))=0$$
So the solutions can be either generated by: $$f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$$
where $c$ generate an infinite amount of solutions, or it can be any function satisfying $f(0)=0$.
However, I can't seem to find a way to check such a solution on the initial equation (we have to check it since we've simply eliminated the impossible solutions and we've only found that the solutions can be somehow generated using the way I've shown).
Note that this generates the trivial solutions $f(x)=0$ and $f(x)=1$ as well.
Is my reasoning correct?
If so, how can I check whether all the generated functions satisfy the initial equation?
Off-topic, but do $f^2(x)$ and $f(x)^2$ denote the same concept? My teacher uses the former while my book uses the latter (for squaring $f(x)$). Which one is correct, or are both of them correct?
functions functional-equations
asked Jun 3 '14 at 18:16
user26486user26486
9,28221948
$endgroup$
$begingroup$
Note that if you plug $y=0$ into your original equation you have $xf(0) = xf(x)f(0) forall x$, so if $f(0)neq 0$ then your solutions are much more tightly constrained.
$endgroup$
– Steven Stadnicki
Jun 3 '14 at 18:22
$begingroup$
Yes, you are right. But, you haven't proved that you already covered all functions.
$endgroup$
– Jlamprong
Jun 3 '14 at 18:22
$begingroup$
@Jlamprong I'm not sure what you mean. I have proved that whatever the solutions the initial equation has, they can be generated using the formula I've given.
$endgroup$
– user26486
Jun 3 '14 at 18:25
$begingroup$
@StevenStadnicki It follows that the solutions can be generated by $$f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$$ or it can be any function satisfying $f(0)=0$, if my reasoning is correct.
$endgroup$
– user26486
Jun 3 '14 at 18:29
$begingroup$
@StevenStadnicki I've edited the question.
$endgroup$
– user26486
Jun 3 '14 at 18:38
add a comment |
$begingroup$
Find all $f:mathbb Rto mathbb R$ such that $forall x,yinmathbb R$ the given equality holds: $$xf(y)+yf(x)=(x+y)f(x)f(y)$$
My try:
whenever $y=0$, we have $$xcdot f(0)cdot(1-f(x))=0$$
So the solutions can be either generated by: $$f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$$
where $c$ generate an infinite amount of solutions, or it can be any function satisfying $f(0)=0$.
However, I can't seem to find a way to check such a solution on the initial equation (we have to check it since we've simply eliminated the impossible solutions and we've only found that the solutions can be somehow generated using the way I've shown).
Note that this generates the trivial solutions $f(x)=0$ and $f(x)=1$ as well.
Is my reasoning correct?
If so, how can I check whether all the generated functions satisfy the initial equation?
Off-topic, but do $f^2(x)$ and $f(x)^2$ denote the same concept? My teacher uses the former while my book uses the latter (for squaring $f(x)$). Which one is correct, or are both of them correct?
functions functional-equations
asked Jun 3 '14 at 18:16
user26486user26486
9,28221948
$endgroup$
Find all $f:mathbb Rto mathbb R$ such that $forall x,yinmathbb R$ the given equality holds: $$xf(y)+yf(x)=(x+y)f(x)f(y)$$
My try:
whenever $y=0$, we have $$xcdot f(0)cdot(1-f(x))=0$$
So the solutions can be either generated by: $$f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$$
where $c$ generate an infinite amount of solutions, or it can be any function satisfying $f(0)=0$.
However, I can't seem to find a way to check such a solution on the initial equation (we have to check it since we've simply eliminated the impossible solutions and we've only found that the solutions can be somehow generated using the way I've shown).
Note that this generates the trivial solutions $f(x)=0$ and $f(x)=1$ as well.
Is my reasoning correct?
If so, how can I check whether all the generated functions satisfy the initial equation?
Off-topic, but do $f^2(x)$ and $f(x)^2$ denote the same concept? My teacher uses the former while my book uses the latter (for squaring $f(x)$). Which one is correct, or are both of them correct?
functions functional-equations
functions functional-equations
asked Jun 3 '14 at 18:16
user26486user26486
9,28221948
asked Jun 3 '14 at 18:16
user26486user26486
9,28221948
edited Jun 3 '14 at 18:35
user26486
asked Jun 3 '14 at 18:16
user26486user26486
9,28221948
asked Jun 3 '14 at 18:16
user26486user26486
9,28221948
asked Jun 3 '14 at 18:16
user26486user26486
9,28221948
9,28221948
$begingroup$
Note that if you plug $y=0$ into your original equation you have $xf(0) = xf(x)f(0) forall x$, so if $f(0)neq 0$ then your solutions are much more tightly constrained.
$endgroup$
– Steven Stadnicki
Jun 3 '14 at 18:22
$begingroup$
Yes, you are right. But, you haven't proved that you already covered all functions.
$endgroup$
– Jlamprong
Jun 3 '14 at 18:22
$begingroup$
@Jlamprong I'm not sure what you mean. I have proved that whatever the solutions the initial equation has, they can be generated using the formula I've given.
$endgroup$
– user26486
Jun 3 '14 at 18:25
$begingroup$
@StevenStadnicki It follows that the solutions can be generated by $$f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$$ or it can be any function satisfying $f(0)=0$, if my reasoning is correct.
$endgroup$
– user26486
Jun 3 '14 at 18:29
$begingroup$
@StevenStadnicki I've edited the question.
$endgroup$
– user26486
Jun 3 '14 at 18:38
add a comment |
$begingroup$
Note that if you plug $y=0$ into your original equation you have $xf(0) = xf(x)f(0) forall x$, so if $f(0)neq 0$ then your solutions are much more tightly constrained.
$endgroup$
– Steven Stadnicki
Jun 3 '14 at 18:22
$begingroup$
Yes, you are right. But, you haven't proved that you already covered all functions.
$endgroup$
– Jlamprong
Jun 3 '14 at 18:22
$begingroup$
@Jlamprong I'm not sure what you mean. I have proved that whatever the solutions the initial equation has, they can be generated using the formula I've given.
$endgroup$
– user26486
Jun 3 '14 at 18:25
$begingroup$
@StevenStadnicki It follows that the solutions can be generated by $$f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$$ or it can be any function satisfying $f(0)=0$, if my reasoning is correct.
$endgroup$
– user26486
Jun 3 '14 at 18:29
$begingroup$
@StevenStadnicki I've edited the question.
$endgroup$
– user26486
Jun 3 '14 at 18:38
$begingroup$
Note that if you plug $y=0$ into your original equation you have $xf(0) = xf(x)f(0) forall x$, so if $f(0)neq 0$ then your solutions are much more tightly constrained.
$endgroup$
– Steven Stadnicki
Jun 3 '14 at 18:22
$begingroup$
Note that if you plug $y=0$ into your original equation you have $xf(0) = xf(x)f(0) forall x$, so if $f(0)neq 0$ then your solutions are much more tightly constrained.
$endgroup$
– Steven Stadnicki
Jun 3 '14 at 18:22
$begingroup$
Yes, you are right. But, you haven't proved that you already covered all functions.
$endgroup$
– Jlamprong
Jun 3 '14 at 18:22
$begingroup$
Yes, you are right. But, you haven't proved that you already covered all functions.
$endgroup$
– Jlamprong
Jun 3 '14 at 18:22
$begingroup$
@Jlamprong I'm not sure what you mean. I have proved that whatever the solutions the initial equation has, they can be generated using the formula I've given.
$endgroup$
– user26486
Jun 3 '14 at 18:25
$begingroup$
@Jlamprong I'm not sure what you mean. I have proved that whatever the solutions the initial equation has, they can be generated using the formula I've given.
$endgroup$
– user26486
Jun 3 '14 at 18:25
$begingroup$
@StevenStadnicki It follows that the solutions can be generated by $$f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$$ or it can be any function satisfying $f(0)=0$, if my reasoning is correct.
$endgroup$
– user26486
Jun 3 '14 at 18:29
$begingroup$
@StevenStadnicki It follows that the solutions can be generated by $$f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$$ or it can be any function satisfying $f(0)=0$, if my reasoning is correct.
$endgroup$
– user26486
Jun 3 '14 at 18:29
$begingroup$
@StevenStadnicki I've edited the question.
$endgroup$
– user26486
Jun 3 '14 at 18:38
$begingroup$
@StevenStadnicki I've edited the question.
$endgroup$
– user26486
Jun 3 '14 at 18:38
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
You're most of the way there; now just show that your piecewise $f$ works for arbitrary $x$ and $y$. You may find it easiest to break the analysis into four cases: $x,yneq 0$; $x=0, yneq 0$; $xneq 0, y=0$; and $x, y=0$. For instance, for the last case, you have $xf(y)+yf(x) = 0cdot c+0cdot c = 0 = (0+0)cdot ccdot c = (x+y)cdot f(x)cdot f(y)$. You should find similar results in the other cases.
But note that not every function satisfying $f(0)=0$ satisfies the equation; what happens if you take e.g. $f(x) = sin x$, $x=y=fracpi4$? What you've shown is that every function with $f(0)=0$ satisfies the equation when we choose one of $x,y$ to be $0$, but the equation has to hold for arbitrary $x$ and $y$. You may find it useful to set $x=y$ and see what the resultant restriction on $f(x)$ is.
Incidentally, the approach that you're taking is the typical way of solving these questions: find special values of $x$ or $y$ that simplify the equation considerably and turn it into a one-parameter equation. For instance, taking $x=0$ here (obviously) simplifies the equation considerably; so does taking $x=y$.
answered Jun 3 '14 at 18:57
Steven StadnickiSteven Stadnicki
41.1k867122
$endgroup$
$begingroup$
What I meant was if the solution can't be generated by $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$, then $f(0)=0$.
$endgroup$
– user26486
Jun 3 '14 at 19:09
$begingroup$
I haven't fully understood the solution yet. When $f(0)neq0$, it's clear that $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$ generates all the solutions. How to go on with the case $f(0)=0$?
$endgroup$
– user26486
Jun 3 '14 at 20:12
$begingroup$
We have that the solution $f(x)=0$ isn't generated by $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$, so there has to be something about the case $f(0)=0$ that creates such a solution.
$endgroup$
– user26486
Jun 3 '14 at 20:21
$begingroup$
@mathh If $f(a)=0$ for some $a not =0$, then putting $y=a$ gives $af(x)=0 , forall x$ so $f(x)=0 , forall x$. Otherwise $f(x) not =0$ for $x not =0$. Putting $x=y$ gives $2xf(x)=2xf(x)^2$. For $x not =0$ we have $xf(x) not =0$ by above so $f(x)=1$. Thus $f(x)=begin{cases} c , c in mathbb{R} , x=0 \ 1 , x not =0 end{cases}$.
$endgroup$
– Ivan Loh
Jun 3 '14 at 20:26
$begingroup$
@IvanLoh It seems like our problem was that we analysed the case $y=0implies xf(0)(1-f(x))=0$ and got lost when we were left with the case $f(0)=0$, which was never able to provide us the rest of the solutions. However, when we analysed the case $x=y$, we could check all the cases and finish our solution. I think that was the problem.
$endgroup$
– user26486
Jun 3 '14 at 20:48
|
show 6 more comments
$begingroup$
Your solution is not complete. You have only identified a necessary condition. It is not obvious, not true, that they are sufficient conditions.
answered Jun 4 '14 at 1:16
Calvin LinCalvin Lin
36.3k349114
$endgroup$
$begingroup$
That's why I emphasized: "However, I can't seem to find a way to check such a solution on the initial equation (we have to check it since we've simply eliminated the impossible solutions and we've only found that the solutions can be somehow generated using the way I've shown)." You shold've read the entire description before commenting.
$endgroup$
– user26486
Jun 4 '14 at 10:55
$begingroup$
Ivan Loh has posted the full solution in the comments. The final solutions are the functions $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$ and $f(x)=0$.
$endgroup$
– user26486
Jun 4 '14 at 11:17
$begingroup$
One of the main questions that were highlighted in brown was "If so, how can I check whether all the generated functions satisfy the initial equation?"
$endgroup$
– user26486
Jun 4 '14 at 12:40
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%2f819565%2ffind-all-f-mathbb-r-to-mathbb-r-such-that-forall-x-y-in-mathbb-r-the-given%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You're most of the way there; now just show that your piecewise $f$ works for arbitrary $x$ and $y$. You may find it easiest to break the analysis into four cases: $x,yneq 0$; $x=0, yneq 0$; $xneq 0, y=0$; and $x, y=0$. For instance, for the last case, you have $xf(y)+yf(x) = 0cdot c+0cdot c = 0 = (0+0)cdot ccdot c = (x+y)cdot f(x)cdot f(y)$. You should find similar results in the other cases.
But note that not every function satisfying $f(0)=0$ satisfies the equation; what happens if you take e.g. $f(x) = sin x$, $x=y=fracpi4$? What you've shown is that every function with $f(0)=0$ satisfies the equation when we choose one of $x,y$ to be $0$, but the equation has to hold for arbitrary $x$ and $y$. You may find it useful to set $x=y$ and see what the resultant restriction on $f(x)$ is.
Incidentally, the approach that you're taking is the typical way of solving these questions: find special values of $x$ or $y$ that simplify the equation considerably and turn it into a one-parameter equation. For instance, taking $x=0$ here (obviously) simplifies the equation considerably; so does taking $x=y$.
answered Jun 3 '14 at 18:57
Steven StadnickiSteven Stadnicki
41.1k867122
$endgroup$
$begingroup$
What I meant was if the solution can't be generated by $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$, then $f(0)=0$.
$endgroup$
– user26486
Jun 3 '14 at 19:09
$begingroup$
I haven't fully understood the solution yet. When $f(0)neq0$, it's clear that $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$ generates all the solutions. How to go on with the case $f(0)=0$?
$endgroup$
– user26486
Jun 3 '14 at 20:12
$begingroup$
We have that the solution $f(x)=0$ isn't generated by $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$, so there has to be something about the case $f(0)=0$ that creates such a solution.
$endgroup$
– user26486
Jun 3 '14 at 20:21
$begingroup$
@mathh If $f(a)=0$ for some $a not =0$, then putting $y=a$ gives $af(x)=0 , forall x$ so $f(x)=0 , forall x$. Otherwise $f(x) not =0$ for $x not =0$. Putting $x=y$ gives $2xf(x)=2xf(x)^2$. For $x not =0$ we have $xf(x) not =0$ by above so $f(x)=1$. Thus $f(x)=begin{cases} c , c in mathbb{R} , x=0 \ 1 , x not =0 end{cases}$.
$endgroup$
– Ivan Loh
Jun 3 '14 at 20:26
$begingroup$
@IvanLoh It seems like our problem was that we analysed the case $y=0implies xf(0)(1-f(x))=0$ and got lost when we were left with the case $f(0)=0$, which was never able to provide us the rest of the solutions. However, when we analysed the case $x=y$, we could check all the cases and finish our solution. I think that was the problem.
$endgroup$
– user26486
Jun 3 '14 at 20:48
|
show 6 more comments
$begingroup$
You're most of the way there; now just show that your piecewise $f$ works for arbitrary $x$ and $y$. You may find it easiest to break the analysis into four cases: $x,yneq 0$; $x=0, yneq 0$; $xneq 0, y=0$; and $x, y=0$. For instance, for the last case, you have $xf(y)+yf(x) = 0cdot c+0cdot c = 0 = (0+0)cdot ccdot c = (x+y)cdot f(x)cdot f(y)$. You should find similar results in the other cases.
But note that not every function satisfying $f(0)=0$ satisfies the equation; what happens if you take e.g. $f(x) = sin x$, $x=y=fracpi4$? What you've shown is that every function with $f(0)=0$ satisfies the equation when we choose one of $x,y$ to be $0$, but the equation has to hold for arbitrary $x$ and $y$. You may find it useful to set $x=y$ and see what the resultant restriction on $f(x)$ is.
Incidentally, the approach that you're taking is the typical way of solving these questions: find special values of $x$ or $y$ that simplify the equation considerably and turn it into a one-parameter equation. For instance, taking $x=0$ here (obviously) simplifies the equation considerably; so does taking $x=y$.
answered Jun 3 '14 at 18:57
Steven StadnickiSteven Stadnicki
41.1k867122
$endgroup$
$begingroup$
What I meant was if the solution can't be generated by $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$, then $f(0)=0$.
$endgroup$
– user26486
Jun 3 '14 at 19:09
$begingroup$
I haven't fully understood the solution yet. When $f(0)neq0$, it's clear that $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$ generates all the solutions. How to go on with the case $f(0)=0$?
$endgroup$
– user26486
Jun 3 '14 at 20:12
$begingroup$
We have that the solution $f(x)=0$ isn't generated by $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$, so there has to be something about the case $f(0)=0$ that creates such a solution.
$endgroup$
– user26486
Jun 3 '14 at 20:21
$begingroup$
@mathh If $f(a)=0$ for some $a not =0$, then putting $y=a$ gives $af(x)=0 , forall x$ so $f(x)=0 , forall x$. Otherwise $f(x) not =0$ for $x not =0$. Putting $x=y$ gives $2xf(x)=2xf(x)^2$. For $x not =0$ we have $xf(x) not =0$ by above so $f(x)=1$. Thus $f(x)=begin{cases} c , c in mathbb{R} , x=0 \ 1 , x not =0 end{cases}$.
$endgroup$
– Ivan Loh
Jun 3 '14 at 20:26
$begingroup$
@IvanLoh It seems like our problem was that we analysed the case $y=0implies xf(0)(1-f(x))=0$ and got lost when we were left with the case $f(0)=0$, which was never able to provide us the rest of the solutions. However, when we analysed the case $x=y$, we could check all the cases and finish our solution. I think that was the problem.
$endgroup$
– user26486
Jun 3 '14 at 20:48
|
show 6 more comments
$begingroup$
You're most of the way there; now just show that your piecewise $f$ works for arbitrary $x$ and $y$. You may find it easiest to break the analysis into four cases: $x,yneq 0$; $x=0, yneq 0$; $xneq 0, y=0$; and $x, y=0$. For instance, for the last case, you have $xf(y)+yf(x) = 0cdot c+0cdot c = 0 = (0+0)cdot ccdot c = (x+y)cdot f(x)cdot f(y)$. You should find similar results in the other cases.
But note that not every function satisfying $f(0)=0$ satisfies the equation; what happens if you take e.g. $f(x) = sin x$, $x=y=fracpi4$? What you've shown is that every function with $f(0)=0$ satisfies the equation when we choose one of $x,y$ to be $0$, but the equation has to hold for arbitrary $x$ and $y$. You may find it useful to set $x=y$ and see what the resultant restriction on $f(x)$ is.
Incidentally, the approach that you're taking is the typical way of solving these questions: find special values of $x$ or $y$ that simplify the equation considerably and turn it into a one-parameter equation. For instance, taking $x=0$ here (obviously) simplifies the equation considerably; so does taking $x=y$.
answered Jun 3 '14 at 18:57
Steven StadnickiSteven Stadnicki
41.1k867122
$endgroup$
You're most of the way there; now just show that your piecewise $f$ works for arbitrary $x$ and $y$. You may find it easiest to break the analysis into four cases: $x,yneq 0$; $x=0, yneq 0$; $xneq 0, y=0$; and $x, y=0$. For instance, for the last case, you have $xf(y)+yf(x) = 0cdot c+0cdot c = 0 = (0+0)cdot ccdot c = (x+y)cdot f(x)cdot f(y)$. You should find similar results in the other cases.
But note that not every function satisfying $f(0)=0$ satisfies the equation; what happens if you take e.g. $f(x) = sin x$, $x=y=fracpi4$? What you've shown is that every function with $f(0)=0$ satisfies the equation when we choose one of $x,y$ to be $0$, but the equation has to hold for arbitrary $x$ and $y$. You may find it useful to set $x=y$ and see what the resultant restriction on $f(x)$ is.
Incidentally, the approach that you're taking is the typical way of solving these questions: find special values of $x$ or $y$ that simplify the equation considerably and turn it into a one-parameter equation. For instance, taking $x=0$ here (obviously) simplifies the equation considerably; so does taking $x=y$.
answered Jun 3 '14 at 18:57
Steven StadnickiSteven Stadnicki
41.1k867122
answered Jun 3 '14 at 18:57
Steven StadnickiSteven Stadnicki
41.1k867122
answered Jun 3 '14 at 18:57
Steven StadnickiSteven Stadnicki
41.1k867122
answered Jun 3 '14 at 18:57
Steven StadnickiSteven Stadnicki
41.1k867122
41.1k867122
$begingroup$
What I meant was if the solution can't be generated by $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$, then $f(0)=0$.
$endgroup$
– user26486
Jun 3 '14 at 19:09
$begingroup$
I haven't fully understood the solution yet. When $f(0)neq0$, it's clear that $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$ generates all the solutions. How to go on with the case $f(0)=0$?
$endgroup$
– user26486
Jun 3 '14 at 20:12
$begingroup$
We have that the solution $f(x)=0$ isn't generated by $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$, so there has to be something about the case $f(0)=0$ that creates such a solution.
$endgroup$
– user26486
Jun 3 '14 at 20:21
$begingroup$
@mathh If $f(a)=0$ for some $a not =0$, then putting $y=a$ gives $af(x)=0 , forall x$ so $f(x)=0 , forall x$. Otherwise $f(x) not =0$ for $x not =0$. Putting $x=y$ gives $2xf(x)=2xf(x)^2$. For $x not =0$ we have $xf(x) not =0$ by above so $f(x)=1$. Thus $f(x)=begin{cases} c , c in mathbb{R} , x=0 \ 1 , x not =0 end{cases}$.
$endgroup$
– Ivan Loh
Jun 3 '14 at 20:26
$begingroup$
@IvanLoh It seems like our problem was that we analysed the case $y=0implies xf(0)(1-f(x))=0$ and got lost when we were left with the case $f(0)=0$, which was never able to provide us the rest of the solutions. However, when we analysed the case $x=y$, we could check all the cases and finish our solution. I think that was the problem.
$endgroup$
– user26486
Jun 3 '14 at 20:48
|
show 6 more comments
$begingroup$
What I meant was if the solution can't be generated by $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$, then $f(0)=0$.
$endgroup$
– user26486
Jun 3 '14 at 19:09
$begingroup$
I haven't fully understood the solution yet. When $f(0)neq0$, it's clear that $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$ generates all the solutions. How to go on with the case $f(0)=0$?
$endgroup$
– user26486
Jun 3 '14 at 20:12
$begingroup$
We have that the solution $f(x)=0$ isn't generated by $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$, so there has to be something about the case $f(0)=0$ that creates such a solution.
$endgroup$
– user26486
Jun 3 '14 at 20:21
$begingroup$
@mathh If $f(a)=0$ for some $a not =0$, then putting $y=a$ gives $af(x)=0 , forall x$ so $f(x)=0 , forall x$. Otherwise $f(x) not =0$ for $x not =0$. Putting $x=y$ gives $2xf(x)=2xf(x)^2$. For $x not =0$ we have $xf(x) not =0$ by above so $f(x)=1$. Thus $f(x)=begin{cases} c , c in mathbb{R} , x=0 \ 1 , x not =0 end{cases}$.
$endgroup$
– Ivan Loh
Jun 3 '14 at 20:26
$begingroup$
@IvanLoh It seems like our problem was that we analysed the case $y=0implies xf(0)(1-f(x))=0$ and got lost when we were left with the case $f(0)=0$, which was never able to provide us the rest of the solutions. However, when we analysed the case $x=y$, we could check all the cases and finish our solution. I think that was the problem.
$endgroup$
– user26486
Jun 3 '14 at 20:48
$begingroup$
What I meant was if the solution can't be generated by $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$, then $f(0)=0$.
$endgroup$
– user26486
Jun 3 '14 at 19:09
$begingroup$
What I meant was if the solution can't be generated by $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$, then $f(0)=0$.
$endgroup$
– user26486
Jun 3 '14 at 19:09
$begingroup$
I haven't fully understood the solution yet. When $f(0)neq0$, it's clear that $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$ generates all the solutions. How to go on with the case $f(0)=0$?
$endgroup$
– user26486
Jun 3 '14 at 20:12
$begingroup$
I haven't fully understood the solution yet. When $f(0)neq0$, it's clear that $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$ generates all the solutions. How to go on with the case $f(0)=0$?
$endgroup$
– user26486
Jun 3 '14 at 20:12
$begingroup$
We have that the solution $f(x)=0$ isn't generated by $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$, so there has to be something about the case $f(0)=0$ that creates such a solution.
$endgroup$
– user26486
Jun 3 '14 at 20:21
$begingroup$
We have that the solution $f(x)=0$ isn't generated by $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$, so there has to be something about the case $f(0)=0$ that creates such a solution.
$endgroup$
– user26486
Jun 3 '14 at 20:21
$begingroup$
@mathh If $f(a)=0$ for some $a not =0$, then putting $y=a$ gives $af(x)=0 , forall x$ so $f(x)=0 , forall x$. Otherwise $f(x) not =0$ for $x not =0$. Putting $x=y$ gives $2xf(x)=2xf(x)^2$. For $x not =0$ we have $xf(x) not =0$ by above so $f(x)=1$. Thus $f(x)=begin{cases} c , c in mathbb{R} , x=0 \ 1 , x not =0 end{cases}$.
$endgroup$
– Ivan Loh
Jun 3 '14 at 20:26
$begingroup$
@mathh If $f(a)=0$ for some $a not =0$, then putting $y=a$ gives $af(x)=0 , forall x$ so $f(x)=0 , forall x$. Otherwise $f(x) not =0$ for $x not =0$. Putting $x=y$ gives $2xf(x)=2xf(x)^2$. For $x not =0$ we have $xf(x) not =0$ by above so $f(x)=1$. Thus $f(x)=begin{cases} c , c in mathbb{R} , x=0 \ 1 , x not =0 end{cases}$.
$endgroup$
– Ivan Loh
Jun 3 '14 at 20:26
$begingroup$
@IvanLoh It seems like our problem was that we analysed the case $y=0implies xf(0)(1-f(x))=0$ and got lost when we were left with the case $f(0)=0$, which was never able to provide us the rest of the solutions. However, when we analysed the case $x=y$, we could check all the cases and finish our solution. I think that was the problem.
$endgroup$
– user26486
Jun 3 '14 at 20:48
$begingroup$
@IvanLoh It seems like our problem was that we analysed the case $y=0implies xf(0)(1-f(x))=0$ and got lost when we were left with the case $f(0)=0$, which was never able to provide us the rest of the solutions. However, when we analysed the case $x=y$, we could check all the cases and finish our solution. I think that was the problem.
$endgroup$
– user26486
Jun 3 '14 at 20:48
|
show 6 more comments
$begingroup$
Your solution is not complete. You have only identified a necessary condition. It is not obvious, not true, that they are sufficient conditions.
answered Jun 4 '14 at 1:16
Calvin LinCalvin Lin
36.3k349114
$endgroup$
$begingroup$
That's why I emphasized: "However, I can't seem to find a way to check such a solution on the initial equation (we have to check it since we've simply eliminated the impossible solutions and we've only found that the solutions can be somehow generated using the way I've shown)." You shold've read the entire description before commenting.
$endgroup$
– user26486
Jun 4 '14 at 10:55
$begingroup$
Ivan Loh has posted the full solution in the comments. The final solutions are the functions $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$ and $f(x)=0$.
$endgroup$
– user26486
Jun 4 '14 at 11:17
$begingroup$
One of the main questions that were highlighted in brown was "If so, how can I check whether all the generated functions satisfy the initial equation?"
$endgroup$
– user26486
Jun 4 '14 at 12:40
add a comment |
$begingroup$
Your solution is not complete. You have only identified a necessary condition. It is not obvious, not true, that they are sufficient conditions.
answered Jun 4 '14 at 1:16
Calvin LinCalvin Lin
36.3k349114
$endgroup$
$begingroup$
That's why I emphasized: "However, I can't seem to find a way to check such a solution on the initial equation (we have to check it since we've simply eliminated the impossible solutions and we've only found that the solutions can be somehow generated using the way I've shown)." You shold've read the entire description before commenting.
$endgroup$
– user26486
Jun 4 '14 at 10:55
$begingroup$
Ivan Loh has posted the full solution in the comments. The final solutions are the functions $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$ and $f(x)=0$.
$endgroup$
– user26486
Jun 4 '14 at 11:17
$begingroup$
One of the main questions that were highlighted in brown was "If so, how can I check whether all the generated functions satisfy the initial equation?"
$endgroup$
– user26486
Jun 4 '14 at 12:40
add a comment |
$begingroup$
Your solution is not complete. You have only identified a necessary condition. It is not obvious, not true, that they are sufficient conditions.
answered Jun 4 '14 at 1:16
Calvin LinCalvin Lin
36.3k349114
$endgroup$
Your solution is not complete. You have only identified a necessary condition. It is not obvious, not true, that they are sufficient conditions.
answered Jun 4 '14 at 1:16
Calvin LinCalvin Lin
36.3k349114
answered Jun 4 '14 at 1:16
Calvin LinCalvin Lin
36.3k349114
answered Jun 4 '14 at 1:16
Calvin LinCalvin Lin
36.3k349114
answered Jun 4 '14 at 1:16
Calvin LinCalvin Lin
36.3k349114
36.3k349114
$begingroup$
That's why I emphasized: "However, I can't seem to find a way to check such a solution on the initial equation (we have to check it since we've simply eliminated the impossible solutions and we've only found that the solutions can be somehow generated using the way I've shown)." You shold've read the entire description before commenting.
$endgroup$
– user26486
Jun 4 '14 at 10:55
$begingroup$
Ivan Loh has posted the full solution in the comments. The final solutions are the functions $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$ and $f(x)=0$.
$endgroup$
– user26486
Jun 4 '14 at 11:17
$begingroup$
One of the main questions that were highlighted in brown was "If so, how can I check whether all the generated functions satisfy the initial equation?"
$endgroup$
– user26486
Jun 4 '14 at 12:40
add a comment |
$begingroup$
That's why I emphasized: "However, I can't seem to find a way to check such a solution on the initial equation (we have to check it since we've simply eliminated the impossible solutions and we've only found that the solutions can be somehow generated using the way I've shown)." You shold've read the entire description before commenting.
$endgroup$
– user26486
Jun 4 '14 at 10:55
$begingroup$
Ivan Loh has posted the full solution in the comments. The final solutions are the functions $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$ and $f(x)=0$.
$endgroup$
– user26486
Jun 4 '14 at 11:17
$begingroup$
One of the main questions that were highlighted in brown was "If so, how can I check whether all the generated functions satisfy the initial equation?"
$endgroup$
– user26486
Jun 4 '14 at 12:40
$begingroup$
That's why I emphasized: "However, I can't seem to find a way to check such a solution on the initial equation (we have to check it since we've simply eliminated the impossible solutions and we've only found that the solutions can be somehow generated using the way I've shown)." You shold've read the entire description before commenting.
$endgroup$
– user26486
Jun 4 '14 at 10:55
$begingroup$
That's why I emphasized: "However, I can't seem to find a way to check such a solution on the initial equation (we have to check it since we've simply eliminated the impossible solutions and we've only found that the solutions can be somehow generated using the way I've shown)." You shold've read the entire description before commenting.
$endgroup$
– user26486
Jun 4 '14 at 10:55
$begingroup$
Ivan Loh has posted the full solution in the comments. The final solutions are the functions $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$ and $f(x)=0$.
$endgroup$
– user26486
Jun 4 '14 at 11:17
$begingroup$
Ivan Loh has posted the full solution in the comments. The final solutions are the functions $f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$ and $f(x)=0$.
$endgroup$
– user26486
Jun 4 '14 at 11:17
$begingroup$
One of the main questions that were highlighted in brown was "If so, how can I check whether all the generated functions satisfy the initial equation?"
$endgroup$
– user26486
Jun 4 '14 at 12:40
$begingroup$
One of the main questions that were highlighted in brown was "If so, how can I check whether all the generated functions satisfy the initial equation?"
$endgroup$
– user26486
Jun 4 '14 at 12:40
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%2f819565%2ffind-all-f-mathbb-r-to-mathbb-r-such-that-forall-x-y-in-mathbb-r-the-given%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
$begingroup$
Note that if you plug $y=0$ into your original equation you have $xf(0) = xf(x)f(0) forall x$, so if $f(0)neq 0$ then your solutions are much more tightly constrained.
$endgroup$
– Steven Stadnicki
Jun 3 '14 at 18:22
$begingroup$
Yes, you are right. But, you haven't proved that you already covered all functions.
$endgroup$
– Jlamprong
Jun 3 '14 at 18:22
$begingroup$
@Jlamprong I'm not sure what you mean. I have proved that whatever the solutions the initial equation has, they can be generated using the formula I've given.
$endgroup$
– user26486
Jun 3 '14 at 18:25
$begingroup$
@StevenStadnicki It follows that the solutions can be generated by $$f(x)=begin{cases}c & cinmathbb R & x=0\ 1 & xneq 0end{cases}$$ or it can be any function satisfying $f(0)=0$, if my reasoning is correct.
$endgroup$
– user26486
Jun 3 '14 at 18:29
$begingroup$
@StevenStadnicki I've edited the question.
$endgroup$
– user26486
Jun 3 '14 at 18:38