The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...
37
votes
2answers
538 views
Looking for a function such that…
There was this question on one of the whiteboards at my company, and I found it intriguing. Maybe it's a dumb thing to ask. Maybe there is a simple answer that I couldn't see. Anyway, here it is:
...
24
votes
2answers
575 views
$f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$
A function that satisfies both $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$ for all real $x$ is known to be the identity over $\mathbb Q$, but is it also the identity over $\mathbb R$? If not, can you provide ...
18
votes
6answers
694 views
Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
Find all polynomials $P$ such that
$P(x^2+1)=P(x)^2+1$
18
votes
3answers
3k views
Prove that this function is bounded
This is an exercise from Problems from the Book by Andreescu and Dospinescu. When it was posted on AoPS a year ago I spent several hours trying to solve it, but to no avail, so I am hoping someone ...
17
votes
6answers
1k views
$f\colon\mathbb{R}\to\mathbb{R}$ such that $f(x)+f(f(x))=x^2$ for all $x$?
A friend came up with this problem, and we and a few others tried to solve it. It turned out to be really hard, so one of us asked his professor. I came with him, and it took me, him and the ...
17
votes
4answers
540 views
Solving functional equation $f(x)^2+f(y)^2=f(x+y)(f(f(x))+f(y))$
I was trying to find functions $f:(0,+\infty)\to(0,+\infty)$ satisfying the following functional equation
$$
f(x)^2+f(y)^2=f(x+y)(f(f(x))+f(y))
$$
The problem is that I can't find here any reasonable ...
17
votes
2answers
358 views
Proving that $f(n)=n$ if $f(n+1)>f(f(n))$
How can we prove that if $f:\mathbb{N}\rightarrow\mathbb{N}$ is a function so that $f(n+1)>f(f(n))$ for all $n\in\mathbb{N}$ then $f(n)=n$ for all $n\in\mathbb{N}$?
16
votes
4answers
641 views
Given $f(f(x))$ can we find $f(x)$?
Given $f(f(x))=x+2$ does it necessarily follow that $f(x)=x+1$? This question comes from a precalculus algebra student.
14
votes
4answers
483 views
Polynomial: $p(x) = p(x+3)$.
Determine polynomial $p(x)$ s.t. $p(x) = p(x+3)$.
Just by looking at the above equation, it immediately appears that p has got to be some kind of constant function. I thought it might also be a ...
14
votes
4answers
451 views
Solving (quadratic) equations of iterated functions, such as $f(f(x))=f(x)+x$
In this thread, the question was to find a $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(f(x)) = f(x) + x$$
(which was revealed in the comments to be solved by $f(x) = \varphi x$ where $\varphi$ is ...
14
votes
3answers
2k views
The easy(?) part of IMO 2011 Problem 3
Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies
$$f(x + y) \leq yf(x) + f(f(x))$$
for all real numbers $x$ and $y$.
How can I prove that ...
14
votes
2answers
418 views
If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere
Prove that if $f\colon\mathbb{R}\to\mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ for all $x,y$, and $f$ is continuous at $0$, then it is continuous everywhere.
If there exists $c \in \mathbb{R}$ ...
13
votes
6answers
562 views
Solution to $1-f(x) = f(-x)$
Can we find $f(x)$ given that $1-f(x) = f(-x)$ for all real $x$?
I start by rearranging to: $f(-x) + f(x) = 1$. I can find an example such as $f(x) = |x|$ that works for some values of $x$, but not ...
13
votes
3answers
264 views
Solving Differential Functional Equation $f(2x)=2f(x)f'(x)$
Find all functions satisfying $f(2x)=2f'(x)f(x)$
Under given condition, can't we find explicit solutions?
13
votes
3answers
612 views
Continuous function satisfying $f^{k}(x)=f(x^k)$
How does one set out to find all continuous functions $f:\mathbb{R} \to \mathbb{R}$ which satisfy $f^{k}(x)=f(x^k)$ , where $k \in \mathbb{N}$?
Motivation: Is $\sin(n^k) ≠ (\sin n)^k$ in general?
13
votes
1answer
743 views
$f(x+y)+f(x-y)=2[f(x)+f(y)]$
Here is the problem: Find all continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy
$$f(x+y)+f(x-y)=2[f(x)+f(y)]\;\;\;(1).$$
Here is my attempt:
Fix $\delta>0$ and let ...
12
votes
3answers
1k views
Is there a name for function with the exponential property $f(x+y)=f(x) \times f(y)$?
I was wondering if there is a name for a function that satisfies the conditions
$f:\mathbb{R} \to \mathbb{R}$ and $f(x+y)=f(x) \times f(y)$?
Thanks and regards!
12
votes
4answers
584 views
Solving for the implicit function $f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$ and $f(1)=1$
How can I find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(1)=1$ and
$$f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$$
for all real numbers $x$ and $y$ with $y\neq0$?
PS. This is ...
12
votes
4answers
404 views
thoughts about $f(f(x))=e^x$
I was thinking, inspired by mathlinks, precisely from this post, if there exists a continuous real function $f:\mathbb R\to\mathbb R$ such that $$f(f(x))=e^x.$$ However I have not still been able to ...
12
votes
1answer
173 views
Functions $f$ satisfying $ f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R}$.
How to prove that the continuous functions $f$ on $\mathbb{R}$ satisfying
$$f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R},$$
are given by
$$f(x)=x+a,a\in\mathbb{R}.$$
Any hints are welcome. Thanks.
11
votes
2answers
351 views
Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?
If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples?
11
votes
4answers
227 views
If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $f(x)\neq x$ for all $x$, must it be true that $f(f(x))\neq x$ for all $x$?
Let $f: \Bbb R → \Bbb R$ be a continuous function such that $f(x)=x$ has no real solution . Then is it true that $f(f(x))=x$ also has no real solution ?
11
votes
8answers
930 views
How to find the function $f$ given $f(f(x)) = 2x$?
I was wondered how to find the function in this equality:
$f(f(x))=2x$. Also $f$ is continuous.
I don't need the answer, how to find it is more important.
11
votes
4answers
313 views
$f(16x)=16f(x) $ and $ f$ is continuous
$f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function such that $f(16x)=16f(x)$ for every real $x$.
Should it be $f(x)=ax$? How can I prove that?
11
votes
3answers
642 views
Find all functions with $f(x + y) + f(x - y) = 2 f(x) f(y)$ and $\lim\limits_{x\to\infty}f(x)=0$.
Determine all functions $f \colon\mathbb{R}\to\mathbb{R}$ satisfying the following two conditions:
(a) $f(x +y) + f(x - y) = 2 f(x) f(y)$ for all $x, y\in\mathbb{R}$;
(b) ...
11
votes
2answers
175 views
Can every real function be represented as two shifted even functions?
I saw the theorem that every function can be represented as the sum of and even and odd function, and this made me wonder: can every function from the reals to the reals, defined on all the reals, be ...
11
votes
3answers
188 views
What was this theorem called
Back at the university we have proven (lot of work) that if $$S(X)C(Y)+C(X)S(Y) = S(X+Y)$$ and $$C(X)C(Y)-S(X)S(Y) = C(X+Y)$$ then $S(X)$ is $\sin(x)$ and $C(X)$ is $\cos(x)$ (or constant $0$, meh). ...
11
votes
1answer
275 views
Find all the function that satisfy : $f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy$
Find all the function that satisfy :
$$f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy$$
I only find $f(0)=0$ but I can't prove $f(x)=2x$
11
votes
1answer
143 views
How find this function $f(x)$
Let $f:\mathbb{R}\to\mathbb{R}$ be continuous such that $\dfrac{f(x)}{x}=f'\left(\dfrac{x}{2}\right)$. Find $f(x)$.
(2):if $f(x)$ on $x\in[a,b] $ be continuous,find all $f(x)$?
I think this is an ...
11
votes
1answer
168 views
How can prove this equation.
if $a+b=c+d=e+f=\dfrac{\pi}{3}$,
$\dfrac{\sin{a}}{\sin{b}}\cdot\dfrac{\sin{c}}{\sin{d}}\cdot\dfrac{\sin{e}}{\sin{f}}=1$,
Prove that:
...
11
votes
2answers
138 views
About the solution of the infinite recurrence $f(x,f(x,f(x,f(x,f(…))))=a$
On internet I found some recreational problems as $$3=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$$ $$\frac{1}{2}=\frac{1}{x+\frac{1}{x+...}}$$ $$2=x^{x^{x^{...}}}$$ And the trick to solve them was just ...
11
votes
3answers
168 views
Functions satisfying $\sum_{n=0}^k(-1)^n\binom{k}{k-n}f^{k-n}(x)=0$.
This question was motivated by this one. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. Define $f^n(x)=f\circ f\circ\cdot\cdot\cdot\circ f$, $n$ times, $f^0(x)=x$ and $k\geq 2$ a ...
10
votes
4answers
441 views
The functional equation $f(x)+f(y)+f(z)+f(x+y+z)=f(x+y)+f(y+z)+f(z+x)$
How to find the all continuous functions $f\colon \mathbb{R}\to\mathbb{R}$ such that $f(x)+f(y)+f(z)+f(x+y+z)=f(x+y)+f(y+z)+f(z+x)$
10
votes
3answers
275 views
$f(x^2) = xf(x)$ implies that $ f(x) = mx$?
Suppose a function $f : \mathbb{R} \to \mathbb{R} $ satisfies the relation $$f(x^2) = xf(x) \ \ \forall x$$ Does this imply $f$ must be a straight line, $f(x) = mx$? If so, why? If not, are there ...
10
votes
2answers
450 views
Entire functions such that $f(z^{2})=f(z)^{2}$
I'm having trouble solving this one. Could you help me?
Characterize the entire functions such that $f(z^{2})=f(z)^{2}$ for all $z\in \mathbb{C}$.
Hint: Divide in the cases $f(0)=1$ and $f(0)=0$. ...
10
votes
3answers
287 views
Existence of a function
I came across this question
Does there exist a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $f(x+y)>f(x)(1+yf(x))$ and $x,y\in \mathbb{R}^+$
and I didn't know how to begin on it.
10
votes
2answers
251 views
When is $f^{-1}=1/f\,$?
I seem to remember a nice article that I read many years ago (perhaps in the American Mathematical Monthly) which investigated the question, "Under what conditions on $f:\mathbb{C}\to\mathbb{C}$ is ...
10
votes
2answers
321 views
very elementary proof of Maxwell's theorem
Maxwell's theorem (after James Clerk Maxwell) says that if a function $f(x_1,\ldots,x_n)$ of $n$ real variables is a product $f_1(x_1)\cdots f_n(x_n)$ and is rotation-invariant in the sense that the ...
10
votes
2answers
277 views
Solution of functional equation $f(x/f(x)) = 1/f(x)$?
I've been trying to add math rigor to a solution of the functional equation in [1], eq. (22). It is:
$$
f\left(\frac{x}{f(x)}\right) = \frac{1}{f(x)}\,,
$$
where you know that $f(0)=1$ and $f(-x) = ...
10
votes
1answer
368 views
Which trigonometric identities involve trigonometric functions?
Once upon a time, when Wikipedia was only three-and-a-half years old and most people didn't know what it was, the article titled functional equation gave the identity
$$
\sin^2\theta+\cos^2\theta = 1
...
9
votes
3answers
421 views
a continuous function satisfying $f(f(f(x)))=-x$ other than $f(x)=-x$
My question is about existence of a non-trivial solution of the functional equation $f(f(f(x)))=-x$ where $f$ is a continuous function defined on $\mathbb{R}$. Also, what about the general one ...
9
votes
1answer
207 views
Is there a real-valued function $f$ such that $f(f(x)) = -x$?
Is there a function $f\colon \mathbb{R} \to\mathbb{R} $ such that $ f(f(x)) = -x$ ?
9
votes
2answers
66 views
Uniqueness of solution for a functional equation
Let $h \in \mathcal{C}:=C([-a,a])$, where $a>0$. Prove that there exists a unique function $f \in \mathcal{C}$ such that
$$
f(x)=\frac{x}{2}f\Big(\frac{x}{2}\Big)+h(x)\quad \forall x \in [-a,a].
$$
...
8
votes
2answers
211 views
Factoring x + y
I'm trying to find functions $f(x)$ and $g(y)$ such that $$f(x)\cdot g(y) = x + y$$
I can't seem to find a single solution to this problem. Anything I try becomes of the form $f(x,y) \cdot g(y) or ...
8
votes
2answers
2k views
If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t
Let $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$.
I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the same as $g(xy) = g(x) + ...
8
votes
1answer
361 views
Polynomials which satisfy $p^{2}(x)-1 = p(x^{2}+1)$
Can we find a polynomial $p(x) \in \mathbb{R}$ such that $\text{deg}\ p(x)>1$ and which satisfies $$p^{2}(x)-1=p(x^{2}+1)$$ for all $x \in \mathbb{R}$.
This question can be very well identified with ...
8
votes
3answers
456 views
Help remembering a Putnam Problem
I recall that there was a Putnam problem which went something like this:
Find all real functions satisfying
$$f(s^2+f(t)) = t+f(s)^2$$
for all $s,t \in \mathbb{R}$.
There was a cool trick to ...
8
votes
3answers
168 views
Find all the functions which satisfy a given functional equation
I was given by a friend of mine the following problem. Find all the functions $f:\mathbb R\to\mathbb R$ which satisfy the following functional equation $$f(y+f(x))=f(x)f(y)+f(f(x))+f(y)-xy.$$
We ...
8
votes
1answer
55 views
Are there associative bijections $f\colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$?
This is my first question and I hope this question is not too brief to be acceptable:
There are well-known bijections $\mathbb{N \times N \to N}$ and well-known associative compositions on countably ...
8
votes
1answer
274 views
Riemann's thinking on symmetrizing the zeta functional equation
In the translated version of Riemann's classic On the Number of Prime Numbers less than a Given Quantity, he quickly derives the zeta functional equation through contour integration essentially as
...
