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Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (...

18
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0answers
610 views

Strange functional equation: $f(x)+f(\cos(x))=x$

BACKGROUND: A while ago, I became obsessed for a period of time with the following functional equation: $$f(x)+f(\cos(x))=x$$ I am only considering the unique real analytic solution to this functional ...
-1
votes
1answer
19 views

Describing conformal maps in terms of a complex functional equation.

Conformal maps have the interesting property that they map circles and points to points and circle. I wonder if the only holomorphic functions with this property are conformal maps. But I realized I ...
4
votes
1answer
83 views

Are functions satisfying a certain inequality monotone

Let $f: (0, \infty) \rightarrow (0,\infty)$ be a continuous function which satisfies the inequality $$f(x) + f(y) \geq 2f(x+y).$$ Is $f$ necessarily monotone?
2
votes
1answer
58 views

A functional equation in a NO exam

Find all functions $f:\Bbb {R} \rightarrow \Bbb {R} $ such that: $f(f(xy-x))+f(x+y)=yf(x)+f(y).$ I recently proved that $f(x+1)=f(x)+f(1).$
0
votes
1answer
28 views

$\max\{x,y\}$ unique solution to functional equation?

I know that the function $f:\mathbb R^2\mapsto \mathbb R,\,\,f(x,y):=\max\{x,y\}$ satisfies the equation $$f(x,y)+f(-x,-y)=\lvert x-y\rvert.$$ I want to prove/disprove if this is the only continuous ...
1
vote
1answer
42 views

Substitution in functional equation

When solving functional equations it can be helpful to substitute another function, say, g(x) rather than x to the original functional equation h(x). Under what condition is this a permissible ...
12
votes
3answers
158 views

Functional Equation $f(x+y)-f(x-y)=2f'(x)f'(y)$

I am trying to solve the equation $f(x+y)-f(x-y)=2f'(x)f'(y)$ for all $f:\mathbb{R}\to\mathbb{R}$ non-constant, differentiable functions. Here is my progress: Any solution must be an even function ...
4
votes
2answers
131 views

How to find $ f(x)$ if $f(1-f(x))=x$ for all $x$ $\in \mathbb{R}$

How can I determine $ f(x)$ if $f(1-f(x))=x$ for all real $x$? I have already recognized one problem caused from this: it follows that $ f(f(x))=1-x $, which is discontinuous. So how can I construct ...
1
vote
0answers
30 views

Functional equation with three variables!

I have a functional equation that is valid for $0\leq a\leq b \leq c$ of the form $$f(b-a)g(c-a) = g(c)\left[f(b) - g(b)\frac{f(a)}{g(a)}\right],$$ with the additional information that $f(x)g(x) > ...
4
votes
3answers
42 views

Proof verification of an exercise involving a functional equation

Let $f: \mathbb{N} \rightarrow \mathbb{R}$ be a function and $a \in \mathbb{R}$ such that $$f(m+n) = f(m) + f(n) + a$$ $$f(2) = 10, f(20) = 118$$ Find $a$ and $f$. I found this exercise at the ...
4
votes
1answer
126 views

Find $f(x)$ satisfying $f(f(x))=x^x$

By inspection my attempts are always wrong. I really have no idea and given up. How to find $f(x)$ satisfying $f(f(x))=x^x$? My attempts: $f(x)=x^x$ $f(x)=x^{1/x}$ $f(x)=\frac{1}{x^x}$ My ...
11
votes
2answers
150 views

A function satisfying $f \left ( \frac 1 {f(x)} \right ) = x$ [duplicate]

Background. This question originates from the problem of finding a function $f$ such that its $n$-th iterate is equal to its $n$-th power, which I asked about here. Now I would like to focus on the ...
2
votes
0answers
40 views

Non-continuous additive map without AC(Axiom of choice) [duplicate]

Let $f:\Bbb R\to \Bbb R$ be a function that $f(a+b)=f(a)+f(b)$. It is pretty easy to see that $f(x)=f(1)x$ is a solution, but this solution is not interesting, so I showed that there exists non-...
0
votes
2answers
81 views

The functional equation $f(xy)=f(x)f(y)$ [duplicate]

Let $f(x)$ be a function that satisfies this functional equation, $f(xy)=f(x)f(y)$. With a little bit of intuition and luck one may come to a conclusion that these are perhaps the solutions of $f(x)$,...
2
votes
1answer
79 views

A specific question about functions

Let $\Bbb R$ be the set of real numbers. Determine all functions $f : \Bbb R \to\Bbb R$ such that, for all real numbers x and y,$$f(f(x)f(y))+f(x+y)=f(xy)$$My attempt: let's first find some partial ...
37
votes
2answers
580 views

Which functions satisfy $f^n(x) = f(x)^n$ for some $n \ge 2$?

Let $n$ be an integer greater than $1$. The notation $f^n$ is notoriously ambiguous: it means either the $n$-th iterate of $f$ or its $n$-th power. I was wondering when the two interpretations are in ...
1
vote
1answer
53 views

Is the Schröder Equation valid for higher dimensional iterated maps?

The Schröder's functional equation is the eigenfunction equation for the composition operator given as: $$ \psi \circ y (x) = s \cdot \psi(x) ~~~~~~~~~~~ (1) $$ The interesting bit about it (at ...
1
vote
4answers
75 views

Find the natural number $a$ for which $\sum^n_{k=1}f(a+k)=16(2^n-1)$, given that $f(x+y)=f(x)\,f(y)$. [closed]

Find the natural number $a$ for which $$\sum^n_{k=1}f(a+k)=16(2^n-1)\,,$$ where the function $f:\mathbb{N}\to\mathbb{N}$ satisfies the relation $$f(x+y)=f(x)f(y)\text{ for all }x,y\in \mathbb{N}...
0
votes
1answer
35 views

How to find the conditional distribution of an estimator given a prior

The problem: Given a known quantity $x$, distributed with known distribution $π(x)$ ~ $N(0,σ^2)$, I'm looking for the distribution of the estimator of $x$, $\hat{x}$ distributed with $p(\hat{x}\mid x)...
4
votes
2answers
110 views

Is there a function such as the derivative of its inverse is the inverse of its derivative?

A differentiable bijection $f$ from $I$ (real interval different from an empty one and a point) to $J$. $f'$ is the derivative of $f$ and a bijection from $I'$ to $J'$. $g$ is the inverse of $f$ ...
11
votes
0answers
147 views

Find a function such that $f^{-1}=f'$

Let $f:\Bbb{R}^+\rightarrow\Bbb{R}^+$ be a differentiable bijection and let $f$ satisfy: $f'=f^{-1}$ (where $f^{-1}$ denotes the inverse of $f$). Find $f$. This comes from a facebook page "...
2
votes
2answers
95 views

Equation of function s.t., $f(x^2)+2f(x)=0$

Are there function $f:\mathbb{R} \to \mathbb{R}$ s.t., $f\not =0$ and $f(x^2)+2f(x)=0$? I tried polynomial. But it is impossible because of its degree. I tried logarithm, but I couldn't.
2
votes
2answers
77 views

Functional Equation $f\Big(y\,\big(f(x)\big)^2\Big)=x^3\,f(xy)$

Let $f:\mathbb{Q}^+\to \mathbb{Q}^+$ satisfies the following equation $$ f\Big(y\,\big(f(x)\big)^2\Big)=x^3\,f(xy)$$ for all positive rationals $x,y$. Show that $f(x)=\dfrac1x$ for all $x\in\...
1
vote
1answer
47 views

Solutions to functional equational of Riemann zeta function

Is it possible to determine all the meromorphic functions $f : \Bbb C \to \Bbb C$ which have a pole only at $s=1$, of order $1$, and such that $$f(s) = 2^s \pi^{s-1} \sin(\pi s / 2) \Gamma(1-s) f(1-...
-1
votes
1answer
81 views

$f(x)=\max_{0\le y\le 1}\frac{|x-y|}{x+y+1}$ [closed]

Let $f:[0,1]\to\mathbb R$ be defined as $$f(x)=\max\left\{\frac{|x-y|}{x+y+1}: 0\le y\le 1\right\}\text{ for }0\le x\le 1$$ Then which of the following statements is correct? $f$ is strictly ...
2
votes
1answer
39 views

Proving that a function is additive in a functional equation

I have the equation $$h(x,k(y,z))=f(y,g(y,x)+t(y,z)), \tag{*}\label{equation}$$ where $h,k,f,g,t$ are continuous real valued functions. $h,f$ are strictly monotone in their second argument. all ...
12
votes
1answer
253 views

Find $f(f(\cdots f(x)))=p(x)$

$\newcommand{\nest}{\operatorname{nest}}$Let's define a function $\nest(f, x, k)$, which takes a function $f$, an input $x$, and a non-negative integer $k$, and calls $f$ on $x$ repeatedly ($k$ times)....
2
votes
2answers
79 views

Is this theorem true?

If $f(x)+f(y)=f(x+y)$, then: $f(x)=a x$ where $a$ is a constant. Is the above statement true? Is there a way of proving it? The application of this theorem is in the last part of page 52 (second ...
2
votes
2answers
62 views

A question about polynomial function equations

Assume $f(x)$ is a polynomial, and satisfies $$f(x^2)=f^2(x),f(0)=1$$ How to find $f(x)$? My try: Let $g(x)=f(x)-1$, and then I want to show the roots of $g(x)$ is greater than its degree, so that $...
3
votes
1answer
53 views

$f(x^2+yf(x))=xf(x+y)$.Find f [duplicate]

Find all real functions $f:R\mapsto R$ satisfying the relation $$f(x^2+yf(x))=xf(x+y).$$ My Answer: Putting $y=0$ we get, $f(x^2)=xf(x)$. o Which implies $\frac{f(x^2)}{x^2}=\frac{f(x)}{x}$. Let $...
5
votes
1answer
138 views

Functional equation $3f(-3x) -f(x) = 3x^2$

I am trying to solve the following functional equation: $f(x)$ is a continuous function, satisfying (1) $f(f(x))=x$ (2) $ 3f(-3x)-f(x)= 3x^2 $ for $x>0$. From (1) & (2), I found that $...
7
votes
2answers
96 views

What is this property to be called?

Two functions, $f$ and $g$, that satisfy the following identity: $$f(g(a_1,...,a_n), g(b_1,...,b_n),...) = g(f(a_1,b_1,....), f(a_2,b_2,...)...)$$ (notice the "transposition" of the arguments), do ...
0
votes
0answers
66 views

When do there exist two functions that satisfy the equation?

How can we pick two non-zero functions $f$ and $g$ so that for all integer $k\ge 0$ $$\sum_{n=1}^\infty f(n)g(n)^k = \left(\sum_{n=1}^\infty f(n)g(n) \right)^k$$ Assume $f$, $g$, and $k$ allow for ...
1
vote
1answer
25 views

Indicator Function appearing when solving functional equation

Let $I_A(x)=\begin{cases} 0 & x \in A\\1 & \text{else} \end{cases}$ where $A \subseteq \mathbb{R}.$ A particularly hard functional equation boiled down to determining the sets $A$ satisfying ...
0
votes
2answers
20 views

Can we plug a linear sequence of real numbers into the sine function such that the resulting sequence will be a one-to-one sequence?

I was thinking about this problem for some time which can be stated in plain words as follows: Can we plug a linear sequence of real numbers into the sine function such that the resulting ...
4
votes
2answers
104 views

Solutions of $f(x+y^{n})=f(x)+[f(y)]^{n}$.

Consider the functional equation $f(x+y^{n})=f(x)+[f(y)]^{n}$ where $f:\mathbb R \to \mathbb R$ and $n$ is given integer $>1$. This equation was discussed yesterday and it was shown that $f$ is ...
0
votes
0answers
49 views

How to solve the functional equation $f(x^n+y^n)=f(x^n)+f(y^n)$ for any positive integar $n$?

I got some problems when solving the functional equation $f(x+y^n)=f(x)+[f(y)]^n, (x,y\in\mathbf{R}) $ for all positive integar $n$. I tried to solve it as following: =================================...
2
votes
1answer
106 views

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $\varphi(f(x+y))=\varphi(f(x))+\varphi(f(y))\quad\forall x,y\in\mathbb{N}$

I was studying about Cauchy's Functional Equation, and suddenly I had the following question in my mind: Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $$\varphi(f(x+y))=\varphi(f(x))...
0
votes
1answer
44 views

Is it possible to solve a PDE that depends on explicit evaluations of only 1 parameter, and if so, how?

A while back, I asked how to solve a very ugly little duckling of a PDE, and if it could be done. No response. This is a better way repeating the question - if I have a PDE, where inside the equation ...
2
votes
2answers
79 views

$0\leq f,g\in C^{1},\int_{a}^{b}\sqrt{f}\geq\int_{a}^{b}\sqrt{g}\implies\int_a^b f\geq\int_{a}^{b}g$?

$f,g$ are differentiable non-negative functions on $[a,b]$ with $ \int_{a}^{b}\sqrt{f(t)}dt\geq\int_{a}^{b}\sqrt{g(t)}dt $. So do we have that $\int_{a}^{b}f(t)dt\geq\int_{a}^{b}g(t)dt$ ? Does ...
2
votes
2answers
98 views

If $\int^{x}_{0}2x(f(t))^2dt = \bigg(\int^{x}_{0}2f(x-t)dt\bigg)^2$ and $f(1) = 1,$ Then $f(x)$ is

If $\displaystyle \int^{x}_{0}2x(f(t))^2dt = \bigg(\int^{x}_{0}2f(x-t)dt\bigg)^2$ and $f(1) = 1,$ Then $f(x)$ is Try: Using $\displaystyle \int^{a}_{0}f(x)dx = \int^{a}_{0}f(a-x)dx$ We can write ...
2
votes
1answer
71 views

About $ f(x) + c \space f(g(x)) = h(x) $

Let $g(x),h(x), f’(x) $ be functions that can be expressed by radicals , log and exp , but $f(x) $ can not. Now consider functional equations like $$ f(x) + c \space f(g(x)) = h(x) $$ Where $c^2 = ...
0
votes
0answers
66 views

How to solve $ f\left(\sqrt[3]{1 - z^3}\,\right)^2 = 1 - f(z)^2 $?

How to Find analytic $f(z)$ such that $$ f\left(\sqrt[3]{1 - z^3}\,\right)^2 = 1 - f(z)^2 $$ Koenigs function can not be used here So I am stuck. How does the riemann surface look like ?
3
votes
3answers
54 views

Summing combinations with repetition

Given $m,n,k\in\mathbb{N}=\{1,2,...\}$, I wonder if it is possible to find a $F:\mathbb{N}^3\to \mathbb{N}$ such that $$ \binom{m+k-1}{k}+\binom{n+k-1}{k}=\binom{F(m,n,k)+k-1}{k}. $$ EDIT: A more ...
2
votes
0answers
90 views

all functions that uphold $f'(x)=f(1/x)$ [duplicate]

Find all functions $y:\mathbb{R}\to\mathbb{R}$ such that $$y'(x)=y\Big(\frac{1}{x}\Big)$$ we can differentiate both sides to get a homogenous second order Euler equation $$y''(x)=-\frac{1}{x^2}y'\Big(...
1
vote
1answer
84 views

BMO 1999 Q5 Functional Eqn where to start

The question states: Consider all functions $f$ from the positive integers to the positive integers such that (i) for each positive integer $m$, there is a unique positive integer $n$ such that $f(n)...
0
votes
0answers
32 views

Find $f,$ given that $\sum_{i=1}^n f (p_i (x_1,…,x_k))=0 $

Reading through some math Olympiads it appears that there is a whole class of problems of this form: Let $p_1,\ldots,p_n\in\mathbb {R}[x_1,\ldots,x_k] $ with $k,n\in \mathbb {N}^*$ Find all $f:\...
20
votes
2answers
368 views

Solving the infinite radical $\sqrt{6+\sqrt{6+2\sqrt{6+3\sqrt{6+…}}}}$

$$\sqrt{6+\sqrt{6+2\sqrt{6+3\sqrt{6+\cdots}}}}$$ This is a modification on the well-known Ramanujan infinite radical, $\sqrt{1+\sqrt{1+2\sqrt{1+3\sqrt{1+\cdots}}}}$, except it cannot be solved by the ...
1
vote
0answers
30 views

Solving PDE where boundary conditions explicitly expressed in equation

How does one solve $$G_{t}(t,x) = \left(\frac{1-p}{2}(x^{k-1}-x)-\frac{1+p}{2}\right)G_{x}(t,0)+\frac{1-p}{2}(x+x^{k-1})G_{x}(t,1)+\left(\frac{1+p}{2}-x\right)G_{x}(t,x).$$ $G$ here is a generating ...
1
vote
1answer
66 views

Functional equation $f(x)=\frac 1T \sum_{t=1}^Tf(a+tx)$ [closed]

I am dealing with the following functional equation: $$ f(x)=\frac 1T \sum_{t=1}^Tf(a+tx)$$ where $a \in \mathbb{R}$ is a parameter. Is there a way to find a possible $f$ that satisfies the above ...