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 (functions) or (elementary-set-theory).

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3
votes
2answers
128 views

(Olympiad Question) $f(2x) + 2f(y) = f(f(x+y))$

This was Problem 1 from the 2019 International Mathematics Olympiad. Find all functions $f$ : $\mathbb{Z}$ $\rightarrow$ $\mathbb{Z}$ that satisfy $f(2x) + 2f(y) = f(f(x+y))$ whenever $x , y$ $\...
2
votes
0answers
64 views

show this function have this $2f(\sqrt{xy})=f(x)+f(y),\forall x,y>0$ [duplicate]

Let $f:R^{+}\to R$ and such foy any $x,y>0$ have $$f(\frac{x+y}{2})+f(\frac{2xy}{x+y})=f(x)+f(y)$$ show that $$2f(\sqrt{xy})=f(x)+f(y),\forall x,y>0$$ I find this result surprising because the ...
1
vote
1answer
78 views

Are there any smooth/analytic solutions to the functional equation $f(x+1)-f(x)=f\left(\frac 1x\right)$?

Inspired by yesterday's closed question with many upvotes, I studied the functional equation $$f(x+1)-f(x)=f\left(\frac 1x\right)\qquad\forall x\in\mathbb R\setminus\{0\}\qquad f\in C^1$$ I have ...
7
votes
2answers
94 views

Given $a,b,c$ positive numbers, is there a function $h$ such that $h(ax+b) = c \cdot h(x)$ for all $x>0$?

Given $a,b,c$ positive numbers, is there a function $h$ (no trivial) such that $$h(ax+b) = c\, h(x)$$ for all $x>0$ ?. I know that taking $h(x)= c^{\frac{x}{b}}$ makes $h(x+b) = c \, h(x)$, and ...
7
votes
0answers
195 views

functional equation $f(x+1)-f(x)=f(\frac 1x)$ [closed]

I am looking for a differentiable function ((not identically null) on $\mathbb {R}$ verifying the functional equation $$\forall x\in \mathbb {R^*},\quad f(x+1)-f(x)=f\left(\frac 1x\right)$$
2
votes
2answers
47 views

Can someone guide me through this functional equation?

I'm not necessarily sure how to approach this problem—or whether it even has a solution—but I would like to know an example of a non-constant function that satisfies this condition: $$f(x,y,z)f(x,z,y)...
2
votes
2answers
73 views

Functional equation - Cyclic Substitutions

Please help solve the below functional equation for a function $f: \mathbb R \rightarrow \mathbb R$: \begin{align} &f(-x) = -f(x) , \text{ and } f(x+1) = f(x) + 1, \text{ and } f\left(\frac 1x\...
5
votes
1answer
89 views

A functional-differential equation

Consider the functional differential equation $$4f\left(\frac x2-f(x)\right)+4~\epsilon~ f(x)f'\left(\frac x2-f(x)\right)=f(x)$$ for all $x\geq0$ together with the initial condition $f(0)=0$ and the ...
-1
votes
3answers
62 views

Differential Equations Question involving $f(x+y)$ [closed]

let $f:\mathbb R \to \mathbb R$ be a differentiable function with $f(0) = 1$ and satisfying the equation $$f(x+y) = f(x)f '(y) + f '(x)f(y)\qquad \forall ~x,y \in \mathbb R$$ then find the value of $\...
0
votes
1answer
19 views

Functional equation with two inputs.

Some context, I was trying to derive something and stumbled upon f(x,y)f(y,x)=1, when x and y does not equal 0. Could someone suggest some tricks or books that I could apply to these types of ...
4
votes
1answer
114 views

How prove this $f=C$ if $4f(x,y)=f(x-1,y)+f(x,y-1)+f(x+1,y)+f(x,y+1)$

Question: if $f:\mathbb{Z}^2\to \mathbb{R}$ is bounded ,and for any $x,y\in \mathbb{Z}$,we have $$4f(x,y)=f(x-1,y)+f(x,y-1)+f(x+1,y)+f(x,y+1)$$ show that $$f\equiv C$$ where $C$ is constant. My try:...
9
votes
2answers
221 views

Minimum possible value of $f(2007)$ where $f(m f(n)) = n f(m)$, $m,n\in \Bbb N$

If $f$ is from positive integers to positive integers and satisfies $f(m f(n)) = n f(m)$ then find the minimum possible value of $f(2007)$. My work so far: $f(1) = 1$ . Proof: Suppose $f(1) = k \...
3
votes
0answers
175 views

Prove that $f(x)=x$. [duplicate]

Let $f:\mathbb{R}\to\mathbb{R}$ be a function that satisfies $f(-x)=-f(x)$ $f(x+1)=f(x)+1$ $f\left(\frac1x\right)=\frac{f(x)}{x^2}$ for $x\neq0$. Prove that $f(x)=x$. I am even interested in the ...
7
votes
2answers
96 views

What are the continuous solutions to the functional equation $f(x) = \tfrac{1}{2}f(x^2)+\tfrac{1}{2}f(2x-x^2)$?

The solution set is a vector space and includes all functions of the form $f(x)=ax+b$. But apart from these observations I have nothing to say about the problem. If it helps, restrict the domain of $f$...
77
votes
1answer
12k views

Overview of basic facts about Cauchy functional equation

The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of ...
10
votes
7answers
11k views

$f(x)f(1/x)=f(x)+f(1/x)$

Find a function $f(x)$ such that: $$f(x)f(1/x)=f(x)+f(1/x)$$ with $f(4)=65$. I have tried to let $f(x)$ be a general polynomial: $$a_0+a_1x+a_2x^2+\ldots a_nx^n$$ which leaves $f(1/x)$ as: $$a_0+...
0
votes
2answers
52 views

Find all functions by condition

How to find all functions continuous on $R$ by condition: $$f(x)+f(2x)=6x+1$$ If I assume that the function is linear. So $f(3x)=6x+1$. But then how to find the coefficients? Through derivative? ...
0
votes
2answers
51 views

A Computational Functional Equation Problem

How do I approach the following contest problem: Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(x)f(f(x)) = 1$. Given that $f(1000) = 999$, compute $f(500)$.
6
votes
1answer
127 views

Find all functions

Find all functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that $$f(xyz)=f(xy+yz+xz)(f(x)+f(y)+f(z))$$ for all non-zero reals $x, y, z$ such that $...
0
votes
1answer
28 views

Find all functions satisfy an equality

The question: Find all functions $f$ defined over $\mathbb{R}$ satisfying the equality: $\forall x,y \in \mathbb{R}$ $$f(y - f(x)) = f(x^{2002} - y) - 2001y f(x)$$ How do I approach (any hints) to ...
0
votes
2answers
76 views

Find all functions $f$

Please help me to solve this example: Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, so that for every $t\in\mathbb{R}$ apply: $$f(-t)=-f(t),$$ $$f(t+1)=f(t)+1, and$$ $$f(\frac{1}{t})=\...
1
vote
3answers
89 views

Functions satisfying: $f(f(x)^2+f(y))=xf(x)+y$

The problem is to find all the continuous functions $f:\mathbb{R}\to \mathbb{R}$ defined by :$f(f(x)^2+f(y))=xf(x)+y$ I'm trying my best to figure out a way to find the expression of this unknown ...
0
votes
0answers
43 views

Two variable function equations

Is there a good resource for learning two variable functions and where can I find some problems about two variable functional equations? I have already searched AOPS and some other simiar websites but ...
1
vote
0answers
50 views

Prove that there exists a continuous function $f(\!x\!):\mathbb{R}\rightarrow\mathbb{R}$ so that the following functional equation holds.

Problem. Prove that there exists a continuous function $f: \mathbb{R}\rightarrow \mathbb{R}$ so that $$4 f(\!x\!)\!f(\!x+ \frac{\pi}{3}\!)\!f(\!x+ \frac{2 \pi}{3}\!)\!-\!f^{3}(\!x\!)\!-\!f^{3}(\!x+ \...
1
vote
0answers
69 views

Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ so that $f(x)f(y)- \frac{4}{9} xy= f(\!x+ y\!)\,(\!\forall x,\,y\in \mathbb{R}\!)$ .

Problem. Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that $f(x)f(y)- \dfrac{4}{9}\,xy= f(x+ y)\,\,(\!\forall x,\,y\in \mathbb{R}\!)$ (1). My above problem given a solution, and I'm ...
2
votes
3answers
68 views

Functional equation for tan

If $f$ is a differentiable function on $\mathbb{R}$ and $f'(0)=2$ satisfying $$f(x+y) = \frac{f(x)+f(y)}{1-f(x)f(y)},$$ then to prove that $f(x)=\tan 2x$. I know that we must prove using the first ...
0
votes
2answers
103 views

Find all functions for $f:\Bbb{N}\to\Bbb{N}$ such that $f\left(m^2+f(n)\right)=f\left(m^2\right) +n$

I would have given my approach but i didnt get anywhere. I just substituted zeroes and got $f(f(n)) =n$ and I'm just lost. Any help would be appreciated
1
vote
1answer
99 views

$f:\mathbb{R_{\geq 0}} \to \mathbb{R_{\geq 0}}$ such that for all $x$ we have $xf(1+xf(y))=f(f(x)+f(y))$

Find all nonnegative real number $a$, such that $f(a)=0$ for any function $f$ satisfying: $xf(1+xf(y))=f(f(x)+f(y))$ with all $x,y$ are nonnegative real number. I don't know why this problem only ...
0
votes
0answers
55 views

Functions $f$ that $f(f(x))=x$, but $f:S^1\to S^1$

Background Denote $e_A$ the identity map from $A$ to itself. Questions such like solving $f$ in the functional equation $f\circ f=e_\mathbb{R}$ or $f\circ f=e_{\mathbb{R}\setminus\{a_1,\ldots,a_n\}}$ ...
0
votes
1answer
59 views

A problem with functions defined on positive integers.

Where [x] denotes the greatest integer number, which does not exceed x. I need some help please. The proof should also be at high school level. Please don’t use hard or complex things.
6
votes
2answers
162 views

Finding all positive real functions satisfying $xf(y)+f(f(y))\leq f(x+y)$

Find function $f: \mathbb{R}_{> 0}\rightarrow \mathbb{R}_{> 0}$ such that: $xf(y)+f(f(y))\leq f(x+y)$ for all positive $x$ and $y$? That problem made me think a lot. This is the first time I ...
1
vote
0answers
69 views

$f(e^x) = e^{f(x)}$, what is f? [duplicate]

Find all functions $f$ and their domains, such that $f(e^x) = e^{f(x)}$ I have verified that the functions below satisfy the equation for certain domains. Would these be the only solutions? But how ...
-1
votes
1answer
77 views

Number of solutions of the equation $e^{f(x)}=f(x)+2$ [closed]

Let $f$ be an everywhere differentiable function, and suppose that $f(x)=0$ has a unique solution, and suppose that $f$ has no local extreme points. What is the number of solutions of the equation $...
10
votes
3answers
408 views

Prove that there is no function $f:\Bbb{R}\to\Bbb{R}$ with $f(0)>0$ such that $\forall x,y\in\Bbb{R}, f(x+y)\geq f(x)+y f(f(x))$

I got this problem: Prove that there is no function $f:\Bbb{R}\to\Bbb{R}$ with $f(0)>0$ such that $\forall x,y\in\Bbb{R}, f(x+y)\geq f(x)+y f(f(x))$. (Hint: the solution involves limits at ...
0
votes
2answers
59 views

Function of a Function Differential Equation [duplicate]

Is there any function, $f(x)\neq x$, for which $f(f'(x))=f'(f(x))$?
8
votes
0answers
174 views

Is this $f(x) = x+1$ the only solution to this functional equation.

I am considering the problem of finding all functions $f:(0,\infty)\rightarrow(0,\infty)$ satisfying the functional equation: $$f\big(xf(y)+f(x)\big) = 2f(x)+xy.$$ I have been able to prove the ...
9
votes
3answers
238 views

If $f(x)$ satisfies $2f (x) = f(xy) + f(x/y)$, find $f(x)$

If $f(x)$ is a continuous and differentiable function which satisfies the function equation If $$2f (x) = f(xy) + f\left(\frac xy\right)\quad \forall x,y \in \mathbb{R}^{+}$$ and $f'(1)=1$ then find $...
3
votes
1answer
88 views

$f:\mathbb{R} \rightarrow \mathbb{R}$, $f(xf(y)+f(x))=2f(x)+xy$

So far I've only got that $f(x) = x + 1 \qquad\forall x \in\mathbb{R}$ is probably the only solution, and that Substitute (1,y): $f(f(y)+f(1))=y+2f(1) \implies f\text{ surjective}$ $f(x)=f(y) \...
11
votes
2answers
349 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)....
14
votes
1answer
863 views

Equation about generating functions and subfactorial

Suppose $G_n(w)$ is a formal power series (really a probability generating function, see the following explanation) of variable $w$, try to solve out $G_n(w)$ for all $n\ge0$ from the formal-power-...
2
votes
2answers
79 views

find functions $f$ such that $f(x)+f(y) = f(g(x,y))$, $g$ is given and symmetric

I want to find solutions $f$ of the following functional equation given a function $g(x,y)$, which is symmetric ($g(x,y)= g(y,x)$) and strictly monotonic $\forall x,y \in $ Reals: $f(x)+f(y) = f(g(x,...
28
votes
3answers
868 views

$f(ax)=f(x)^2-1$, what is $f$?

Suppose $f(ax)=(f(x))^2-1$ and suppose that $f$ is analytic in some neighborhood of $x=0$. Expanding in power series, we get $a=1+\sqrt{5}$ or $1-\sqrt{5}$. We take positive $a$. If $f\neq{\rm const}$ ...
-1
votes
1answer
66 views

Solve the functional equation $f(x+1)-f(x)=x*\sin(x) $ [closed]

Solve $f(x+1)-f(x)=x*\sin(x) $
5
votes
3answers
159 views

Cauchy's functional equation — additional condition

Consider the function $f:R \to R$ $$f(x+y)=f(x)+f(y)$$ which is known as Cauchy's functional equation. I know that if $f$ is monotonic, continuous at one point, bounded, then the only solutions are $f(...
1
vote
3answers
137 views

If,for all $x$,$f(x)=f(2a)^x$ and $f(x+2)=27f(x)$,then find $a$

I am finding this problem confusing : If,for all $x$,$f(x)=f(2a)^x$ and $f(x+2)=27f(x)$,then find $a$. When $x=1$ I have that $f(1)=f(2a)$ using the first identity. Then when $x=2a$ I have by the ...
1
vote
1answer
41 views

Rational Functional Equations

Suppose $f(x)$ is a rational function such that $3 f \left( \frac{1}{x} \right) + \frac{2f(x)}{x} = x^2$ for all $x \neq 0$. Find $f(-2)$. I tried substituting different values of $x$ to get a system ...
0
votes
1answer
28 views

which is the function of (3) based in this equation

If we know that $(f\circ f)(x)=4x+3$, with $f(0)=4$, what is $f(3)=?$ I have found that $f(f(x))= 16x+15$, but I don't know where to go from there.
1
vote
0answers
49 views

I can compute a function $F(x)$ such that $F(x(1/2-x))= F(x)/2$, It is analytic on a filled Julia set.

I have been studying a function $F(x)$ obeying $F(p(x))=F(x)/2$. I did numerical work for $p(x)=x(1/2-x)$, although a similar functional equation could be solved for any polynomial with an attractive ...
10
votes
3answers
424 views

$f(x^2) = 2f(x)$ and $f(x)$ continuous

I ran into a problem recently where I obtained the following constraint on a function. $$f(x^2) = 2f(x) \,\,\,\, \forall x \geq 0$$ and the function $f(x)$ is continuous. Can we conclude that $f(x)$ ...
1
vote
1answer
30 views

Uniqueness of the solution of non-linear ODE of second order

Let $n$ be an integer with $n>3$ and $f \colon [0,\infty ) \to \mathbb{R}$ be a solution of $t^{1-n}(t^{n-1}f'(t))'=f(t)|f(t)|^{\frac{4}{n-2}}$ with initial values $f(0)=a$ and $f'(0)=0$. Then, is ...