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).

1
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0answers
55 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
60 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
88 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
76 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
49 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
56 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
152 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 ...
0
votes
0answers
63 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 ...
0
votes
1answer
74 views

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

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
397 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
51 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
173 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
235 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
83 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
340 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
859 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
76 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
797 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
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1answer
65 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
149 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
135 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 ...
3
votes
5answers
4k views

A difficult problem on functions [closed]

I've been trying to solve the following problem but can't wrap my head around it. Let $x$, $f(x)$, $a$, $b$ be positive integers. Furthermore, if $a > b$, then $f(a) > f(b)$. Now, if $f(f(x)) = ...
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.
0
votes
0answers
48 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
422 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
27 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 ...
3
votes
1answer
46 views

Restricted Cauchy equation on a non-dense domain

It is really well known that if $f: \mathbf{R}\to \mathbf{R}$ is continuous and $$ \forall x,y \in \mathbf{R},\,\,\,\,f(x+y)=f(x)+f(y) $$ then $f$ is linear, i.e., there exists $a \in \mathbf{R}$ such ...
2
votes
1answer
64 views

How many functions can be defined from natural number to natural number such that LCM(f(n),n) -HCF(f(n),n) is less than 5.

How many functions $f:\mathbb{N}\to\mathbb{N}$ can be defined from natural number to natural number such that $$\text{LCM}(f(n),n) -\text{HCF}(f(n),n) <5$$ LCM is least common multiple HCF is ...
0
votes
0answers
59 views

If $f: N\rightarrow N $ be such that $f(f(n))+f(n+1)=n+2$ for all $n\in N$ find $f(1)$ and $f(2)$. [duplicate]

Putting $n=1$, we get $f(f(1))+f(2)=3$. Thus there are two possibilities. Either $f(f(1))=1, f(2)=2$ or $f(f(1))=2, f(2)=1$. Also, we observe that $f(f(n))=n+2-f(n+1)\leq n+1$ and $f(n+1)=n+2-f(f(n))\...
1
vote
2answers
78 views

How do I find a function $f$ such that $f(x^2)=2f(x)$? [duplicate]

Does there exist a continuous function $f$ such that $f(x^2)=2f(x)$ and $f(0)=0$? How do you solve this? I understand that this is nothing like a normal equation, because you can't solve for $x$, or $...
1
vote
0answers
97 views

What is a function $f$ such that $f(f(x))=2f(x)$ and $f(0)=1$?

Is there a function $f$ such that $f(f(x))=2f(x)$ and $f(0)=1$? I don't know how to attack this problem. How do I solve an equation for a function?
2
votes
2answers
34 views

$f(x,y) - f(x,z) = g(y,z)$ implies $f(x,y) = a(x) + b(y)$

A result I need is: If $f(x,y) - f(x,z) = g(y,z)$ for all $(x,y,z)$, then $f(x,y) = a(x) + b(y)$ for some functions $(a,b)$. This seems almost obvious, and I've constructed a proof, but that proof ...
0
votes
1answer
180 views

real analysis -functional equation

Be $f:\mathbb R^ +\mapsto\mathbb R$ a function that satisfies the following conditions: a)$ f(f(f(x)))+2x=f(3x)$ for every $x\gt 0$; b) $\lim_{x \to \infty} (f(x)-x)=0$. This was proposed by ...
2
votes
1answer
56 views

Find all functions $f:(0,\infty)\rightarrow(0,\infty)$ subject to the conditions: [duplicate]

Find all functions $f:(0,\infty)\rightarrow(0,\infty)$ subject to the conditions: $f(f(f(x)))+2x=f(3x)$ for all $x>0$ and $\displaystyle\lim_{x\to\infty}(f(x)-x)=0$ I tried as follows: Suppose $...
10
votes
1answer
338 views

Which positive continuous functions satisfy $F(x) = F(e^x)-F(e^{-x})$ for $x\geq 0$?

There is at least one such function. It is the cdf of the equilibrium probability distribution for the chaotic sequence $x(n+1) = |\log x(n)|$ with $x(1) = 2$. Its graph (approximation) is pictured ...
4
votes
3answers
232 views

Complicated rational number functional equation

Let $\mathbb{Q}^+$ denote the set of positive rational numbers. Let $f : \mathbb{Q}^+ \to \mathbb{Q}^+$ be a function such that $f \left( x + \frac{y}{x} \right) = f(x) + \frac{f(y)}{f(x)} + 2y$ for ...
2
votes
1answer
67 views

Find all function $f$ : $f(x+y)=e^{2xy}f(x)f(y)$

Problem: Suppose $f\colon\mathbb{R}\to\mathbb{R^+}$ is differentiable function which satisfies $f'(0)=1$ and $$\forall x, y \in \mathbb{R}, \quad f(x+y)=e^{2xy}f(x)f(y)$$ Where $\mathbb{R^+}$ is set ...
6
votes
1answer
100 views

Differential Equation with inverse function $\frac{1-f^{-1}\left(\frac{f(x)}{x}\right)}{1-x} = 1- \frac{f(x)}{xf'(x)}$

$$\frac{1-f^{-1}\left(\frac{f(x)}{x}\right)}{1-x} = 1- \frac{f(x)}{xf'(x)}$$ I know $f(x) = ax+b$ is a solution. How can I find other solutions?
17
votes
2answers
630 views

Solve the functional equation $f(xf(y)+yf(x))=yf(x)+xf(y)$

Let $f:\mathbb{R}\to \mathbb{R}$ and such for any real numbers $x,y$ we have $$f(xf(y)+yf(x))=yf(x)+xf(y)$$ Find $f(x)$. I have let $x=y=0$ have $$f(0)=2f(0)\Longrightarrow f(0)=0$$ and I guess the ...
8
votes
3answers
2k views

Find all functions $f$ such that $f(x)+f(\frac{1}{1-x})=x$

I would like to find all functions $f:\mathbb{R}\backslash\{0,1\}\rightarrow\mathbb{R}$ such that $$f(x)+f\left( \frac{1}{1-x}\right)=x.$$ I do not know how to solve the problem. Can someone ...
2
votes
1answer
423 views

solve functional equation: $[f(x)]^2-[f(y)]^2$=$f(x+y)f(x-y)$

i am trying to solve following problems and please guys help me suppose that,there is given following equation $[f(x)]^2-[f(y)]^2$=$f(x+y) \cdot f(x-y)$ there was said that,it requires some knowledge ...
2
votes
4answers
308 views

$2f(x)=f(y) \Rightarrow 2f(tx)=f(ty)$

Find all continuous and strictly monotonic function $f:[0,\infty)\to \Bbb R$ such that: If there is a pair $(x,y)\neq (0,0)$ such that $2f(x)=f(y)$ then $2f(tx)=f(ty)$ for all $t>0$; There is at ...
0
votes
3answers
46 views

Functional equation with three variables

I have a functional equation with three variables. $f(x,y,z)$ is a real function with three variables where y is different from z i.e., $f(x,y,z)$ defined only for $y \neq z$. This function satisfies ...
2
votes
2answers
71 views

Proof verification: system of functional equation

A problem in Putnam Competition 1992(?). The question asked: Prove that, the only solution of the system of functional equation with respect to $f:\mathbb Z\to\mathbb Z$:$$ \begin{cases} f(f(n))=n\\...
0
votes
0answers
32 views

Only zero in kernel of every element of a subspace of the dual does not imply density of said subspace for non-reflexive Banach spaces

Let X be a Banach space and $M \subset X'$ a subspace of its dual space. If X is reflexive we know the following statements are equivalent: (i) $M$ is dense in $X'$ (ii) $x \in X$ and $\phi(x)=0$ ...
5
votes
1answer
171 views

Another functional equation: $f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor$

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
0
votes
1answer
60 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
78 views

Find the sum of all values of $f(2017)$ given $f^{f(a)}(b) f^{f(b)} (a) = [f(a+b)]^2$.

Let $f:\mathbb N\rightarrow \mathbb N$ be an injective function such that $$f^{f(a)}(b) f^{f(b)} (a) = [f(a+b)]^2$$ for all $a,b \in \mathbb N$. Let $S$ be the sum of all possible values of $f(2017)$...
2
votes
1answer
64 views

Find all functions such that $f(1+xf(y))=yf(x+y)$ where $x,y \in R^+$

Find all functions run over positive real numbers such that $f(1+xf(y))=yf(x+y)$ where $x,y\in R^+$ MY ANSWER: Putting $x=y=0$,we get, $f(1)=0$ Putting $x=0$ we get, $f(1)=yf(y)$ or,$yf(y)=0$ ...