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 ...

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1answer
67 views

The functional equation $f(f(x)+xf(y))=xf(y+1)$

I'm trying to solve the functional equation $f(f(x)+xf(y))=xf(y+1)$. Up to now I found that $f(f(0))=0$ when $x=0$ and that $f(y+1)=f(f(y)+f(1))$ by setting $x=1$. Also $f(x)=x$ is an apparent ...
-1
votes
4answers
56 views

How can one show that $f(n)>n$

Given a function $f\colon \Bbb N^\ast\to \Bbb N^\ast$ such that we have $f(f(n))=f(n+1)-f(n)~\forall~n\in\Bbb N^\ast$. Show that: If $f(n+1)-f(n)>1$ then $f(n)>n$. Since $f(n+1)>f(n)+1$ I ...
7
votes
5answers
147 views

Find all functions $f$ such that $f(x-f(y)) = f(f(x)) - f(y) - 1$

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $f(x-f(y)) = f(f(x)) - f(y) - 1$. So far, I've managed to prove that if $f$ is linear, then either $f(x) = x + 1$ or $f(x) = -1$ must be ...
3
votes
2answers
139 views

Functional equation $f(xy)=f(x)+f(y)$ and continuity

Prove that if $f:(0,\infty)→\mathbb{R}$ satisfying $f(xy)=f(x)+f(y)$, and if $f$ is continuous at $x=1$, then $f$ is continuous for $x>0$. I let $x=1$ and I find that $f(x)=f(x)+f(1)$ which ...
2
votes
1answer
36 views

discontinuous solutions of functional equation

This is a followup to this question. It's well known that Cauchy's functional equation, $$f(x+y) = f(x) + f(y),$$ has discontinuous solutions. In fact, any discontinuous solution is discontinuous ...
0
votes
2answers
53 views

Can you solve the following functional equation?

I found the following functional equation: Find all functions $f : \Bbb R \rightarrow \Bbb R $ such that: $xf(x) - yf(y) = (x - y)f(x + y) $ for all $x, y \in \mathbb R $ Could you please help me? I ...
2
votes
1answer
47 views

Functional Equation similar to Cauchy's

Find all functions $f:\mathbb{Q}\rightarrow \mathbb{Q}$ such that for any $x,y\in{}\mathbb{Q}$ we have $$\{f(x)\}+\{f(y)\}=\{f(x+y)\}.$$ Note that $\{t\}$ denotes the fractional part of $t$ for ...
2
votes
3answers
101 views

Functional equation $f\left(\frac{1}{x}\right)+(x+1)f(x)=1$

Find all functions $f$ such that $f\left(\frac{1}{x}\right)+(x+1)f(x)=1,\space x\neq0$.
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2answers
40 views

Does induction find all solutions?

Induction shows that an equality holds for all values of $n$. It doesn't show that this is the only equality or formula for the expression that may hold true, correct? For example, say a question asks ...
1
vote
0answers
58 views

Is it possible to compute $\Gamma(\rho_k)$ or $\zeta(3+2\omega_k)$, when $\rho_k=\frac{1}{2}+i\omega_k$ is a nontrivial zero of Riemann zeta function?

I believe that I can to use the equation (1) (from reference [1]), a well known equation involving the Gamma function, Riemann zeta function and the theta function $\psi(x)=\sum_{n=1}^\infty ...
3
votes
0answers
59 views

Functional Equation $f(m + f(n)) = f(m) - n$

Well, similar questions have already been asked. But they are not identical and the solution methods offered there are not the same one as here. Anyway, I want to solve functional equation $f(m + ...
2
votes
1answer
29 views

Functional equation $f(x+y)-f(x)-f(y)=\alpha(f(xy)-f(x)f(y))$ is solvable without regularity conditions

I was reviewing this question and got motivated to solve this general problem: Find all functions $f:\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$, ...
2
votes
2answers
74 views

Differentiability of $f(x+y) = f(x)f(y)$ [duplicate]

Let $f$: $\mathbb R$ $\to$ $\mathbb R$ be a function such that $f(x+y)$ = $f(x)f(y)$ for all $x,y$ $\in$ $\mathbb R$. Suppose that $f'(0)$ exists. Prove that $f$ is a differentiable function. This is ...
2
votes
1answer
53 views

Solution of the functional equation $g(x)g(z) = g(x+z)+g(x-z)$

What is the solution for the following functional equation? $g(x)g(z) = g(x+z)+g(x-z)$ The solution given is: $g(z) = 2\cos(z)$. In the derivation of the result (using Taylor expansion), there is ...
4
votes
4answers
60 views

Functions satisfying the functional equation $[1-f(x)f(y)]f(x+y)=f(x)+f(y)$

How to prove that there is no real function defined on $\mathbb{R}$, continuous at $0$ and not always vanishing satisfying the functional equation $$[1-f(x)f(y)]f(x+y)=f(x)+f(y) \tag{E}$$
2
votes
2answers
55 views

Prove that functional equation doesn't have range $\Bbb R.$

Prove that any solution $f: \mathbb{R} \to \mathbb{R}$ of the functional equation $$ f(x + 1)f(x) + f(x + 1) + 1 = 0 $$ cannot have range $\mathbb{R}$. I transformed it into $$ f(x) = \frac ...
1
vote
2answers
70 views

Finding $f(n)=f(f(n-1))+f(f(n+1))$

Determine whether a function exists from the positive integers to the positive integers which satisfies the equation: $$f(n)=f(f(n-1))+f(f(n+1))$$. My guess is that this function does not exist, as ...
0
votes
1answer
44 views

continuous function and functional equation

Let $ g:\mathbb{R} \to \mathbb{R}$ be a continuous function: $g(x) = g(x+1)$. Show that $g$ satisfies the equation: $g(x)=\frac{1}{4} \left(g(\frac{x}{2})+g(\frac{x+1}{2})\right)$ for all $x$. Is it ...
1
vote
0answers
28 views

Brute force a formula based on numbers and the result?

I have thought of a very funny thing. In World of Warcraft, all weapons have some Damage Per Second variable specified. I want to know how they calculate that result, based on the result and some ...
1
vote
2answers
35 views

Does the equation $f(x)+g(y)=x^2+xy+y^2$ have solutions in real functions $f$ and $g$?

Does the equation $$f(x)+g(y)=x^2+xy+y^2 \mbox{ } \forall x,y \in \mathbb{R}$$ have solutions in real functions $f$ and $g$?
1
vote
2answers
55 views

if $2f\left(\frac{x}{x^2+x+1}\right)=\frac{x^2}{x^4+x^2+1}$ then what is $f(x)$?

assume that: $$2f\left(\frac{x}{x^2+x+1}\right)=\frac{x^2}{x^4+x^2+1}$$ Then what is $f(x)$?
3
votes
4answers
80 views

How to solve the functional equation $f(x+a)=f(x)+a$

Are there any other solutions of the functional equation $f(x+a)=f(x)+a$ ($a=\mathrm{const}$, $a\in\mathbb{R}\setminus\left\{0\right\}$) apart from $f(x)=x+C$ ($C=\mathrm{const}$)? Edit: $a$ is a ...
2
votes
2answers
216 views

Find all the functions satisfying this criterion

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$\left|f(x)-f(y)\right|=2\left|x-y\right|$$
1
vote
2answers
69 views

Find all functions such that $f: R \longrightarrow R, \forall x \in R, f(x)f(x^2-1)=\sin(x) $.

Find all functions such that $$f: \mathbb{R} \longrightarrow \mathbb{R} \\ f(x)f(x^2-1)=\sin(x), \quad\forall x \in \mathbb{R}$$ That is a difficult problem for me. Help me please.
4
votes
2answers
95 views

Functional equation $x\space f(x^2) = f(x)$

How can I logically lead to the answer from the following conditions? $$ \left\{ \begin{align} & x \, f(x^2) = f(x) \text{ for all } x > 0, \\ & f(x) \text{ is continuous}, \\ &f(1) = ...
4
votes
3answers
142 views

Find $f$ if $ f(x)+f\left(\frac{1}{1-x}\right)=x $

Find $f$ if $$ f(x)+f\left(\frac{1}{1-x}\right)=x $$ I think, that I have to find x that $f(x) = f\left(\frac{1}{1-x}\right)$ I've tried to put x which make $x = \frac{1}{1 - x}$, but this equation ...
7
votes
1answer
105 views

Prove $f(x)=0$ when $f(2x^2-1)= f(x)\cdot 2x$

Prove that $f(x)=0$ for $$x\in[-1, 1]$$ $f-continuous$ and for all $x$: $$f(2x^2-1)= f(x)\cdot 2x$$ It is simple for integer numbers. Another fact that I've noticed that $$f(2(-x)^2-1)= f(-x)\cdot ...
1
vote
0answers
35 views

Functional equation for $\sum_{n=1}^{\infty} \sinh(cn)^{-s}$?

Does anyone know of any kind of functional equation (or closed form) for $\sum\limits_{n=1}^{\infty}\sinh(cn)^{-s}$, where $c$ is an arbitrary constant? I've been messing around with it off and on for ...
0
votes
0answers
28 views

Time delay equation

If $x(t)=\displaystyle\frac{1+(x(t-T))^m}{(x(t-T)))^m}$ for all $t$ where $T$ is constant and $x(t)=x_s$ is the solution to the above equation, why can I write that: ...
3
votes
1answer
55 views

Given $f(x+y)=f(x)+f(y)$ and $f(0)=0$, what can we say about f(x)

Given $f(x+y)=f(x)+f(y)$ and $f(0)=0$, what can we deduce about $f(x)$? I intend to say that $f(x)=x$, but find difficult to prove it. Is my guess correct, or wrong?
1
vote
3answers
85 views

Find all function satisfying $f(f(n))+f(n)=2n+3k$

Find all functions $f:\mathbb{N_{0}} \rightarrow \mathbb{N_{0}}$ satisfying the equation $f(f(n))+f(n)=2n+3k,$ for all & $n\in \mathbb{N_{0}}$, where $k$ is a fixed natural number. A friend of ...
0
votes
1answer
37 views

Integral functional equation.

$f(x)= \Big(\int _1^x g_1(t)g_2(t)dt\Big)\Big(\int _1^x g_3(t)g_4(t)dt\Big)-\Big(\int _1^x g_1(t)g_3(t)dt\Big)\Big(\int _1^x g_2(t)g_4(t)dt\Big) $ $\forall$ $x$ $\in R$ Where $g_1(x),g_2(x),g_3(x)$ ...
1
vote
0answers
89 views

Equidistant sets and extremal points

Let $\lVert\cdot \rVert$ be any norm on $\mathbb{R}^n$ and let $x\in \mathbb{R}^n$ be a point. Let $\textrm{ext}(B_{\lVert x\rVert})$ denote the set of all extremal points of $B_{\lVert x ...
0
votes
1answer
106 views

Is $f(x)=e^x$ the only solution to $f(f'(x))=f'(f(x))$? [duplicate]

Is $f(x)=e^x$ the only solution to $f(f'(x))=f'(f(x))$? In particular I'm interested in the qualitative properties of the such solutions.
1
vote
1answer
40 views

How to prove $X(t)$ is differentiable?

Suppose $X(t)\in M_n(\Bbb R), t\in\Bbb R$ and is continuous, invertible at every point on the real line, if the equation $$X(t)X(s)=X(t+s)$$ holds for all $t,s\in\Bbb R$, prove that there exists a ...
2
votes
0answers
25 views

Show that retarded argument functional operator is surjective

I want to show that for every $\lambda \neq 0$ and $g\in L^{2}(\mathbb{R})$, there is a $f \in L^{2}(\mathbb{R})$ such that $$e^{-|x+1|}f(x) - \lambda f(x+1) = g(x)$$ I tried looking at the operator ...
2
votes
1answer
47 views

Find all real polynomials $P(x)$ which satisfy the equation$ P(x)P(-x)=P(x^2-1)$

Find all real polynomials $P(x)$ having only real zeros and which satisfy the equation $$P(x)P(-x)=P(x^2-1)$$ Please explain me the process and refer some books to learn polynomials. Thanks ...
0
votes
0answers
19 views

How to convert $\ max \int_{-\infty}^{\infty} f(x) dx =\ min \int_{-\infty}^{\infty} g(x) dx $

Let $f$ or $g$ have a given condition. ( example $\int_{- \infty}^{\infty} f(x) exp(-x^2) dx = 1$ Or $g(g(x)) = x^3$ Or a differential equation for one of $f,g$. ) I want to find a general way - if ...
1
vote
0answers
69 views

Minimizing $\int_{0}^{1} (1+x^2)f(x)^2 dx$ for $f(f(x)) = x^2$

What is $$\min_{f\in D} \int_{0}^{1} (1+x^2)f(x)^2\mathrm dx,$$ where $D$ is the collection of all continuous real functions from $[0,1]$ such that in the interval $[0,1]$ we have $f(f(x)) = x^2 $. ...
0
votes
1answer
31 views

Functional equaliton on continuous function $f:[0,1]\to \mathbb{R}$

Let $f:[0,1]\to \mathbb{R}$ be continuous, and suppose that $f(0)=f(1).$ Show that there is a value $x\in [0, 1998/1999]$ satisfying $f(x)=f(x+1/1999)$.
6
votes
1answer
91 views

How can I find $f(3)$ if $f(2x)=f^2(x)-2f(x)-\frac{1}{2}$?

If $f(2x)=f^2(x)-2f(x)-\frac{1}{2}$ and $f(1) = 2$ then find $f(3)$. Can you give me any hint that I can start with?
1
vote
2answers
70 views

Express $y$ from $\ln(x)+3\ln(y) = y$

i am finding a inverse function to $$y=\frac{e^\frac{x^3}{3}}{x^3}$$ First step: $$x=\frac{e^\frac{y^3}{3}}{y^3}$$ Next: ... $$(xy^3)^3 = e^{{y}^3}$$ and now in this step i have no idea how can i ...
0
votes
0answers
22 views

Existence of an inverse function in this functional equation

Let $V,W$ each be connected and separable sets. Suppose we have continuous functions $F:\mathbb{R}\times V\rightarrow \mathbb{R}$, $G:\mathbb{R}\times V\rightarrow \mathbb{R}$, $f:V\times W\rightarrow ...
2
votes
1answer
25 views

Functional inequality $f(y)+xf(x)≤yf(x)+f(f(x))$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function sucht that: $$ f(y)+xf(x)≤yf(x)+f(f(x)) $$ for all $x,y\in\mathbb{R}$. Show that $$ f(x)+yf(x+y)≤0 $$ for all $x,y\in\mathbb{R}$. I tried some ...
2
votes
1answer
27 views

Cauchy's functional equation for involutions

It is well known that Cauchy's functional equation for $f:\mathbb{R}\to\mathbb{R}$ $$ f(x+y)=f(x)+f(y)\space\forall x,y\in\mathbb{R} $$ admits highly pathological solutions if no further conditions ...
2
votes
1answer
45 views

Functional equation $f(x^3) - f(x^3 - 2) = (f(x))^2 + 12$

$$f(x^3) - f(x^3 - 2) = (f(x))^2 + 12$$ Given the functional equation above, I am trying to find the value of $f(3)$. I do not remember the exact statement of the problem precisely, so I am not sure ...
1
vote
1answer
41 views

Solve the functional equation (medium-hard)

Find all real functions $f(x)$ $\mathbb{R} \to \mathbb{R}$ such that $x^2f(yf(x)) = y^2f(x)f(f(x))$ Obviously, let $y=0$ you instantly get, $f(0) = 0$. Also, a relation is: $f(yf(x)) = ...
0
votes
2answers
78 views

How can one show that$f(x^n)=nf(x)$

How can one show that if $f(xy)=f(x)+f(y)$ holds for all for real $x$ and $y$ that $$f(x^n)=nf(x).$$ How can i prove that $f(\frac{1}{x})=-f(x)$ To calculate f(1) do i need to pute x=1 ? Do i need ...
1
vote
1answer
18 views

Determine formula to calculate inverse range between two numbers

I've got a measurement for a graphics processing filter that increases intensity as the value decreases. For whatever reason, this filter's default value (minimum) is 1.75, and I've set the maximum ...
0
votes
0answers
41 views

Uniqueness question from functional equation

Let $f(x)$ and $f_2(x)$ be real and continuous on the interval $[0,\infty[$. Let $f(1) = f_2(1) = 1 , f(2) = f_2(2) = e$. Let $g(x)-g(x-1) = f(x-1)$. Let $ f ' (x) = exp(g(x) - g(1)).$ and $f '' ...