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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4
votes
1answer
221 views

Cauchy's functional equation - a generalisation? (do additive maps have to be continuous?)

If a map $f : \mathbb{R} \to \mathbb{R}$ is additive, in the sense that $f(x + y) = f(x) + f(y)$, then it is simple to show that $f$ is $\mathbb{Q}$-linear, buy it does not need to be ...
0
votes
2answers
37 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 ...
2
votes
1answer
89 views

How to solve the functional equation : $T(n)=(\log n)T(\log n)+n$

I want to solve the following functional equation using any ways: $$T(n)=(\log n)T(\log n)+n$$
3
votes
0answers
57 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 + ...
0
votes
2answers
98 views

Solving the functional equation $f(x + y) + g(x-y) = \lambda g(x) f(y)$

Let $\lambda$ be a nonzero real constant. Find all functions $f,g : \mathbb R \rightarrow \mathbb R$ that satisfy the functional equation for all $x,y \in\Bbb R$: $$f(x + y) + g(x-y) = \lambda ...
2
votes
1answer
27 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 $y$, ...
8
votes
3answers
264 views

Solve this functional equation:

Functional equations such as this one appear only once every several years on exams, so I feel it's hard to have a sure-fire way to approach the problem, unlike, say, solving a series convergence ...
6
votes
1answer
362 views

Find all continuous functions $f$ : $\mathbb{R} \rightarrow \mathbb{R}$ satisfying: $f(xy)+f(x+y)=f(xy+x)+f(y)\quad\forall x,y \in \mathbb{R}$

Find all continuous functions $f$ : $\mathbb{R} \rightarrow \mathbb{R}$ satisfying: $$f(xy)+f(x+y)=f(xy+x)+f(y)$$ $\forall x,y \in \mathbb{R}$ I have tried that : $P(y;x)-P(x;y)$: ...
1
vote
1answer
124 views

solving a functional equation using given values

Moderator Note: This is a current contest question on Brilliant.org. The current contest ends on 13 October 2013, after which time this question will be unlocked. A Function $f$ from the ...
4
votes
3answers
2k views

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

I want to prove the following claim: If $f:(0,\infty)\to\mathbb{R}$ satisfying $f(xy)=f(x)+f(y)$, and if $f$ differentiable on $x_0=1$, then $f$ differentiable for all $x_0>0$. Thank you.
1
vote
2answers
47 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 ...
3
votes
2answers
396 views

Prove that $f'$ exists for all $x$ in $R$ if $f(x+y)=f(x)f(y)$ and $f'(0)$ exists

A function $f$ is defined in $R$, and $f'(0)$ exist. Let $f(x+y)=f(x)f(y)$ then prove that $f'$ exists for all $x$ in $R$. I think I have to use two fact: $f'(0)$ exists $f(x+y)=f(x)f(y)$ How to ...
2
votes
1answer
77 views

$F(F(x)+x)^k)=(F(x)+x)^2-x$

I have no idea about this problem. But I feel we have to use chain rule of differentiation here. The Function $F(x)$ is defined by the following identity: $F(F(x)+x)^k)=(F(x)+x)^2-x$ The value of ...
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 ...
0
votes
1answer
38 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
2answers
66 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 ...
1
vote
0answers
23 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
31 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 $g$?
4
votes
2answers
230 views

solving functional equation $f(x+y) +f(x)f(y)=f(x)+f(y)+f(xy)$ for all real numbers [duplicate]

The functional equation to be solved is $f(x+y) +f(x)f(y)=f(x)+f(y)+f(xy)$. Domain: Reals, Codomain: Reals.I found about 4 possible solutions to the equation but ran into a fundamental problem with ...
1
vote
2answers
51 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
73 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
212 views

Find all the functions satisfying this criterion

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

Equation with a definite integral - can I differentiate it?

I have an equation like this: $$te^{t} = \int\nolimits_0^t e^\tau u(\tau)d\tau$$ I don't really know how to solve it.. Would it be possible to differentiate both sides of the equation? If so, how ...
2
votes
2answers
109 views

How do I prove this function is monotonic?

Let $f:\mathbb R\to \mathbb R$ be a function such that $f(x+y)=f(x)+f(y)$ and $f(xy)=f(x)f(y)$ for every $x,y\in \mathbb R$ and $f(1)=1$. In order to prove this function is 1-1, I just need to prove ...
1
vote
2answers
63 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
84 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) = ...
3
votes
3answers
137 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
96 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
33 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 ...
1
vote
2answers
66 views

Demonstrating solutions to two functional equations

Find examples (the more the better) of functions $f: \mathbb{Z} \to \mathbb{C}$ satisfying the relations $f(x+y) = f(x) + f(y)$ $f(xy) = f(x)f(y)$ I have only $f(x)= ax$ $f(x)= x^a$ This task ...
0
votes
0answers
25 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: ...
1
vote
3answers
81 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 ...
3
votes
1answer
53 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
0answers
87 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
29 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 $g_1(x),g_2(x),g_3(x)$ ...
0
votes
1answer
100 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
39 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 ...
1
vote
1answer
44 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 ...
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 ...
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
66 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 $. ...
31
votes
7answers
6k views

Find $f(x)$ such that $f(f(x)) = x^2 - 2$

Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$. Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but ...
0
votes
1answer
30 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
90 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
67 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
21 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
20 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 ...
0
votes
2answers
77 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 ...