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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6
votes
2answers
113 views

Finding a function $h$ that satisfies $h \left ( \frac{x}{x^2+h(x)} \right )=1$

Someone gave me a random maths problem to solve: Given that $h \left ( \dfrac{x}{x^2+h(x)} \right )=1$, what is $h(x)$ The restrictions given were: $h(x) \neq constant$ $\exists \frac{dh}{dx}$ ...
0
votes
1answer
24 views

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$ subject to $\int_\mathbb{R} f(x)\,dx = 1$.

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$, where $\alpha$ is a real number, subject to $\int_\mathbb{R} f(x)\,dx = 1$. I have no idea ...
1
vote
2answers
34 views

A smooth function satisfying these functional constraints

I am looking for any function on a square $$f:[-1,1]\times [-1,1] \rightarrow [0,1]$$ with the following properties: The function $f$ is as smooth as possible, e.g. differentiable almost everywhere. ...
2
votes
1answer
41 views

Substitution with functional equations

I've found this nice introduction worksheet that I started to work through with the goal to get a better understanding of functions and finding them in equations. I've gotten so far but in this one ...
2
votes
2answers
108 views

Is a function $f:\mathbb R\to\mathbb R$ such that $f(x+y)=f(x)+f(y)$ always continuous? [duplicate]

Is there a function $f:\mathbb R\to\mathbb R$ such that $f(x+y)=f(x)+f(y)$ which is not continuous? I have proved that if it's continuous in one point $a\in\mathbb R$ then it's continuous on all ...
4
votes
1answer
63 views

If $h(x)=f(g(f(x)))$ is bijective, what do we know about $f,g$?

Question: If $h(x)=f(g(f(x)))$ as a function $\mathbb R \rightarrow \mathbb R$ is bijective, what do we know about $f,g$, which are also functions $\mathbb R \rightarrow \mathbb R$? Is my proof ...
0
votes
2answers
41 views

Solve the functional equation $ q \, \frac{f(x+1)}{f(x)}=\frac{h(x+1)}{h(x)}. $

Let $f(x),h(x)$ be two differentiate on $\mathbb{R}$ functions, $f(0)=h(0)=1$. Solve the functional equation $$ q \, \frac{f(x+1)}{f(x)}=\frac{h(x+1)}{h(x)}, $$ here $q$ is a constant. For ...
0
votes
1answer
72 views

Find a function $\Phi$ such that $ \Phi(x)^{T}\Phi(y)=\exp(-\|x-y\|^2/(2\sigma^2))$

It's a question from HW: Suppose we have $ \Phi:\mathbb{R}^p \to \mathbb{R}^\infty $ that satisfies: $$ \Phi\left(x\right)^{T}\Phi\left(y\right)=\exp\left(-\frac{\left\Vert x-y\right\Vert ...
2
votes
1answer
60 views

Solving the functional equation $2f(x)-f(1/x)=3x$

If $$2f(x)-f(1/x)=3x$$ how would I find $f(x)$? I have tried various linear and other functions but I do not know how to start this
-1
votes
1answer
54 views

Result related to $f (x+y+z) =f (x)f (y)f (z)$ [closed]

If $f (x+y+z) =f (x)f (y)f (z) $for all real $x,y,z$ and $f(2)=4$ and $f'(0)=3$. Then, how to find $f(0)$ and $f'(2)$?
0
votes
0answers
66 views

How to do this estimate

If $a,b$ are two vectors in $\mathbb R^n$ satisfy the following relation \begin{equation} \frac{|a|^2}{1+(1-|a|^2)^{\frac{1}{2}}}\geq \frac{|a|^2-2\langle ...
4
votes
2answers
66 views

Functional equation on integers

Is there a function $f$ such that $$f(x,y,n)=f(x+y,y-x,n+1)$$ $$f(x,y,n)\neq f(x+1,y,n)$$ $$f(x,y,n)\neq f(x,y+1,n)$$ where $x,y$ are integers, $n$ is a positive integer and the range of $f$ is a ...
6
votes
2answers
166 views

Interesting functional equation: $f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$

Solve for the function f(x): $$f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$$ I'm not able to solve this. [For instance, I tried solving for $f(\frac{x}{x+f(x)})$, but this doesn't lead me ...
0
votes
3answers
63 views

Solving the functional equation $f(a + b) = a + f(b)$

How would you solve: $$f(a+b)=a+f(b) ?$$ It seems similar to the Cauchy equation $$f(a+b)=f(a)+f(b),$$ but I'm not sure what to do with this. I have a feeling the only solution is $f(k)=k$ but idk. ...
2
votes
1answer
90 views

Find all function $f: \mathbb{Q} \to \mathbb{Q}$ such that $f(x+f(x)+2y)=2x+2f(f(y))$ [closed]

Find all function $f: \mathbb{Q} \to \mathbb{Q}$ such that $f(x+f(x)+2y)=2x+2f(f(y))$ for all $x,y \in \mathbb{Q}$.
0
votes
1answer
28 views

Each non-empty interval $I \subset \mathbb{R}$ we have $f(I)=f(\mathbb{R})$ when $f$ is additive

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an additive function such that $f(1)=0$. Then for each non-empty interval $I \subset \mathbb{R}$ we have $f(I)=f(\mathbb{R})$. Since $f$ is additive ...
5
votes
3answers
148 views

Can we obtain $f(y+x)=y+f(x)$ from $f(x^2+f(x)^2+x)=f(x)^2+x^2+f(x)$?

Find all function $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that $$f(m^2+f(n))=f(m)^2+n.$$ Let $P(x,y)$ be the assertion: $f(x^2+f(y))=f(x)^2+y \; \forall x,y \in \mathbb{Z}^+.$ $P(x,x)$ ...
-4
votes
2answers
106 views

Nontrivial entire $f(z)$ never equal to $0$ [closed]

I'm looking for nonconstant entire functions $f(z)$ such that $f(z)\neq 0$ for any $z$. More specifically I'm looking for nontrivial cases. So $\exp(z),\exp(z^2),...$ is not what I am looking for. ...
1
vote
3answers
56 views

$f(f(y)+1)=y+f(1)$ is bijective.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(xf(y)+x)=xy+f(x), \; \forall x,y \in \mathbb{R}.$$ I read a solution in finding this function. It states that setting $x=1$ ...
1
vote
0answers
19 views

Derivative of sum of two functional derivatives with different ranges

I have a functional of the the following form, $(o<a<1)$ : $F(g(x)) = \int_0^a \! g(x) x \, \mathrm{d}x. + \int_a^1 \! (g(x)-k)x^2 \, \mathrm{d}x. $ I want to find $ \frac{\partial ...
-1
votes
1answer
17 views

Fredholm-like equation

I have the following equation: $$\lambda=\displaystyle\int_{a}^b f(x)g(x)dx$$ Where $\lambda$ is a constant and I know the expresión for f(x). Is there any way of extracting the fucntion g(x)? I ...
0
votes
2answers
126 views

If $f(x+y^3)=f(x)+[f(y)]^3$ and $f'(0)\ge0$, what can $f(10)$ be? [closed]

A real valued function satisfies the condition: $f(x+y^3)=f(x)+[f(y)]^3$ for all real $x$, $y$. If $f'(0)\ge0$ how to find $f(10)$ ?
0
votes
0answers
37 views

Transcendental Functional Equations

Given $f^k(x) = f \circ f \circ f\circ ...(x)$ composed $k$ times, Do the functional equations $f^k(x) = g(x)$, where $g(x)$ is a basic transcendental elementary function, for example, the inverse ...
2
votes
4answers
86 views

Find all functions $\mathbb{R}^{+}\rightarrow \mathbb{R}$

Find all functions $f$: $\mathbb{R}^{+}\rightarrow \mathbb{R}$ such that $$f\left ( \frac{x}{y} \right )= f(x)+f(y)-f(x)f(y)$$ for all $x,y\in\mathbb{R}^{+}$. Here, $\mathbb{R}^{+}$, denotes the ...
8
votes
3answers
243 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 ...
0
votes
0answers
15 views

Find a derivative of equation that contains Fourier series

I need to find a derivative of follow equation $$ \left(r_{0} + \sum [a_{i}\cos(i\phi) + b_{i}\sin(i\phi)] \right)({\sin\phi-k\cos\phi}) - b = 0 $$ I know the derivative of $\left(r_{0} + \sum ...
0
votes
1answer
28 views

Representation of a Functional Equation.

Let, $f:\mathbb{R}^{J+1}\to\mathbb{R}$, $g:\mathbb{R}^J\to\mathbb{R}$ and consider the following equation: $$f(x,g(x))=g(x).$$ I would like to know if the following statement is true (or find ...
1
vote
0answers
56 views

Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ for which $f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy))$ [duplicate]

Problem Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy $$f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy))$$ for all $x,y\in\mathbb{R}$. This is a contest math problem, and I have very little ...
2
votes
2answers
62 views

A difficult problem on functions

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)) = ...
0
votes
0answers
18 views

A Somewhat Complicated Functional Equation

Given a function on $\mathbb R^2$ $$C(x,y)=e^x\big(e^y N(d_1)-N(d_2)\big) \tag1$$ where $N$ is the cumulative standard normal distribution function (an increasing function) and ...
4
votes
1answer
84 views

Way to solve basic functional equations

Is there any general way to solve basic functional equations? For example we have algebraic ways to solve algebraic equations (find $x$)! But for functional equations like : $$f(x) + f(x-1) = 0$$ ...
1
vote
1answer
45 views

Polynomials $f(x)$ such that $f(x)f(x-1)+f(x^2)=0$

How can I find all polynomials $f(x)$ such that $f(x)f(x-1)+f(x^2)=0?$ I am self-studying functional equations, but don't know how to start this one. A hint would suffice.
1
vote
4answers
81 views

If $f(f(x)) = x $ has at most 1 solution, then so does $f(x) = x$.

Let f be defined on [0,1] and its values are between 0 and 1. If $f(f(x)) = x $ has at most 1 solution, then $f(x) = x$ has at most 1 solution. Please, give me a hint how to prove this.
1
vote
1answer
49 views

Easily computable function that is one-to-one and onto with 2 or more inputs

I am looking for a function that will take take n inputs and have 1 output, and knowing the 1 output and function you can get back the n inputs, without error and computationally easy in both ...
7
votes
2answers
81 views

Solve the functional equation $ \dfrac{f(x)}{f(y)}=f\left( \dfrac{x-y}{f(y)} \right) $

Solve the functional equation $$ \frac{f(x)}{f(y)}=f\left( \frac{x-y}{f(y)} \right), $$ here $f: \mathbb{R} \to \mathbb{R}$ and $f$ is differentiable at $x=0.$ By set $x=y$ we get $f(0)=1$. ...
7
votes
3answers
137 views

Functions proof.

Find all functions $$f: \mathbb{Z} \rightarrow \mathbb{Z}$$ such that $$f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a)$$ for all integers $$a, b, c$$ satisfying $$a+b+c=0$$ I have no idea how to ...
8
votes
4answers
118 views

Is $f(x) = Cx\log x$ the only solution to $f(xy) = xf(y) + yf(x)$?

I was studying $L(x) = x \log x$ function and found that it satisfies the following functional equation for positive $x, y$: $$ f: \mathbb R^+ \to \mathbb R\\ f(xy) = x f(y) + y f(x) $$ I have a ...
3
votes
0answers
61 views

Solving an equation for a function

I am trying to do some proof and in connection with that this question arose: Can you find a decreasing function so that $$ 1-\frac{f(x)}{f(ax)} = (1-a)^2x^2 $$ where $0\leq a \leq 1$ and $x$ is ...
1
vote
0answers
35 views

Solve the system of functional equations.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \begin{cases} f(x(1+f(x)))=f(x)^2,\\ f(x(1-f(x)))=f(x) f(-x),\\ f(x(-1+f(-x)))=f(x)f(-x). \end{cases} I have found only that for ...
2
votes
2answers
46 views

Determine function such that $f\left(\sqrt{x_1^2+(f(x_1))^2}\right) = f\left(\sqrt{x_2^2+(f(x_2))^2}\right)$ for every $x_1,x_2$.

Determine a numerical no constant function $f$ such that for all $x_1$, $x_2$ in its domaine of definition, the equality $$f\left(\sqrt{x_1^2+(f(x_1))^2}\right) = ...
11
votes
0answers
102 views

How to solve non-linear differential equation [duplicate]

How to solve non-linear differential equation $$y'(x) = y(y(x)), \quad y\colon\mathbb{R}\to\mathbb{R}?$$ Of course, $y(x)\not\equiv 0$. If we substitute $y(x) = Ax^n$, we get complex $n$ and $A$. Any ...
32
votes
3answers
996 views

Is it true that this function $f(n)=n^{13}$?

Assume strictly monotone increasing function; such that $f:N^{+}\to N^{+}$, $h$ for all $n\in N^{+}$, $$f(f(f(n)))=f(f(n))\cdot f(n)\cdot n^{2015}$$ Prove or disprove:$f(n)=n^{13}$ ...
6
votes
1answer
67 views

Is there a constructive discontinuous exponential function? [duplicate]

It is well-known that the only continuous functions $f\colon\mathbb R\to\mathbb R^+$ satisfying $f(x+y)=f(x)f(y)$ for all $x,y\in\mathbb R$ are the familiar exponential functions. (Prove ...
3
votes
1answer
45 views

Functional equation $f(f(y))+f(x-y)=f(xf(y)-x)$

Find all functions $f$ defined on the real and takes real values ​​such that $$f(f(y))+f(x-y)=f(xf(y)-x)$$ for all $x,y\in \mathbb{R} $ My approach: Let be x=y=o, so $f(f(0))+f(0)=f(0) \to ...
6
votes
1answer
126 views

Solution to the functional equation $f(x^y)=f(x)^{f(y)}$

Consider the functional equation problem $$ f: \Bbb{R} \rightarrow \Bbb{R}$$ $$ f(a^b) = f(a)^{f(b)},$$ when $a,b \in \Bbb{R}, a,b \ge 0.$ So far the only solution I have is the trivial $$ f(x) ...
5
votes
2answers
256 views

find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y \in \mathbb{R}^n$

find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying \begin{equation*} f(x+y)+f(x-y)=2f(x)+2f(y)~\forall x,y \in \mathbb{R}^n. \end{equation*} My attempt: I manage to show ...
0
votes
1answer
28 views

Recurrence equation $f(g\cdot x)/f(x)=t$

I am trying to solve equation $\frac{f(g x)}{f(x)}=t$ and I'm out of ideas. Any suggestions? In my problem, $f$ is a continuous and weakly increasing function.
0
votes
2answers
40 views

The functional equation $x(x+1)+C(x)=(x+1)(x+2)+C(x+1)=(x+2)(x+3)+C(x+2)=…$

Consider the functional equation $$x(x+1)+C(x)=(x+1)(x+2)+C(x+1)=(x+2)(x+3)+C(x+2)=...$$ The equality continues to infinity. Is there $C(x)$ that satisfies all the equality? If there is, what is it? ...
2
votes
1answer
80 views

An awkward Functional Equation

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(f(y)+x^2+1)+2x=y+(f(x+1))^2$$ for all $x,y \in \mathbb{R}.$ I proved that $f$ is bijective, but I am stuck there. Any help please?
1
vote
1answer
112 views

Functional equation $f(x)=f(x-1)+f(x-a)$

Please help to solve functional equation for real numbers We have: $a\in \mathbb{R}$ and $a>1$ $f_a(x)=1$ if $x<a$ $f_a(x)=f_a(x-1)+f_a(x-a)$ for $x\ge a$ @update Actually we have to find ...