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
34 views

Solve differential equation

How can we solve (if a closed form expression for f(x) can be found) the following first-order linear differential equation? $$f'(x)=f(x)\cdot (\cos x+\tan x)$$ I have found that one function which ...
5
votes
1answer
56 views

Functional equation: Finding $f(100)$

A polynomial of degree 98 such $f (k)=1/k$ for $k=1,2,3...,98,99$ exists. How to find $f(100)$? What are the possible methods ?
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0answers
12 views

Criteria when bigger number of functions can be obtained from smaller number

It is known that $$ A_1(x_1, x_2) = \partial \varphi(x_1, x_2)/\partial x_1, $$ $$ A_2(x_1, x_2) = \partial \varphi(x_1, x_2)/\partial x_2 $$ holds if and only if $$ \partial A_1/\partial ...
0
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1answer
22 views

Find functions that satisfy this equation.

Give some examples of functions, $F$ and $G$ such that $$x=\sqrt{F(x)+G(x)\sqrt{F(x+n)}}-\sqrt{F(x+n)}.$$ $n$ can be a constant. [Edit]: with $n\gt{0}$
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0answers
16 views

Functional equation + differential equation = way of finding solution?

Question I was wondering about the following: Let's say there is a differential equation whose solution is $f$ And $f$ also satisfies a functional equation. Can anyone construct an (non-trivial) ...
0
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1answer
30 views

A functional equation that is equal to 7x

I wish to find all of the functions $f:\mathbb R \to \mathbb R$ such that $$ f(x) + 3 f\left( \frac {x-1}{x} \right) = 7x $$ for all nonzero $x$. I have tried plugging in $\frac{x-1}{x}$, but that ...
1
vote
0answers
33 views

Example of a well defined functional integral?

So I was playing around with the notion of a functional integral. Basically given a set $S$ of functions we can define $$ \int_{f \in S} L(f) $$ As the sum of of every function $f$ evaluated by ...
2
votes
4answers
91 views

Prove that $\lim_\limits{x\to 0}{f(x)}=0$

Let $f:\mathbb{R}\mapsto \mathbb{R}$ be a function such that: $$2\cdot f(x)-\sin(f(x))=x, \forall x\in \mathbb{R}$$ Prove that $\lim_\limits{x\to 0}{f(x)}=0$. I think I need to use the sandwich ...
1
vote
3answers
87 views

What is meaning of this question and how to solve it?

I am stuck with understanding the meaning of the question, which states: Show that $\cos(n\theta)=f_n(\cos\theta)$ for polynomials $f_n(x)$ satisfying $$f_{n+1}(x)=2xf_n(x)-f_{n-1}(x) \tag{1}$$ ...
4
votes
2answers
88 views

Determine all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $xf(y)+yf(x)=(x+y)f(x^2+y^2)$ for all $x,y\in\mathbb{N}$ (contest question)

The question below is from the 2002 Canada National Olympiad. I have found one family of functions but need help in finding (or proving the non-existence) of others. Suggestions on how to improve the ...
5
votes
4answers
392 views

Find a polynomial from an equality

Find all polynomials for which What I have done so far: for $x=8$ we get $p(8)=0$ for $x=1$ we get $p(2)=0$ So there exists a polynomial $p(x) = (x-2)(x-8)q(x)$ This is where I get stuck. How do I ...
5
votes
2answers
108 views

given $2f(x) + f(1-x) = x^2$ find $f(-5)$

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. A function $f$ has property that $2f(x)+ f(1-x) = x^2$ ...
6
votes
2answers
108 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
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1answer
19 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 ...
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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
38 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
103 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
61 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
40 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
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1answer
69 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
56 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
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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 $f'(2)$?
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0answers
65 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
64 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
155 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
2answers
47 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
79 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 ...
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0answers
40 views

A functional equation of the form f(f(x)) [duplicate]

What is the solution to this equation? $f(f(x)) = x^2-x+1$
5
votes
3answers
145 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: $f(x^2+f(y))=f(x)^2+y \; \forall x,y \in \mathbb{Z}^+.$ $P(x,x)$ ...
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votes
2answers
104 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
55 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
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0answers
18 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
15 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 ...
-4
votes
2answers
87 views

Find the value of this $2f\left ( \frac{1}{2} \right )$ [closed]

IF $f(f(x))=1-x$, Find $$2f\left ( \frac{1}{2} \right )=??$$ help guys, I really tried but I couldn't.
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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
36 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
84 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
235 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
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0answers
53 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
57 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
17 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
79 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
42 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
76 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
46 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
73 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
134 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 ...