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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2
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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)) = ...
31
votes
8answers
1k views

How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$?

How to get $f(x)$, if we know that $f(f(x))=x^2+x$? Is there an elementary function $f(x)$ that satisfies the equation?
8
votes
4answers
278 views

if the function $f(f(n))+f(n)=2n+2014$,find the $f$

let the function $f:N^{+}\to N^{+}$,and such $$f(f(n))+f(n)=2n+2014$$ Find the $f(n)$ My try: let $n=1$,then we have $$f(f(1))+f(1)=2016$$ let $f(1)=a$,then $$f(a)+a=2016$$ and let ...
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 ...
1
vote
1answer
36 views

Periodic Function With A Given Functional Equation

Let $f$ be a real valued function defined for all real numbers $x$ such that for some positive constant '$a$' the equation $f(x+a)=1/2+\sqrt{f(x)-(f(x))^2}$ holds for all $x$. Then prove that $f$ is ...
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:$f(n)=n^{13}$ ...
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
1answer
114 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 ...
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.
2
votes
1answer
83 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: $$T(n)=(\log n)T(\log n)+n$$
1
vote
1answer
463 views

Solution of Cauchy functional equation which has an antiderivative

Let $f\colon\mathbb R\to\mathbb R$ be a function such that $$f(x+y)=f(x)+f(y)$$ for any $x,y\in\mathbb R$ i.e., it fulfills Cauchy functional equation. Additionally, suppose that $F'=f$ for some ...
3
votes
0answers
393 views

What special role plays the function $\pi^{\frac x\pi}$ in analysis?

I have tried to redefine some special functions in the most "natural" way, that is the way which allows to simplify the relations the most. I would call these functions "parelementary". The ...
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 ...
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 $f(0)=1$. ...
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?
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 ...
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) = ...
0
votes
0answers
21 views

Extract independent parameters

I have a two-variable function which depends also on a number of parameters (six to be exact) $f(x,y; c_1, c_2, c_3, \dots, c_6)$. The explicit form is quite complicated so I will not give it here. It ...
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 ...
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 ...
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 ...
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 ...
15
votes
1answer
2k views

Find continuous functions such that $f(x+y)+f(x-y)=2[f(x)+f(y)]$

Here is the problem: Find all continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy $$f(x+y)+f(x-y)=2[f(x)+f(y)]\;\;\;(1).$$ Here is my attempt: Fix $\delta>0$ and let ...
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 ...
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? ...
1
vote
1answer
52 views

Commutative version of hyper operators.

As I understand it, addition and multiplication are defined on the reals as having identity elements 0 and 1 and being commutative and associative. Multiplication is also distributive over addition. ...
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 ...
2
votes
2answers
87 views

All solutions to functional equation $f(x+1)-f(x)=1$

I was thinking of the possibility of finding all solutions other than $f(x)=x$ for the functional equation: $f(x+1)-f(x)=1$ If there are other solutions, what will be some restrictions for the ...
7
votes
6answers
218 views

Solve the following equation: $\sqrt {x + \sqrt {4x + \sqrt {16x + \sqrt {64x + 5}}}} - \sqrt x= 1$

A past examination paper had the following question that I found interesting. I tried having a go at it but haven't come around with any solutions. How would one go about tackling it? $$\sqrt {x + ...
0
votes
1answer
194 views

Solving this set of quadratic equations

I have a set of quadratic equations of the form: \begin{equation*} 2S_0q_0(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_0 = 0 \\ 2S_1q_1(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_1 = 0 \\ \vdots \\ ...
0
votes
0answers
23 views

Stability of Cauchy exponential functional equation

If f : R → R is a function satisfying |f(x + y)- a^xy f(x) f(y)| ≤ δ for all x,y ∈ R and for some positive δ, where a is a positive real constant, then show that either the function $f(x) ...
8
votes
1answer
90 views

Complicated real to real 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 $x,y \in \mathbb{R}.$ So far I have proved that $f$ is bijective. How should I continue?
1
vote
1answer
38 views

Integral equation involving Planck radiation formula

I am stuck in solving the following integral equation: $$\sigma T^4=\pi\int_{\lambda_0}^{\lambda_1}d\lambda W_{\lambda,T}$$ where: ...
4
votes
1answer
51 views

A functional equation with no term outside functions

Find all $f:\mathbb{N} \rightarrow \mathbb{N}$ satisfying $$f(m-n+f(n))=f(m)+f(n)$$ for all $m,n \in \mathbb{N}.$ I have no idea about how to find them, because there are no terms outside of the ...
4
votes
1answer
57 views

Simple Derivation of Functional Equation Question (L'Hospital's Rule)

First, the question is: $f$ is a differentiable function and $f : R \rightarrow R$ $xf(x)-yf(y)=(x-y)f(x+y)$ $f'(2x)=?$ My approach for problem is using L'Hospital's rule: $$ ...
3
votes
3answers
202 views

What function satisfies $f(x)+f(−x)=f(x^2)$?

What function satisfies $f(x)+f(−x)=f(x^2)$? $f(x)=0$ is obviously a solution to the above functional equation. We can assume f is continuous or differentiable or similar (if needed).
0
votes
2answers
83 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 ...
0
votes
1answer
60 views

Trigonometric functional equations. I need hints for this problem

Find all functions $ f,g : \mathbb R \to \mathbb R $ that satisfy the functional equation $$g(x)f(y) = f \left(\frac{x + y}{2}\right)^2-f\left(\frac{x-y}{2}\right)^2$$ for all $x,y \in \mathbb R$.
2
votes
1answer
78 views

Proof regarding a probability generating function (Poisson)

Let $f(s)$ be the probability generating function ($pgf$) of a non-negative, integer valued random variable. It is also given that $f(1-p+ps)f(p) = f(ps)$. Prove that $f(s) = e^{\lambda(s-1)}$ for ...
1
vote
0answers
49 views

Cauchy's function

An exercise in Real and Complex analysis (exercise 18, page 195,chapter 9) Show that if a function $f$ on the real line $\mathbb{R}$ satisfies \begin{equation*} f(x+y)=f(x)+f(y) ...
0
votes
1answer
36 views

Pexerized Dalembert funtional equation..

Let $\lambda$ be a nonzero real constant. Find all functions $f,g: \Bbb R \rightarrow \Bbb R$ that satisfy the functional equation $f(x+y)+g(x−y)=\lambda f(x)g(y)$. I try this : Let $y=0$ in the ...
2
votes
4answers
77 views

Solve: $f(x+\frac{1}{y}) + f(x-\frac{1}{y}) = 2f(x).f(\frac{1}{y})$

Here i have one functional equation: If $$f(x+\frac{1}{y}) + f(x-\frac{1}{y}) = 2f(x)\cdot f(\tfrac{1}{y})\text{ for all x},y\in\mathbb{R}-{0}$$ and $f(0) = \frac{1}{2}$ , then find the value of ...
2
votes
4answers
85 views

Solving for $f(2004)$ in a given functional equation

Given that $$f(1)=2005$$ and $$f(1)+f(2)+...f(n) = n^{2}f(n)$$ for all $n>1$. Determine the value of $f(2004)$. My progress: I first substituted $n-1$ into the equation to get ...
0
votes
5answers
82 views

Suppose $f$ is a real function satisfying $f(x+f(x))$ = $4f(x)$ and $f(1) = 4$. Then the value of $f(21)$?

Should I proceed with just putting the value of $f(1)=4$ in the first equation or there will be a different way of solving this ?
2
votes
2answers
81 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 ...
0
votes
0answers
32 views

D'Alembert's functional equation. I need to solve this problem

Let $λ$ be a nonzero real constant. Find all functions $f,g \colon \mathbb{R} \to \mathbb{R}$ that satisfy the functional equation $$ f(x + y) + g(x-y) = \lambda f(x) g(y) $$ for all $x,y \in ...
0
votes
1answer
174 views

Finding $\alpha$ such that $f(\alpha(x+y))=f(x)+f(y)$

Problem taken from the link: http://web.mit.edu/rwbarton/Public/func-eq.pdf I am stating the question here For which $\alpha$ does there exists a nonconstant function $f: \mathbb{R} \to \mathbb{R}$ ...