Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).

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8
votes
1answer
315 views

About $2$-periodic continuous solutions of $f(x)+f(x+1)=f(2x+1)$

Suppose I want to find all the continuous solutions to the functional equation $$f(x)+f(x+1)=f(2x+1),\tag{E1}$$where $f$ is a continuous and $2$-periodic function defined on the dyadic rationals. I ...
17
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3answers
3k views

The easy(?) part of IMO 2011 Problem 3

Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies $$f(x + y) \leq yf(x) + f(f(x))$$ for all real numbers $x$ and $y$. How can I prove that $...
12
votes
2answers
2k views

On sort-of-linear functions

Background A function $ f: \mathbb{R}^n \rightarrow \mathbb{R} \ $ is linear if it satisfies $$ (1)\;\; f(x+y) = f(x) + f(y) \ , \ and $$ $$ (2)\;\; f(\alpha x) = \alpha f(x) $$ for all $ x,y \in \...
0
votes
1answer
48 views

designing an equation that compares two values and returns a probability

Given two values, I'm trying to come up with a formula that will return 50% if both values are equal, 25% if the first value is half the second, 75% if the second is half the first. In other words: ...
1
vote
3answers
410 views

Finding an $f(x)$ that satisfies $f(f(x)) = 4 - 3x$

I need to find $f(f(x)) = 4 - 3x$ In other examples, such as $f(2)$, I can see that the result equates to $-2$ or $f(x^2)$ becomes $-3x^2 + 4$. Do I really just substitute $f(x)$ for $x$ and ...
3
votes
1answer
296 views

Starting out with functional equations

I am thinking of starting learning about various functional equations and ways to solve them, any help as to which books could be of help to me? I have some knowledge about some basic functional ...
3
votes
1answer
409 views

Finding a function that satisfies constraints numerically

I have the following system of equations for function $p(y)$ and I need help debugging my solution: $$\begin{align} 0&=\log(p(y))+1-\lambda-\gamma y^2-\eta \left(\frac{e^{-y^2/2}}{\sqrt{2\pi}}\...
12
votes
7answers
4k views

How to find the function $f$ given $f(f(x)) = 2x$?

I was wondered how to find the function in this equality: $f(f(x))=2x$. Also $f$ is continuous. I don't need the answer, how to find it is more important.
1
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1answer
2k views

Solving an equation with a “nested” function

In a little calculation I'm doing for fun, I've come across this equation involving a function of two arguments which is nested on the right side: $$f(t_1 + t_2, K) = f\bigl(t_2, f(t_1, K)\bigr)$$ I'...
25
votes
2answers
744 views

$f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$

A function that satisfies both $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$ for all real $x$ is known to be the identity over $\mathbb Q$, but is it also the identity over $\mathbb R$? If not, can you provide ...
1
vote
1answer
232 views

Any example of functions are automorphism?

I am looking for functions fulfilling $f(x+y) = f(x) + f(y)$ and $f(x*y) = f(x)*f(y)$. I can only find $f(x)=x$, any more? Any example of functions are automorphism?
6
votes
3answers
2k views

Classifying Functions of the form $f(x+y)=f(x)f(y)$ [duplicate]

Possible Duplicate: Is there a name for such kind of function? The question is: is there a nice characterization of all nonnegative functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(x+...
11
votes
3answers
338 views

Existence of a function

I came across this question Does there exist a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $f(x+y)>f(x)(1+yf(x))$ and $x,y\in \mathbb{R}^+$ and I didn't know how to begin on it.
2
votes
1answer
434 views

How to solve algebraically the equation $x = \frac{1}{2}\cos\left(\frac 2 3 \sin\left(\frac 3 4 x\right)\right) + 1$

How to solve this trigonometric equation $x = \frac 1 2 \cos\left(\frac 2 3 \sin\left(\frac 3 4 x\right)\right) + 1$ ? The iterative solution seems to be 1.417. Can anybody suggest an algebraic ...
0
votes
1answer
389 views

Complexity of $T(n)=\sqrt{n}T(\sqrt{n})+n$

I tried to find the complexity of this recursion equation: $T(n)=\sqrt{n}T(\sqrt{n})+n$, by doing couple of iterations and getting a general idea, but I completely got lost. I'd really love your ...
4
votes
1answer
178 views

Question in solving $\phi(t)=\phi(2t)+\phi(2t-1)$, $\phi\ne0$

Actually one can resort to the two-scale equation in multiresolution analysis. Perform Fourier transformation on both side of $\phi(t)=\phi(2t)+\phi(2t-1)$, it turns out that $$\hat\phi(\omega)=\frac{...
5
votes
1answer
406 views

Iterative Functional Equation

Find all functions $ f: \mathbb{N} \rightarrow \mathbb{N}\; $ satisfying $$ f(f(f(n))) + 6f(n) = 3f(f(n)) + 4n + 2001 , \forall n \in\mathbb{N} $$ After some trial and error I assumed the ...
10
votes
2answers
591 views

Functional equation of irreducible characters

I am preparing to an exam in representations of finite groups. I am trying to tackle a problem regarding a characterization of irreducible characters: Let $f$ be a complex-valued function on a finite ...
4
votes
2answers
261 views

$f(z_1 z_2) = f(z_1) f(z_2)$ for $z_1,z_2\in \mathbb{C}$ then $f(z) = z^k$ for some $k$

Same as my previous question except domain is complex. I tried assuming that the function was analytic, so for $z_1=z_2=z$ , $f(z^2) = f(z)^2$ $$\sum_{n=0}^\infty a_n z^{2n}=\left(\sum_{n=0}^\infty ...
4
votes
1answer
427 views

Solving (and proving) a combinatorial functional recursive equation

I have a sequence of functions $f_k(n)$ defined as follows: $f_1(n)=n^{n-2}$ $f_k(n)=\sum_{i=1}^{n-1}f_{k-1}(i)\cdot(n-i)^{n-i-2}\cdot{n-k \choose n-i-1}$ My goal is to find and prove a closed-form ...
1
vote
1answer
59 views

Transform the sample to make it more similar to a given

$X=\{x_{i}\}$ and $Y=\{y_{i}\}$ are numeric samples: $y_i \ge 0, x_i \ge 0, i \in [0..N]$. I need to find the mapping $F(X)=\{F(x_i)\}$ with fairly simple formula such that: Euclidean distance $\rho(...
0
votes
1answer
109 views

General solution to homogeneous difference equation

With a given example $$ a_{n-1} = ca_{n-2} $$ general solution: $$ a_{n} = c . c . a_{n-2} $$ $$ = c . c . a_{n-3} $$ $$ = c^n a_0 $$ Question: Find the general solution for the ...
0
votes
3answers
245 views

Recurrence relation - How to solve this recurrence relation

a person invests 1000 at a bank at 4 percent compound interest compounded annually and every year government and bank charges amounting to C are deducted and if An is the value of the investment at ...
1
vote
2answers
173 views

Finding 2^2^2^2^2 … n times

f(1) = 2 f(2) = 2^2 f(3) = 2^(2^2) f(4) = 2^(2^(2^2)) etc how can I find a closed form solution to this? Thanks.
0
votes
1answer
210 views

Solving general solution of recurrence relation by iteration

$$a_{n-1} = ca_{n-2} $$ Hence $$a_n = c \cdot c \cdot a_{n-2} $$ $$ = c \cdot c \cdot c \cdot a_{n-3} $$ ...... $$ = c^na_0 $$ Why is there a iteration on the constant $c$ ?
3
votes
2answers
268 views

Understanding difference equation

I was given an example $$R_n = R_{n-1} + R_{n-2} $$ This equation is given as an second-order equation. Why is it so?
11
votes
2answers
262 views

Can every real function be represented as two shifted even functions?

I saw the theorem that every function can be represented as the sum of and even and odd function, and this made me wonder: can every function from the reals to the reals, defined on all the reals, be ...
8
votes
3answers
339 views

Solving $\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$

I recently came across this equation : $$\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$$where $f \in \mathcal{C}^1(\mathbb{R}, \mathbb{R})$. I've done the following, but I'm stuck at ...
3
votes
1answer
105 views

All functions with the property $ k \mid f(m+n) \iff k \mid f(m)+f(n)$

Let $\mathbb N$ be the set of all positive integers. How can one find all functions $f: \mathbb N \to \mathbb N$ such that $$ k \mid f(m+n) \iff k \mid f(m)+f(n)$$ For all positive integers $k$.
5
votes
3answers
151 views

Looking for a function $f$ such that $f(i)=2(f(i-1)+f(\lceil i/2\rceil))$

I'm looking for a solution $f$ to the difference equation $$f(i)=2(f(i-1)+f(\lceil i/2\rceil))$$ with $f(2)=4$. Very grateful for any ideas. PS. I've tried plotting the the initial values into "...
3
votes
1answer
257 views

A differential-functional equation: $f'(f^{-1}(x)) = 1/g(x)$

Problem: Given $g(x)$, solve the equation $f'(f^{-1}(x)) = \frac{1}{g(x)}$ for an invertible and differentiable function $f(x)$. So far I have tried setting $y = f^{-1}(x) \Leftrightarrow x = ...
3
votes
2answers
2k views

Converting polar equation to cartesian coordinate polar equation and back again?

OK, so I have the following polar equation: $r = Θ/20$ And I would like to translate this a little to the right, and down from the polar origin. Now, I figure since I know cartesian coordinate ...
0
votes
2answers
51 views

Reversing bijections defined via conditional expressions

Let's say that I have a variable $j$ defined by the following formula: $$j=\frac{n(n+2) + m}{2}$$ where $n$ and $m$ are two parameters, both integers, that satisfy the following conditions: $n\in \...
1
vote
1answer
216 views

Functional Equation Analysis

Given: $$F(F(n)) = n$$ $$F(F(n + 2) + 2) = n$$ $$F(0) = 1$$ where n is a non-negative integer. $$F(129) = ?$$ How can we solve such kind of functional equations? Is there any simpler approach ...
3
votes
1answer
231 views

$f(1-f(x))=f(x)$

Find all continuous $f:[0,1] \rightarrow [0,1]$ such that $f(1-f(x))=f(x)$.
5
votes
1answer
509 views

$f(x+f(y))=f(x-f(y))+4xf(y)$

Find all functions $f:R\rightarrow R$ which satisfy $f(x+f(y))=f(x-f(y))+4xf(y)$ $\forall x,y \in R$. I strongly suspect $0$ and $x^2+C$ to be the only solutions but, as is almost the case with ...
0
votes
1answer
123 views

$x^3[f(x+1)-f(x)]=1$

Possible Duplicate: $x^3[f(x+1)-f(x-1)]=1$ Given that f is continuous and $x^3[f(x+1)-f(x)]=1$, determine $\lim_{x\rightarrow \infty}f(x)$ explicitly.
0
votes
1answer
131 views

$x^3[f(x+1)-f(x-1)]=1$

Given that $x^3[f(x+1)-f(x-1)]=1$, determine $\lim_{x\rightarrow \infty}(f(x)-f(x-1))$ explicitly.
3
votes
0answers
87 views

Function Shape Reference

I'm wondering if their exists a visual/behavioral reference for the fundamental families of functions. I'm not a mathematician so excuse my language if I'm being overly vague. I would like to have a ...
1
vote
2answers
147 views

How to solve the following system?

I need to find the function c(k), knowing that $$\sum_{k=0}^{\infty} \frac{c(k)}{k!}=1$$ $$\sum_{k=0}^{\infty} \frac{c(2k)}{(2k)!}=0$$ $$\sum_{k=0}^{\infty} \frac{c(2k+1)}{(2k+1)!}=1$$ $$\sum_{...
1
vote
1answer
195 views

conversion of a powerseries $-3x+4x^2-5x^3+\ldots $ into $ -2+\frac 1 x - 0 - \frac 1 {x^3} + \ldots $

This is initially a funny question, because I've found this on old notes but I do not find/recover my own derivation... But then the question is more general. Q1: I considered the function $ f(...
4
votes
1answer
506 views

Find a function such that $f(\log(x)) = x \cdot f(x) $

I recently read an article in which the author describes how to find some functions that obey to certain recursion relationships. If we want to find a function that satisfies, for example, $f(x^a) = ...
5
votes
1answer
239 views

Generalization of cos: is this function known?

Consider a function $f_1$ defined by $f_1(x)=1-x+o(x)$ and $f_1(2x)=f_1(x)^2 + 0$. It's simple to find that $f_1(x)=e^{-x}$ (for example by writing series near $x=0$). Consider a function $f_2$ ...
6
votes
1answer
485 views

Iterated polynomial problem

Polynomial $P$ satisfies $P(n)>n$ for all positive integers $n$. Every positive integer $m$ is a factor of some number of the form $P(1),P(P(1)),P(P(P(1))),\ldots $. Prove that $P(x)=x+1$.
2
votes
0answers
426 views

Solving $f(f(x))=g(x)$ equations [duplicate]

Possible Duplicate: Square root of a function (in the sense of composition) I'm interested in solving equations of the form $f(f(x))=g(x)$ for $x\in\mathbb{R}$ where $g(x)$ is a known function. ...