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
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: $$T(n)=(\log n)T(\log n)+n$$
7
votes
0answers
70 views

What are the densities of branches of the euclidean tree?

The Euclidean algorithm shows how all coprime pairs of positive integers can be uniquely obtained from the pair $(1,1)$ by applying the two operations $(a,b) \to (a+b,b)$ and $(a,b) \to (a,a+b)$. (or ...
10
votes
1answer
461 views

Find all the function that satisfy : $f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy$

Find all the function that satisfy : $$f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy$$ I only find $f(0)=0$ but I can't prove $f(x)=2x$
100
votes
7answers
4k views

Continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?

I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success: Find a continuous ...
1
vote
1answer
43 views

Funcional Equations:I'm confused [duplicate]

I need help with this: "Find functions $f$, $g : \mathbb{Z} \rightarrow \mathbb{Z}$, knowing that $g$ is injective and such that: $$f(g(x)+y) = g(f(x)+x), \mbox{ for all } x, y \in \mathbb{Z}.$$ Or : ...
0
votes
2answers
78 views

What's the solution of the functional equation

I need help with this: "Find all functions $f$, $g : \mathbb{Z} \rightarrow \mathbb{Z}$, with $g$ injective and such that: $$f(g(x)+y) = g(f(x)+y), \mbox{ for all } x, y \in \mathbb{Z}.$$
7
votes
4answers
765 views

If $f(x/2)=f(x)/2$, then $f(x)=f'(0)x$

Let $f:\mathbb R \to \mathbb R$ be differentiable such that $f(x/2)=f(x)/2$ for any $x\in \mathbb R$. How can I prove that $f(x)=f'(0)x$, for any $x\in \mathbb R$? It seems easy, but I don't know why, ...
0
votes
2answers
148 views

How to create equations to measure time spending in executing algorithms?

I made a program with two functions to calculate factorial. The first uses loops to made de calculations, and the second uses recursive calls to get the same result. The same program measures the ...
1
vote
1answer
102 views

Are all functions on vectors in $GF(2^n)$ representable as polynomial functions?

In $GF(2)$, any function from an n-dimensional vector to a number is equal to a polynomial function of n variables. (See proof below.) Question: Is this true for other $GF$, especially $GF(2^8)$? ...
0
votes
1answer
88 views

simplifying “$\prod_{a\in A}\sum_{b\in B_a}{h(a,b)}$”

Is this equality correct? For finite sets $A$ and $B_a$ (where $a\in A$), we have: $$\prod_{a\in A}\sum_{b\in B_a}{h(a,b)}=\sum_{f\in \prod_{a\in A}B_a}\quad \prod_{a\in A}{h(a,f(a))}$$
1
vote
1answer
77 views

What's the solution of the functional equation?

I need help with this: "Find functions $f$, $g : \mathbb{Z} \rightarrow \mathbb{Z}$, knowing that $g$ is injective and such that: $$f(g(x)+y) = g(f(x)+x), \mbox{ for all } x, y \in \mathbb{Z}.$$
6
votes
2answers
3k views

How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?

This is part of a homework assignment for a real analysis course taught out of "Baby Rudin." Just looking for a push in the right direction, not a full-blown solution. We are to suppose that ...
7
votes
3answers
830 views

Find all the functions which satisfy a given functional equation

I was given by a friend of mine the following problem. Find all the functions $f:\mathbb R\to\mathbb R$ which satisfy the following functional equation $$f(y+f(x))=f(x)f(y)+f(f(x))+f(y)-xy.$$ We ...
9
votes
2answers
89 views

Uniqueness of solution for a functional equation

Let $h \in \mathcal{C}:=C([-a,a])$, where $a>0$. Prove that there exists a unique function $f \in \mathcal{C}$ such that $$ f(x)=\frac{x}{2}f\Big(\frac{x}{2}\Big)+h(x)\quad \forall x \in [-a,a]. $$ ...
5
votes
1answer
294 views

Strictly monotone functions

What are the strictly monotone functions $f\colon (0,\infty)\to (0,\infty)$ which satisfy $x= f(\tfrac{x^2}{f(x)})$ for $x>0$. I cannot find any other than $f(x)=x$.
1
vote
1answer
35 views

Plotting for solution for $y=x^2$ and $x^2 + y^2 = a $

Consider the system $$y=x^2$$ and $$x^2 + y^2 = a $$for $x>0$, $y>0$, $a>0$. Solving for equations give me $y+y^2 = a$, and ultimately $$y = \frac {-1 + \sqrt {4a+1}} {2} $$ (rejected ...
2
votes
2answers
685 views

Attractive fixed-point?

I'm sorry in advance for how specific this problem is. I've been trying to make it as generic as possible but this is as far as I could get. I want to prove that a fixed point is attractive. I didn't ...
2
votes
1answer
87 views

Connecting functional identity of a function with its image set

Okay, now I really need real help, maybe the task is not too heavy but I do not know the easy way to solve this. Let´s start with the problem, now. Suppose that we have some function $f: \mathbb N ...
2
votes
2answers
190 views

How to find $f(x)+f(x+1) = e^x$?

Let $x,a,b$ be real numbers and $f(x)$ a (nongiven) real-analytic function. How to find $f(x)$ such that for all $x$ we have $f(x)+af(x+1)=b^x$ ? In particular I wonder most about the case $a=1$ ...
5
votes
1answer
162 views

Question about a functional equation

We are looking at a theorem which characterizes the affine term structure (ats) models in interes rate theory. What follows is from "Filipović, D. (2009): "Term-structure models: A graduate course", ...
3
votes
2answers
131 views

Average of function, function of average

I'm trying to find all functions $f : \mathbb{R} \to \mathbb{R}$ such that, for all $n > 1$ and all $x_1, x_2, \cdots, x_n \in \mathbb{R}$: $$\frac{1}{n} \sum_{t = 1}^n f(x_t) = f \left ( ...
8
votes
1answer
183 views

A functional equation problem

Let $f$ be a function which maps $\mathbb{Q}^{+}\to\mathbb{Q}^{+}$. And it satisfies $$ \left\{ \begin{array}{l} f(x)+f\left(\frac{1}{x}\right)=1\\ f(2x)=2f(f(x)) \end{array}\right. $$ Show that ...
4
votes
2answers
730 views

Finding all continuous functions such that $f^2(x + y) − f^2(x − y) = 4f(x)f(y)$

I've been working on the following homework problem: Find all continuous functions $f : \mathbb{R} → [0,∞)$ such that $f^2(x + y) − f^2(x − y) = 4f(x)f(y)$ for all real numbers $x, y$. The first ...
6
votes
2answers
333 views

Proving a function is constant when $f(x)f(y) + f(\frac{a}{x})f(\frac{a}{y}) = 2f(xy)$

I've been working on the following homework problem: Consider a function $f : (0,∞) → \mathbb{R}$ and a real number $a > 0$ such that $f(a) = 1$. Prove that if $f(x)f(y) + ...
8
votes
1answer
204 views

Maps of $\mathbb{R}^3$ preserving the cross product

Given a map $\phi:\Bbb R^3 \rightarrow \Bbb R^3$ such that for all $a,b \in \Bbb R^3$: $$\phi(a \times b)=\phi(a) \times \phi(b)$$ Is $\phi$ necessarily a rotation around the origin or the map ...
1
vote
1answer
97 views

If $f(cx, cy) = f(x, y)$ can we always find $g$ such that $g(\frac{x}{y}) = f(x, y)$?

Motivation: A differential equation $\frac{dy}{dx} = f(x, y)$ is said to be homogeneous if we can find $g$ such that $g(\frac{x}{y}) = f(x, y)$, which allows us to solve the equation using the ...
6
votes
2answers
558 views

Uniqueness of Tetration

Let $f(0)=1$ and $f(x+1)=2^{f(x)}$ Also let f be infinitely differentiable. Then does f exist and is it unique? If f is merely continuous, then any continuous function such that f(0)=1 f(1)=2 ...
2
votes
2answers
324 views

Find the equation of this bijection from $ \mathbb{R} $ to $ (0,1) $.

I need some help (hints or an answer) in finding the actual equation of this bijections from the reals $ \mathbb{R} $ to $ (0,1) $. We may assume that the radius of the circle is $ \dfrac{1}{2} $. ...
1
vote
1answer
35 views

Show that functions are affinely dependent

Let $u(x),v(x)$ be continuous bounded functions on $\mathbb{R}$ such that for any Borel probability measures $\mathbb{P}_{1},\mathbb{P}_2$ on $\mathbb{R}$ $$ \int u(x) \, \mathbb{P}_1(dx) \leqslant ...
2
votes
1answer
111 views

How bad can perturbing a functional equation really make things?

A long time ago, I found occassion to find solutions to a functional equation of the following form $f(x-y) = f(x) - f(y) + \delta $ with $\delta \in \mathbb{R}.$ Using the same exact techniques as ...
13
votes
3answers
632 views

Solution of functional equation $f(x/f(x)) = 1/f(x)$?

I've been trying to add math rigor to a solution of the functional equation in [1], eq. (22). It is: $$ f\left(\frac{x}{f(x)}\right) = \frac{1}{f(x)}\,, $$ where you know that $f(0)=1$ and $f(-x) = ...
2
votes
2answers
38 views

Ratio's and Max Size

If I have a video of size: width: 640 height: 480 and a screen of size: width: 1280 height: 720 what is the equation ...
2
votes
0answers
29 views

Multiple integrals satisfying functional equations

If $f:\mathbb{R}_+\to\mathbb{R}_+$ is a positive function which is locally $L^1$ with $\int_0^t{f(s)ds}=at^n$ for all $t\geq0$ then, by differentiating, it follows that $f(t)=a$ if $n=1$ and $f=0=a$ ...
4
votes
1answer
665 views

Functional equation book for olympiad

what may be the good suggestions in olympiad functional equations for a beginner for . I have heard of this book by B.J.Venkatachala but do not whether it will be suitable for me or not. Anybody ...
26
votes
6answers
3k views

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$

Find all polynomials $P$ such that $$P(x^2+1)=P(x)^2+1$$
-1
votes
2answers
222 views

Solve the functional equation $f(1+xf(y))=yf(x+y)$

Problem Find all functions $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$ such that: $$f(1+xf(y))=yf(x+y)$$ for all $x,y \in \mathbb{R^+}$ Progress I can only prove $f$ is a surjective function. I ...
6
votes
1answer
576 views

non-continuous function satisfies $f(x+y)=f(x)+f(y)$

As mentioned in this link, it shows that For any $f$ on the real line $\mathbb{R}^1$, $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable. But how to show there exists such ...
2
votes
1answer
84 views

Functional inequality

Let $S$ be a semigroup such that $S\ne S+S$ and let $f:S\to \Bbb C$ be an unbounded function satisfying $$ |f(s_1)f(s_2)-f(t_1)f(t_2)|\le 1 $$ for all $s_1, s_2, t_1, t_2 \in S$ such that ...
5
votes
3answers
342 views

Find the value of the function at the given point.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function satisfying the conditions $$\begin{align*} (1)&f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}\\ (2)&f(0)=1\\ (3)&f'(0)=-1 ...
5
votes
1answer
161 views

Concerning nonlinear functional equations

There's a problem I've been working on for awhile that involves some hefty functional equations. For example, I may have something along the lines of $$ f(x)f(x) =x+1+f(x+1) $$ I've tried several ...
1
vote
1answer
54 views

Functional inequality

Let $f, g:\Bbb R \to \Bbb R$ be bounded functions satisfying $$ |f(x+y)-f(x)g(y)|\le \frac 1 4 $$ for all $x, y\in \Bbb R$. Prove or disprove $$ |f(x)||1-g(y)|\le \frac 1 4 $$ for all $x, y\in \Bbb ...
3
votes
2answers
186 views

Find all $f:\mathbb{R}^+\to \mathbb{R}^+$ satisfies that:

Find all $f:\mathbb{R}^+\to \mathbb{R}^+$ satisfies that: $$f(x)\cdot f(yf(x))=f(y+f(x))$$ $\forall x,y \in \mathbb{R}^+$
1
vote
0answers
112 views

Periodicity of a function.

If we have two functions $f:\mathbb{R} \to \mathbb{R}$ and $g :\mathbb{R} \to \mathbb{R}$ such that the period of $f$ is 7 and that of $g$ is 11, then the period of $F\left ( x \right ) = f( ...
-1
votes
1answer
642 views

Combine two equation

I have two equations with this format: $$Ds= A+A^2+\alpha_1\tag{1}$$ and $$Ds= M+M^2+\alpha_2 \tag{2}$$ Knowing that $(1)$ explains 72% of $Ds$ and $(2)$ 20%. I want to combine these two equations ...
11
votes
2answers
312 views

When is $f^{-1}=1/f\,$?

I seem to remember a nice article that I read many years ago (perhaps in the American Mathematical Monthly) which investigated the question, "Under what conditions on $f:\mathbb{C}\to\mathbb{C}$ is ...
3
votes
2answers
392 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 ...
3
votes
1answer
215 views

Cauchy functional equation

Is there $U\subset \Bbb R^2$ with Lebesgue measure $0$ such that $$f(x+y)=f(x)+f(y)$$ for all $(x, y)\in U$ implies $f(x+y)=f(x)+f(y)$ for all $(x, y)\in\Bbb R^2$ ?
-1
votes
1answer
70 views

Exercise of functions of a real variable

Let $f, g\colon\mathbb{R}\rightarrow\mathbb{R}$ functions so that for all $\,x, y\in\mathbb{R}$, $$f(x+y)+f(x-y)=2f(x)g(y).$$ Prove that if $f$ is not constant zero function and $|f(x)|\leqslant ...
3
votes
3answers
331 views

Find all functions for which $\ x \cdot f(xy)+f(-y)=x \cdot f(x)$

Does anyone have idea for solution (for all non-zero numbers)? $$\ x ≠ 0,y ≠ 0$$ $$\ f: R \setminus \{0\} → R$$ $$\ x \cdot f(xy)+f(-y)=x \cdot f(x)$$ Thanks!
4
votes
1answer
224 views

Solve system of equations with sin, cos, tg

I am trying to solve this system of equations but without any results. How can I solve this system of equations (in real numbers)? $$\sin^2 x + \cos^2 y = \tan^2 z$$ $$\sin^2 y + \cos^2 z = \tan^2 ...