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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8
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1answer
174 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 ...
3
votes
2answers
616 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
313 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
199 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
89 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 ...
0
votes
0answers
37 views

Proving Join Dependencies in MVD

I have a question regarding natural join operations in multivalued dependencies. I know that a join operator joins two tables on similair attributes, however I have a hard time to figure out how to ...
6
votes
2answers
471 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
265 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
32 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
107 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 ...
11
votes
3answers
460 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
37 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
25 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 $f=0=a$ ...
3
votes
1answer
425 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 ...
23
votes
6answers
2k views

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

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
-2
votes
2answers
184 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 ...
2
votes
1answer
81 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
280 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
142 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
53 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
168 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
107 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
412 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 ...
10
votes
2answers
293 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
1answer
202 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
58 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
323 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
213 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 ...
19
votes
2answers
402 views

Proving that $f(n)=n$ if $f(n+1)>f(f(n))$

How can we prove that if $f:\mathbb{N}\rightarrow\mathbb{N}$ is a function so that $f(n+1)>f(f(n))$ for all $n\in\mathbb{N}$ then $f(n)=n$ for all $n\in\mathbb{N}$?
3
votes
1answer
2k views

Solve the recurrence for $T(n) = T(\sqrt n) + 2$. Assume that T(n) is constant for $n\leqslant 2$.

Trying to work out the following question, but I'm stuck.. Can someone direct me please. Using a change of variables $$ \text{Let}\ m = \lg\ n \\ S (m) = T (2m)\\ T (2^{m}) = T ...
6
votes
1answer
99 views

Can we give a definition of the cotangent based on a functional equation?

I've recently learned that the cotangent satisfies the following functional equation: $$\dfrac1{f(z)}=f(z)-2f(2z)$$ (true for $f(z)\neq 0$). Can we solve this equation for real or complex ...
1
vote
0answers
69 views

Functional equation inspired by moment generating function

A user gave the following nice answer http://math.stackexchange.com/a/161584/5031 My question is that although it is clear $\log M_X(t)=Ct^2$ is a form that satisfies the condition ...
-3
votes
2answers
255 views

find all function f,g that satisfy

Find all function $f,g$ that satisfy: $$g(x)-g(y)=\frac{1}{6} (x-y)(f(x)+f((x+y)/2)+f(y))$$ For $y=0$ we have an equation in $f$: $$4(x-y)(f(x/2)-f(x/2+y/2))=xf(y)-yf(x)$$ How can i do it?
5
votes
1answer
109 views

Uniqueness of solution of functional equation

I have a function $f(x_1,x_2) \colon \mathbb{R}^2_{+} \to \mathbb{R}_{+}$ positive homogenous: $$ f(\lambda x_1, \lambda x_2) = \lambda f(x_1,x_2), \; \forall \lambda > 0 $$ and such that ...
1
vote
1answer
197 views

Books on functional equations

Could you help me please with some functional equations? I need some online books with exercises and some explanations. I'm interested in the Cauchy functional equation and Jensen functional equation. ...
2
votes
3answers
126 views

Iterations of $f(x)=\dfrac{ax+b}{cx+d}$

Consider $f(x)=\dfrac{ax+b}{cx+d}$, where $c\neq0$ and $f(x)$ is not equal to a constant. Is it necessarily true that $f^{[n]}(x)=f(x)$ for some natural number $n > 1$?
2
votes
2answers
520 views

Functional Equation. $f(mn)=f(m)f(n)$ and …

I want to prove the following. We have a function $f: \mathbb{Z} \to\mathbb{R}$ s.t. (1) $f(mn) = f(m)f(n)$ (2) $f(m+n) \leq f(m) + f(n)$ (3) $0 \leq f(x) \leq 1$ then $f(m+n) \leq \max\big(f(m), ...
5
votes
2answers
85 views

$f,g:[a,b]\to [a,b]$ are continuous, $f\circ g=g\circ f$ and $f$ is 1-1. Show that $\exists t\in [a,b]$ so that $f(t)=g(t)=t$

Using the Intermediate Value Theorem on $h(x)=f(x)-x$ and $r(x)=g(x)-x$ I can easily show that $\exists t_f,t_g\in[a,b]$ so that $f(t_f)=t_f$ and $g(t_g)=t_g$. It remains to show that $t_f=t_g$. Since ...
1
vote
0answers
121 views

Question about ratios and combinatorics

In this question that I posted yesterday (11/15): I am solving a programming puzzle that consists of finding all the possible ways to build a brick wall of $48$" $\times$ $10$" (width $\times$ height ...
1
vote
1answer
164 views

A functional equation

$$ f(x^2-1)+2f\left(\frac{2x-1}{(x-1)^2}\right)=2-\frac{4}{x}+\frac{3}{x^2}, \ x>1. \ \ f(x) = ? $$ Don't know how to solve such equiations, help me please. Thank you.
2
votes
2answers
338 views

Properties of solutions of the functional equation $f(kx) = kf(x)$

My problem is that I have a function: $f\colon\mathbb R\to\mathbb R$ with the property that $f(kx) = kf(x)$ for all $k,x \in\mathbb R$. a) I shall show that $f(0)=0$ b) If $f$ is not the ...
1
vote
1answer
112 views

Given $2xf(x)+(x-3)f(\frac{1}{1-x})=4x^2-10x-\frac{1}{2}$, find $f(x)$.

Given$$2xf(x)+(x-3)f\left(\frac{1}{1-x}\right)=4x^2-10x-\frac{1}{2}$$ Find $f(x)$. This's the first time I see this kind of question, I have no idea. Please help. Thank you.
5
votes
3answers
113 views

find $f(x)$ when $3f(x-6)-2f(x-9)=x^2-54$

I can easily show that with the assumption $f$ is a polynomial $f(x)=x^2$. But without that assumption how can I prove that $f(x)=x^2$???. I have tried many change of variables $x=u+k$ but to no ...
5
votes
2answers
129 views

Find function $f :\mathbb{R} \to \mathbb{R}$ satisfy that :

Find function $f :\mathbb{R} \to \mathbb{R}$ satisfy that : $$f(1)=1$$ $$f(x+y)=f(x)+f(y)+2xy$$ $$f\left(\frac{1}{x}\right)=\frac{f(x)}{x^4} \hspace{5pt}\forall x \neq 0$$
2
votes
1answer
156 views

A functional equation related to the exponential function

Suppose that $f:\mathbb{R}\rightarrow (0,\infty)$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ are two functions and satisfies the following relation \begin{equation} f(xy)=f(y)^{g(x)} \end{equation} ...
0
votes
3answers
52 views

Prove: we always have at least x>0 is a $x^3+bx^2+cx-d^2=0$ 's root

Prove: we always have at least one x>0 is a $x^3+bx^2+cx-d^2=0$ 's root (b, c, d are real numbers and $d≠0$)
0
votes
3answers
176 views

Find all function $f$ such that $f(x)+f(\frac1x)=\frac1a; a$ is constant

Which function verified that: $f(x)+f(\frac1x)=\frac1a; a$-constant value?
8
votes
1answer
86 views

Are there associative bijections $f\colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$?

This is my first question and I hope this question is not too brief to be acceptable: There are well-known bijections $\mathbb{N \times N \to N}$ and well-known associative compositions on countably ...
0
votes
1answer
173 views

Find all function such that $f(x)-f(y) = (x -y)g(\sqrt{xy})$

Find all functions $f, g$ that satisfy the functional equation $$ f(x)-f(y)= (x -y)g(\sqrt{xy}) \quad \forall\ x,y>0. $$
6
votes
3answers
189 views

Is there a non-constant function $f:\mathbb{R}^2 \to \mathbb{Z}/2\mathbb{Z}$ that sums to 0 on corners of squares?

A problem in the 2009 Putnam asks about functions $f:\mathbb{R}^2 \to \mathbb{R}$ such that whenever $A,B,C,D$ are corners of some square we have $f(A)+f(B)+f(C)+f(D)=0$. Without spoiling the problem ...