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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).

3
votes
0answers
50 views

Differential Equation with inverse function $\frac{1-f^{-1}\left(\frac{f(x)}{x}\right)}{1-x} = 1- \frac{f(x)}{xf'(x)}$

$$\frac{1-f^{-1}\left(\frac{f(x)}{x}\right)}{1-x} = 1- \frac{f(x)}{xf'(x)}$$ I know $f(x) = ax+b$ is a solution. How can I find other solutions?
1
vote
2answers
47 views

A problem on number of solutions of a functional equations

Find all functions $f:R \rightarrow R$ such that $f(0)=1$ and for all $x\neq -1$ : $f(x)=8f(2x+1)$ (I have found only one solution: $1/(x+1)^3$. Method was by iterated substitution of $2x+1$ ...
5
votes
0answers
39 views

Does there exist a function $f_{\Box,\Box}(\Box)$ making the formula $a + (b \oplus c) = (f_{b,c}(a)+b) \oplus (f_{c,b}(a)+c)$ true?

Let $a$ and $b$ denote the resistances of two resistors. If they're put in series, the total resistance is $a+b$. If they're put in parallel, the total resistance is $$a \oplus b := \frac{1}{\frac{1}{...
1
vote
2answers
100 views

Solution of the equation $f(ax) = bf(x)$

Given the equation $f(ax) = bf(x)$, with $a, b > 0$, demonstrate that the solution is: $$f(x) = g(\log x)x^{\frac{\log b}{\log a}}$$ where $g(x) = g(x + \log a)$ is an arbitrary periodic function ...
0
votes
0answers
12 views

How to extract associative binary operations from a class of binary operations on $\mathbb{R}$

I would like to find (all) associative binary operations of the form $$u_{1}*u_{2}=\ln{\left[e^{u_{1}}+e^{u_{2}}\right]}+Q\left(u_{2}-u_{1}\right),$$ where $Q$ is an arbitrary function. My effort: ...
-1
votes
0answers
4 views

How to calculate odds with variance?

Is there a generator for this calculating an odds variance theory? Calculating 1/720,000, or a way to figure out the odds with variance? Like out of 100 out of 720,000, what would the odds be at that ...
1
vote
1answer
62 views

Prove that $\forall x \in \mathbb{R}, f(x)=0$ [duplicate]

Suppose $f\colon\mathbb{R}\to\mathbb{R}$ is differentiable function and satisfies $$\forall x \in \mathbb{R}, \vert f'(x)\vert \leq \vert f(x)\vert, \quad f(0)=0.$$ Prove or disprove $f(x)=0$ How ...
2
votes
1answer
68 views

About the linear functional equations: f(x+a) = bf(x) and f(ax) = bf(x).

About the linear functional equations: $f(x + a) = bf(x)$ and $f(ax) = bf(x)$, Marek Kuczma e Polyanin A.D. they got the respective solutions (http://eqworld.ipmnet.ru): $f(x) = g(x)b^{x/a}$, where $...
15
votes
2answers
408 views
+50

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

Let $f:\mathbb{R}\to \mathbb{R}$ and such for any real numbers $x,y$ we have $$f(xf(y)+yf(x))=yf(x)+xf(y)$$ Find $f(x)$. I have let $x=y=0$ have $$f(0)=2f(0)\Longrightarrow f(0)=0$$ and I guess the ...
1
vote
3answers
254 views
+100

$2f(x)=f(y) \Rightarrow 2f(tx)=f(ty)$

Find all continuous and strictly monotonic function $f:[0,\infty)\to \Bbb R$ such that: If there is a pair $(x,y)\neq (0,0)$ such that $2f(x)=f(y)$ then $2f(tx)=f(ty)$ for all $t>0$; There is at ...
1
vote
1answer
13 views

Suppose $L$ has a regular parametrix . Assume $U$ is a distribution given in an open set $\Omega \subset R^d$ and $L(U)=f$ , then $U$ is $C^{\infty}$

Suppose $L$ has a regular parametrix . Assume $U$ is a distribution given in an open set $\Omega \subset R^d$ and $L(U)=f$ , with $f$ a $C^{\infty}$ function in $\Omega$ , then $U$ agrees with a $C^{\...
0
votes
2answers
62 views

Existence of $f(x)$

Suppose $g(x)$ is cubic which has two local extrema. Is there differentiable function $f(x)$ which satisfies $\forall x \in \mathbb{R}, g(f(x))=x$ exist? I know if I make $f$ piecewise inverse of $g$...
1
vote
1answer
69 views

I would like to go for search for this given functional equation using either java or python $f(x+1) = f(x)^2-1$

I would like to go for search for this given functional equation using either java or python $$ f(x+1) = f(x)^2-1 $$ $$ f(0) = 1 $$ I don't know where to start. I know how to graph in pycharm. I know ...
1
vote
1answer
36 views

Functional equation $f((xf(x))^2 + f(y))=-x^4 + y$

Problem Functional equation Suppose $f\colon\mathbb{R}\to\mathbb{R}\quad$ $\forall x, y \in \mathbb{R}, f((xf(x))^2 + f(y))=-x^4 + y$ What I found : Put $x=y=0,$ then $f(f(0))=0$ And put $x=f(0)...
0
votes
4answers
80 views

Existence of the function $f(x)$

Let $f\colon\mathbb{R}\to\mathbb{R}$ be differentiable function that satisfies $$ f(0)=1 \\ \forall x \in \mathbb{R}, \quad f(x+1)=\exp(3x^2+1)f(x)$$ I think a function $f$ exist which satisfies ...
0
votes
0answers
33 views

Functional Equation with two conditions

Let function $$ f\colon \mathbb{R}\to\mathbb{R}$$ is continuous function and satisfies both conditions : $$ \forall x\in\mathbb{R}, f(f(x))=x \tag{1}$$ $$ \forall x\in\mathbb{R^+}, \int_{-x}^0f(t)dt-\...
1
vote
0answers
32 views

Solve the functional equation $f\left(x\right) = 1 - \left(1 - f\left(x+1\right)\right)^{\frac{x}{x+1}}$

Trying to find a concave function defined on the positive reals, satisfying some inequalities, I came up with the following relation $f\left(x\right) = 1 - \left(1 - f\left(x+1\right)\right)^{\frac{x}...
5
votes
2answers
105 views

How to solve this D.E $y''(\frac{x}{2})+y'(\frac{x}{2})+y(x)=x$

I know how to slove $y''(x)+y'(x)+y(x)=x$ But I couldn't solve this $$y''(\frac{x}{2})+y'(\frac{x}{2})+y(x)=x$$ any hint to help me? Thanls
0
votes
3answers
42 views

Functional Equation with three variables

I have a functional equation with three variables. $f(x,y,z)$ is a real function with three variables where y is different from z i.e., $f(x,y,z)$ defined only for $y \neq z$. This function satisfies ...
0
votes
1answer
30 views

Differentiating Exponential Functional Equation [duplicate]

The Functional Equation satisfied by the exponential $f(x)=e^{kx}$ is of the form: $$ f(x+y)=f(x)f(y), \quad f(0)=1, f'(0)=k $$ Show that $f'(x) = kf(x)$. Attempt I tried applying Chain Rule to ...
0
votes
2answers
47 views

Question on Function of Function.

$$f(x)=\frac{x+2}{1-2x}$$ $$g(x)=\frac{2x+1}{2-x}$$ Find $$(fofofo...ofOgogo...og)=\frac{1}{x}$$ {fofo... are 101 times and gogo.. are 100 times} Then Find $x$? I calculated as follows Since $$(...
1
vote
3answers
79 views

Is there any functional equation $f(ab+cd)= f(a)+f(b)+f(c)+f(d)$?

I am looking for a real, continuous function that satisfies the functional equation $$ f(ab+cd)= f(a)+f(b)+f(c)+f(d) $$ where $a,b,c,d$ are real. This is equivalent to a function satisfying these two ...
4
votes
2answers
103 views

find all possible functions : $f(a)f(b)-6ab=\frac{3}{2}f(a+b)$

I'm trying to find all possible functions that satisfy this functional equation: $f(a)f(b)-6ab=\frac{3}{2}f(a+b),$ $f\in \mathbb{R}.$ My attempt : $a=b=0$ then $f(0)=0$ or $1.$ But I don't ...
0
votes
1answer
36 views

Functional equations in one variable.. [closed]

How do you solve the functional equation involving only one variable...what if and if not given that $f(x)$ is a polynomial... Say for example $f(x)=f(x-1) +2x$
1
vote
0answers
30 views

$‎\lim_{n\to\infty} (xf(n) + \sum_{k=1}^n f(k) - f(k+x))‎$‎ is‎ ‎convergent for $x\geq 1$. [closed]

‎‎Let $f:[1‎, +‎\infty)\rightarrow\mathbb{R}$ be a function such that ‎$‎\lim_{n\to\infty} (f(n) - f(n+1)) = 0‎$‎‎. ‎Suppose that $\lambda:[1‎, +‎\infty)\rightarrow\mathbb{R}$‎ ‎is an exp-convex ...
2
votes
1answer
124 views

Find functions $f,g$ such that $f(x+y)=g(x·y)$ for all $x,y$? [closed]

My approach towards this question was that first I putted x=0 and then y=0 which yields f(x)=f(y)and f(x)=g(0), and again for x=1 & y=1 it gives g (x)=f(1+x), and so on. My query is that what ...
2
votes
1answer
89 views

IMO 1993 b2 proof

"Let $\mathbb{N}=\{ 1,2,3,\ldots \}$. Determine if there exists a strictly increasing function $f:\mathbb{N}\to\mathbb{N}$ such that 1) $f(1)=2$ 2) $f(f(n))=f(n)+n$ for all $n$." Of the solutions ...
4
votes
0answers
70 views

Solving the equation $f(x)=f^{-1}(x)$. [duplicate]

Exactly under what conditions would the equality $f(x)=f^{-1}(x)$ hold? The proof attached below considers a special case when $f$ is strictly increasing. The theorem then says that the set of ...
4
votes
2answers
86 views

Functional problems: Find all functions such that $f(x)f(y) = f(xy + 1) + f(x - y) – 2$

Find all functions such that $f(x)f(y) = f(xy + 1) + f(x - y) – 2$ for all $x, y $ are real numbers. I put y=0 into the equation and get $(f(0)-1)f(x)=f(1)-2$. If $f(0)≠1$, then $f(x) = (f(1)-2)/...
0
votes
2answers
56 views

Power-like functional equation

I would like to know what are the especifications of a functional equation that give us a power function as a solution. For example, if $f:\Bbb R \to \Bbb R$ is continuous and monotonic, such that $$...
2
votes
0answers
49 views

Functions satisfying $f(2x)+f(3x)\leq k f(x)$

Given $a<0$ such that $k:=2^{a}+3^{a}\in (0,1)$ and define $f(x):=x^{a}$ for all $x>0$. Then, $f(2x)+f(3x)=x^{a}(2^{a}+3^{a})=kf(x)$. Somebody know another example of a function $f:(0,+\infty)\...
2
votes
1answer
70 views

Find all function $f:\ \Bbb{R}\ \longrightarrow\ \Bbb{R}$ such that : $f(ax)f(by)=f(ax+by)+cxy$ where $a,b,c>0$

If $f:\ \Bbb{R}\ \longrightarrow\ \Bbb{R}$ and $a,b,c>0$, then find all function such that : $$f(ax)f(by)=f(ax+by)+cxy,\quad \text{where } a,b,c>0 \text{ for all } x,y\in \Bbb{R}.$$ My attempt ...
0
votes
0answers
25 views

Additive and Bijective function on the real line

I was trying to solve the following functional equation: $f(f(x-y))=f(x)-f(y)$. And I concluded that $f$ must be additive and bijective. The question is: Let $f:\mathbb{R} \to \mathbb{R}$ be an ...
5
votes
1answer
117 views

Finding f$(x)$ from a given functional equation [on hold]

Find all continuous functions $f:(0,1)\rightarrow\mathbb{R}$ such that $$f\left(\dfrac{x-y}{\ln x-\ln y}\right)=\dfrac{f(x)+f(y)}{2},$$ and $x\neq y$. I tried by solving putting $y=\frac{x}{2}$ but ...
1
vote
2answers
60 views

$f: \mathbb{R} \to \mathbb{R},\space\space\space f(x)-2f(\frac{x}{2})+f(\frac{x}{4})=x^2,\space\space$find $f(3)$ in terms of $f(0)$.

$f: \mathbb{R} \to \mathbb{R},\space\space\space\space f(x)-2f(\frac{x}{2})+f(\frac{x}{4})=x^2,\space\space\space\space$ Find $f(3)$ in terms of $f(0)$. My approach: $$f(x)-2f(\frac{x}{2})+f(\frac{...
2
votes
2answers
90 views

Find all functions $f : \mathbb{(0,\infty})\to\mathbb{R}$ such that $f(x+y) = xf(y)+ yf(x)$

Find all functions $f : \mathbb{(0,\infty)}\to\mathbb{R}$ such that $f(x+y) = xf(y)+ yf(x)$ if $f$ is continuous at $x=1$ This problem was looking quite easy at first but the domain of positive ...
6
votes
1answer
71 views

If $f(x)f(y)+f(xy)\le -\frac{1}{4},\forall x,y\in[0,1)$, show that $f(x)=-\frac{1}{2}$

Let $f:[0,1) \to \mathbb{R}$ be a function such that $$f(x)f(y)+f(xy)\le -\dfrac{1}{4} \quad \forall\, x,y\in[0,1).$$ Show that $$f(x)=-\dfrac{1}{2} \quad \forall\, x \in[0,1).$$ I have proved that ...
2
votes
2answers
60 views

Find $x$ where $f(e^{(x+1)})=x-\ln(x)$ approaches one.

Given that $$ x \in [1,\infty) \quad f(e^{(x+1)})=x-\ln(x) $$ and $$ \lim_{x \to a} f(x)=1 $$ Find $a$. I got to the point: $$ \ln(a)-\ln(\ln(a)-1)=2 $$ But from there on I could not get to $a=e^...
0
votes
0answers
22 views

How to find a monotone increasing function given its relationship with its inverse?

I am trying to find all functions $$f : [0, 1] \rightarrow [0, 1]$$ continuous monotone strictly increasing such that $$f(v) = v - \frac{v}{2} \cdot f^{-1}(\frac{v}{2}).$$ However I have no idea how ...
0
votes
2answers
133 views

Solve functional equation $f(z)=c+zf(z^2)$ with series expansion?

Let the functional equation $(1)$ be given as $$ f(z)=c+zf(z^2) \tag{1}$$ where $c \in\mathbb R$ and $c \neq 0$. How can this functional equation be solved with series expansion (power, Taylor or ...
6
votes
1answer
68 views

How to prove $f$ is $C^\infty$

Suppose $f:U \subseteq \mathbb{R}^2 \rightarrow \mathbb{R}$ is continous and $$(x^2+y^4)f(x,y)+(f(x,y))^3=1 \: \text{for all} \: (x,y) \in U. $$ Prove $f$ is $C^\infty$. This kind of exercise ...
2
votes
2answers
433 views

$2+ f(x)f(y)=f(x)+f(y) +f(xy) $, if $f(2)=5$ find $f(5)$

This question has been asked before, however I am interested in seeing why my approach to finding a solution does not work. $2+ f(x)f(y)=f(x)+f(y) +f(xy) $, if $f(2)=5$ find $f(5)$ What I have ...
0
votes
1answer
41 views

The condition for the existence of a symmetric form for the reflection formula $f(1-x)= \chi (x) f(x)$

Suppose we have a functional equation in the form $$f(1-x)=\chi (x) f(x)$$ with given function $\chi (x)$. What is the condition on the function $\chi (x)$ so that we can write this reflection ...
3
votes
1answer
73 views

Find all real functions such that $(x + 1)f(xf(y)) = xf(y(x + 1))$

Find all real functions of real variable such that $$(x + 1)f(xf(y)) = xf(y(x + 1))$$ Let $a=f(0)$. For $y=0$ we get $(x+1)f(ax) = ax$, so if $a\ne 0$ we get $$f(x) = {ax\over x+a}$$ which is actual ...
0
votes
0answers
19 views

Difficulties with Inner products and polarization identities

I am discussing the general inner product space. Here is what Polarization Identities mean. I denote the inner product by $(x,y)$. I am having a difficult time with the polarization identities. Of ...
6
votes
2answers
179 views

Multivariate polynomial functional equation

I’m having some difficulties solving the following functional equation: Find all polynomials $P(x,y)\in\mathbb{R}[X,Y]$ for which: $P(x,y)$ is homogeneous (so $\exists n\in\mathbb{N}, \...
2
votes
1answer
58 views

Functional equation in single-variable calculus

Suppose we know that $f(x) \in C^2$ and $f(x)$ defined for all real numbers. Furthermore, $f(x)$ has following property: $$ \forall x,y \in \Bbb R \quad f(x+y) - f(x) = yf'(x + \frac{y}{2})$$ How to ...
2
votes
2answers
156 views

An equation of rational functions

I'm trying to get the set of solutions of the following equation, whose unknowns are the rational functions $f$ and $g$ : $\forall x\in\mathbb{R}$ such that the LHS and RHS are both defined, $$f(x)f(...
0
votes
0answers
24 views

Cauchy's functional equation from complex to reals

I am looking for the general continuous solutions $f:D \rightarrow \mathbb R$ of the multiplicative Cauchy functional equation $f(x)f(y) = f(xy)$ for the domain $D=\{x \in\mathbb C: |x|<1\}$. (...
4
votes
3answers
122 views

Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y,z \in \Bbb{R} $ , $f(xf(x)+f(y))=x^2+y$

Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y, \in \Bbb{R} $ , $f(xf(x)+f(y))=x^2+y$ We can easily get a strong condition $f(f(y))=y $ by setting $x=0$ . By this equation we ...