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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1answer
23 views

Fuzzy's Involution Generalization

As a consequence from my personal attempt to generalize the Fuzzy Logic's negation, I've got the following functional equation: \begin{align*} &f(x) + f(\alpha + \beta - x) = 2*f\left(\frac{\alpha+...
2
votes
2answers
88 views

Functional equation $P(X)=P(1-X)$ for polynomials

I have encountered the following problem : Find all polynomials $P$ such as $P(X)=P(1-X)$ on $\mathbb{C}$ and then $\mathbb{R}$. I have found that on $\mathbb{C}$ such polynomials have an even ...
2
votes
3answers
59 views

Proving a function to be a difference

I am concerned with a function $f:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ such that $$f(f(x_1,x_2),f(x_3,x_2))=f(x_1,x_3)$$ for any $x_1,x_2,x_3\in \mathbb{R}$, and $$f(x,0)=x$$ $$f(x,x)=0$$ ...
3
votes
1answer
79 views

Functional inequality $\sum_{1\le i<j\le n}f(x_i+x_j)\ge \frac{n(n-1)}{2}f(a_1x_1+a_2x_2+…+a_nx_n)$

Let $n\in\mathbb N, n\ge 2$. Does there exist a set of non-zero real numbers $a_1, a_2,..,a_n$ with this condition: If function $f: \mathbb R \rightarrow \mathbb R$ satisfies the inequality $$...
11
votes
1answer
707 views

Equation about generating functions and subfactorial

Suppose $G_n(w)$ is a formal power series (really a probability generating function, see the following explanation) of variable $w$, try to solve out $G_n(w)$ for all $n\ge0$ from the formal-power-...
5
votes
2answers
81 views

Increasing function with $f'(x)=f(f(x))$ [duplicate]

Is there a strictly increasing function $f: \mathbb{R}\rightarrow \mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x$?
17
votes
2answers
3k views

$f(a+b)=f(a)+f(b)$ but $f$ is not linear

Can you show me a continuous function $ f \colon \mathbb{R}^n\to\mathbb{R}^m\\ $ that satisfies $f(a+b)=f(a)+f(b)$ but is not linear? We have that $$f(0)=f(0+0)=2f(0)\implies f(0)=0\\ f(x-x)=f(0)=f(...
0
votes
0answers
25 views

Functional differential equation separatrix

I've been spinning my wheels with the following differential equation, and would greatly appreciate any guidance on ways to attack it. I have $u(x) \geq 0$ for all $x$. Further, $x \geq 0$. The ...
2
votes
1answer
108 views

Another functional equation: $f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor$

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
2
votes
1answer
63 views

Find all pairs of functions $(f,g)$, $\forall x, y \in \mathbb{R}, f\left(x(y+1)\right)+y=xf(y)+f\left(x+g(y)\right)$

Find all pairs of functions $f,g: \mathbb R \rightarrow \mathbb R$ such that 1). $f\left(xy(y+1)\right)+y=xf(y)+f\left(x+g(y)\right)$ for all real $x,y$; 2). $f(0)+g(0)=0.$ My work so far:...
35
votes
8answers
3k views

Square root of a function (in the sense of composition)

There are some math quizzes like: find a function $\phi:\mathbb{R}\rightarrow\mathbb{R}$ such that $\phi(\phi(x)) = f(x) \equiv x^2 + 1.$ If such $\phi$ exists (it does in this example), $\phi$ can ...
0
votes
1answer
38 views

Let $f$ be a continuous and positive function on $\mathbb{R}_{+} $ such that $\lim_{x \to \infty} <1$

Let $f$ be a continuous and positive function on $\mathbb{R}_{+}$ such that $\displaystyle\underset{x \to \infty}{\lim} \frac{f(x)}{x} <1$. Prove the equation $$f(x)=x$$ has at least one solution ...
1
vote
1answer
53 views

Functional inequality $f(x_1+x_2)\ge f(x_1)+f(x_2)$

Given a function $f$ on the interval $0\le x \le 1$. We know that this function is non-negative and $f(1)=1$. Moreover, for any two numbers $x_1$ and $x_2$ such that $x_1\ge 0, x_2 \ge 0$ and $x_1+x_2\...
0
votes
3answers
112 views

How to solve this functional equation: $2f(x) = f(x-1)+f(x+1)$? [closed]

After some calculations, I came up with this functional equation: $f(x-1)+f(x+1)=2f(x)$. I found linear function is one possible answer, but don't know how to derive it. I don't know much about the ...
0
votes
1answer
17 views

finding proper coefficient for the two graphs to intersect at one point only

We have two functions such as $y=\ln(x)$ and $y=cx^{1/2}$ and I look for the proper positive coefficient $c$ which satisfies that the graphs of the functions above intersects at only one point. If we ...
2
votes
3answers
47 views

Functions validity.

Why does writing a function differently make it valid for a originally invalid input? $e.g:$ $$f(x) = \frac{1} {(\frac1x+2)(\frac1x-3)} \implies x≠0$$ Which may alternatively be written as: $$f(...
0
votes
2answers
37 views

$(x + y + xy)/2 = f( f^{[-1]}(x) + f^{[-1]}(y) )$

Consider $(x + y + xy)/2 = f( f^{[-1]}(x) + f^{[-1]}(y) )$ Where $f^{[-1}]$ denotes the functional inverse of $f$. How to find $f$ ? How about the more General idea of finding $f$ for a given $g$? ...
2
votes
3answers
103 views

Functional equation involving sine function

Let $f: \Bbb R \to \Bbb R$ be a continuous function such that $\sin x + f(x)= \sqrt 2 f\left( x- \frac {\pi} {4} \right) $. Find $f$. I noticed that a solution for $f$ is the cosine function. I don't ...
0
votes
0answers
24 views

General solution for this PDE?

let $f$ be a function maps $\mathbb{R}^2$ to $\mathbb{R}$. let: $u=f^{(1,0)}(x,y)$ $v=f^{(0,1)}(x,y)$ which are partial derivatives w.r.t the first & second argument of $f$. solve $f(h, \...
1
vote
4answers
82 views

If $f(1-x) + 2f(x) = 3x$, what is $f(0)?$

I did the following: $$f(1-0) + 2f(0) = 3\cdot 0$$ $$f(1) + 2f(0) = 0$$ This reminds me of the equation of the straight line in the plane, then: $$\left< \begin{pmatrix} {1}\\ {2} \end{pmatrix}...
5
votes
2answers
87 views

Functional Equation of iterations

Problem: Let $f : \mathbb{Q} \to \mathbb{Q}$ satisfy $$f(f(f(x)))+2f(f(x))+f(x)=4x$$ and $$f^{2009}(x)=x$$ ($f$ iterated $2009$ times). Prove that $f(x)=x$. This is a contest type problem ...
1
vote
0answers
89 views

Find a non-constant real-analytic function $f(x)$ such that for $x\in\Bbb R,\;f(2^x) = f(4^x + 2^{x+1} + 2) - f(4^x + 1)$

Let $f(x)$ be a non-constant real-analytic function and for real $x$ it satisfies : $f(2^x) = f(4^x + 2^{x+1} + 2) - f(4^x + 1)$ Before you ask if this simplifies by writing $2^x = y$ note that $2^...
7
votes
3answers
150 views

Problem based on $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$

Let $$f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}$$ for all real $x$ and $y$. Also, $f'(0) = -1$ and $f(0)=1$. What is then the value of $f(2)$? I got it as $-1$ by using some algebraic ...
0
votes
1answer
40 views

Converting parametric function into cartesian

I am trying to convert the parametric function $x(t) = a\cdot(t - \sin(t)) + b\cdot\cos\left(\frac{t}{2}\right)$ $y(t) = a\cdot\cos(t) + b\cdot\sin\left(\frac{t}{2}\right)$ into a cartesian form. ...
4
votes
1answer
167 views

How to prove $f(x) = \ln x$ continuous by proving first that $f(x)$ continuous at $1$, and then by using $\ln (xy) = \ln(x) + \ln(y)$. [duplicate]

I have a question concerning the proof of the continuity of $f(x) = \ln x$. I read in a comment by Pedro Tamaroff to ncmathsadist's answer to this question that this can be proved in two steps: ...
0
votes
1answer
9 views

How do I determine periodicity of a function through a system of functional equations?

I was given these equations, $f(k+x) = f(k-x)$, $f(2k+x)= -f(2k-x)$ . k is assumed a constant. I was asked to comment whether $f(x)$ is even or odd. By solving I came to the equation, $f(-x)=-f(x)$,...
10
votes
2answers
155 views

Find all functions $f:\mathbb R \to \mathbb R$ satisfying $xf(y)-yf(x)=f\left( \frac yx\right)$

Find all functions $f:\mathbb R \to \mathbb R$ that satisfy the following equation: $$xf(y)-yf(x)=f\left( \frac yx\right).$$ My work so far If $x=1$ then $f(1)=0$ If $y=1$ then $f\left(\frac1x\...
2
votes
1answer
85 views

How prove this sequences have limts and find this value?

let $f:R\longrightarrow R$,and $f(x)=x-\dfrac{x^2}{2}$,and $$L_{1}(x)=f(x),L_{2}(x)=f(f(x)),L_{3}(x)=f(L_{2}(x)),\cdots,L_{n}(x)=f(L_{n-1}(x))$$ and let $$a_{n}=L_{n}\left(\dfrac{17}{n}\right)$$ ...
4
votes
1answer
49 views

Find all functions differentiable and convex

Find all functions $f:[0, \infty) \rightarrow [0, \infty)$, differentiable and convex, so that $f(0)=0 \tag1$ and $ \ f'(x)\cdot f(f(x))=x, \forall x \tag2$ Obviously, $f(x)=x$ is a solution, ...
-2
votes
1answer
46 views

Two functions with all real numbers questions [closed]

Find all functions $f: \mathbb R \to \mathbb R$ such that, for all real numbers $x$, $$xf(x)+f(1-x)=x^3-x$$ Find all functions $f: \mathbb R \to \mathbb R$ such that, for all real numbers $x$ and $y$, ...
0
votes
1answer
38 views

How to solve Functional Equations

I have a midterm tomorrow and have been able to cover all other topics except this. I don't even have an idea how to start these questions. If someone could give me some tips I would very much ...
3
votes
0answers
67 views

Solving a functional equation $2 f(2x)=f(x)(1+\cos(x))+f(x+\pi)(1-\cos(x))$

I am trying to solve the following functional equation, which appears in some of my physics calculations : $f(x)=\frac{1}{2}\left(f(\frac{x}{2})(1+\cos(\frac{x}{2}))+f(\frac{x}{2}+\pi)(1-\cos(\frac{x}...
31
votes
3answers
1k 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:$f(n)=n^{13}$ Put $...
6
votes
2answers
99 views

Is there a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(f(x))=x+1$?

Is there a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(f(x))=x+1$? If so, can you give an example?
9
votes
3answers
116 views

Mean Value theorem functional equation

I need help solving the following functional equation. Suppose $\,f : \mathbb{R} \to \mathbb{R}$ is differentiable and $$f'\Big(\frac{x + y}{2}\Big) = \frac{f(x) - f(y)}{x -y}$$ holds for all $x \...
0
votes
1answer
45 views

Banach fixed-point theorem for a recursive functional equation

I was asked to prove that the functional equation $$ x(t) = \begin{cases} \frac{1}{2} x(3t) + \frac{1}{2},& 0 \leq t \leq \frac{1}{3}\\\ f(t), & \frac{1}{3} < t\leq \frac{2}{3}\\\ \frac{1}{...
0
votes
2answers
48 views

$f(yf(\frac{x}{y})) = \frac{x^4}{f(y)} $

Solve Functional Equation$$f: \mathbb{R}_+ \to \mathbb{R}_+; \; f(yf(\frac{x}{y})) = \frac{x^4}{f(y)} $$ I'm stuck in the beginning. Any hint will be helpful.
7
votes
3answers
407 views

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

Let $f:\mathbb{R}\to\mathbb{R^*}$ be a function such that $f(x+y)+f(x-y)=2f(x)f(y),\forall x,y\in\mathbb{R}$. Prove that $f(x)=1,\forall x\in\mathbb{R}$. I have managed to prove the following: 1) $f(...
2
votes
2answers
66 views

The 'prime logarithm'

Lately I've been thinking about the functional equation $$f(ab) = f(a) + f(b)$$ but not in the usual sense where continuity or differentiability are assumed. It's clear that $f(1) = 0$, by letting $a =...
2
votes
2answers
113 views

solve equations - find a short piece of a wire

Problem: A piece of wire 20 feet long is cut into two pieces so that the sum of the squares of the lengths of the two pieces is 202 square feet. What is the length, in feet, of the shorter piece of ...
2
votes
4answers
162 views

How to solve $f'(x)=f'(\frac{x}{2})$

How do we solve this given $f'(0)=-1$. It does not look separable. I can integrate both sides but end up with a functional equation with is not helpful.
1
vote
0answers
15 views

Functional extrema and the Euler-Lagrange equation

For a functional of the form: $$S(q)=\int_{t_{1}}^{t_{2}}L(q,\dot{q})dt$$ where $\dot{q}=\frac{\partial q}{\partial t}$ , one finds that extrema are reached (to first order) for the condition : $$...
1
vote
1answer
40 views

Solution of a sublinear elliptic problem.

In a lecture notes, the author showed the problem $\tag{$P$}$ $\begin{cases} -\Delta u = |u|^{q-2}u \textrm{ in } \Omega, \\ u(x)= 0 \textrm{ in } \partial\Omega, \end{cases}$ where $\Omega \...
17
votes
4answers
919 views

very elementary proof of Maxwell's theorem

Maxwell's theorem (after James Clerk Maxwell) says that if a function $f(x_1,\ldots,x_n)$ of $n$ real variables is a product $f_1(x_1)\cdots f_n(x_n)$ and is rotation-invariant in the sense that the ...
1
vote
1answer
23 views

Checking my understanding of Cauchy's functional equation.

Cauchy's functional equation is given as $$f(x+y)=f(x)+f(y)$$ Wikipedia states that the solution to this functional equation with $x\in\mathbb Q$ is $f(x)=cx$, where $c$ is an "arbitrary ...
0
votes
1answer
149 views

Prove an additive function has property f(x)=x

So I am given a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x+y)=f(x)+f(y)$ for all real $x$ and $y$, is continuous at $x=0$, and $f(1)=1$. I need to show that $f(x)=x$ for all real $...
23
votes
3answers
2k views

Given $f(f(x))$ can we find $f(x)$?

Given $f(f(x))=x+2$ does it necessarily follow that $f(x)=x+1$? This question comes from a precalculus algebra student.
-1
votes
1answer
41 views

Solving an equation involving complex numbers.

I tried solving the problem on my own. I would like to know if I have made any mistakes. If I have indeed made a mistake, I would appreciate it if someone corrects it and explains what it is. Also, I ...
4
votes
1answer
754 views

Limit of a function satisfying an inequality

If $f(x)+f(y)\leq f(x+y)$ and $f:\mathbb{R}\to\mathbb{R}$, then can we find $\lim_{x\to 0} \frac {f(x)}{x}$? I am not sure whether the question is correct.Thank you.(I tried this idea: $f(x)=f(x+y-y)\...
0
votes
1answer
29 views

Solving functional equation $\frac{f(s, t)}{f(s, m)} = \frac{1}{1+s(m-t)}$

I want to find a functional equation $f(s,x)$ such that $$\frac{f(s, t)}{f(s, m)} = \frac{1}{1+s(m-t)}$$ If it helps the context I need this in is where $t$ is a member of a set of real number and $m$...