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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196 views

Do there exist functions such that $f(f(x)) = -x$? [duplicate]

I am wondering about this. A well-known class of functions are the "involutive functions" or "involutions", which have that $f(f(x)) = x$, or, equivalently, $f(x) = f^{-1}(x)$ (with $f$ bijective). ...
3
votes
2answers
222 views

Functions minimized at the median of their arguments

I am doing research on problems of location of a public facility on a network which lead me to the following question. Is there an interesting way to characterize the class of functions $f : \mathbb{...
0
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1answer
704 views

Functions and Mapping question?

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying the condition $$2f(x) = f(x + y) + f(x + 2y)$$ for all real numbers $x$ and all non-negative real numbers $y$. I just ...
1
vote
0answers
47 views

regarding a set of integro-PDE

Let say I have a minimal example that contains most of the mathematics I am looking for. The example is as follows: \begin{align} &\frac{\partial u^0}{\partial t}(x,t)+\frac{\partial^2 u^0}{\...
4
votes
4answers
825 views

Let $f(x)$ be a 3rd degree polynomial such that $f(x^2)=0$ has exactly $4$ distinct real roots

Problem : Let $f(x)$ be a 3rd degree polynomial such that $f(x^2)=0$ has exactly four distinct real roots, then which of the following options are correct : (a) $f(x) =0$ has all three real roots ...
2
votes
1answer
61 views

Solving functional equation

Let $a\in\mathbb{C}^*$ with $|a|\not=1$. Let $m\in\mathbb{Z}$. Find all functions $g:\mathbb{C}^*\rightarrow\mathbb{C}^*$ and constants $c\in\mathbb{C}^*$ such that $g(x)=g(a^mx)c^m$. I know one ...
10
votes
2answers
373 views

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$, $f(x)f(yf(x))=f(x+y)$

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$$$f(x)f(yf(x))=f(x+y)$$ A start: set y=0 to get $f(x)f(0)=f(x)$. So $f(0)=1$ unless $f$ is identically zero.
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3answers
120 views

If x and y are different integers , and if $2005 +x =y^2 ; 2005+y =x^2 $ then find xy…

Problem : If $2005 +x =y^2 ; 2005+y =x^2$ then find xy... My approach : Let $2005 +x =y^2 .....(i) ; 2005+y =x^2 ......(ii) $ Now from (i) we get : $ y = \sqrt{x + 2005}$ Now putting this ...
0
votes
2answers
382 views

What method is used to find the expression of a function?

I've found some difficulties in this exercise please could you give me help? Let $f$ be a continuous function in $\mathbb R$ such that $$\forall(x;y)\in\mathbb R, f (x+y) + f(x-y) = 2(f(x)+f(y)).$$ ...
2
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1answer
105 views

Functional/Differential Equation

In the midst of a calculation too long (and too irrelevant) to describe here, I've been forced to confront the following equation: $$1-p-f(f(p))-f(p)f'(f(p))=0$$ Here $f$ is a differentiable ...
2
votes
2answers
97 views

Functional equation, find functions

Find all of the functions defined on the set of integers and receiving the integers value, satisfying the condition: $$f(a+b)^3-f(a)^3-f(b)^3=3f(a)f(b)f(a+b)$$ for each pair of integers $(a,b)$.
1
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1answer
124 views

solving a functional equation using given values

Moderator Note: This is a current contest question on Brilliant.org. The current contest ends on 13 October 2013, after which time this question will be unlocked. A Function $f$ from the positive ...
3
votes
2answers
239 views

How find all $f(x)$ such $f(x\cdot f(y))=\cdots $

Let $k$ be a given real number. Find all the functions $f:\mathbb R\longrightarrow\mathbb R$ such that $$f(x\cdot f(y))=y\cdot f(x)+kxy\,.$$ My try: let $x=y=0$ then $$f(0)=0$$ and $x=1,y=1$, then $$...
3
votes
1answer
281 views

How to identify this power series as $k\sin(k/x)$?

In this question, a functional equation is solved for functions with a power series. We find a recursive formula: (copied from the answer by user achille hui) \begin{align} ( 2^1 - 3 ) a_2 &= 0\\ ...
0
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1answer
66 views

Does $f(\mathbf x _1 + \mathbf c ,…,\mathbf x _n + \mathbf c)=f(\mathbf x _1 ,…,\mathbf x _n)$ imply…

I'm trying to prove the following claim: Let $\mathbf x _1,...,\mathbf x_n\in \mathbf R ^p$ and $f:\mathbf R ^p \times ... \times \mathbf R ^p \ \ \text{(n times!)}\rightarrow \mathbf R.$ Suppose ...
10
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4answers
855 views

How find this function $f(1+xy)=f(x)f(y)+f(x+y)$

let $f:\mathbb R\longrightarrow \mathbb R$,and for any real numbers $x,y$ have $$f(1+xy)-f(x+y)=f(x)f(y)$$ and $f(-1)\neq 0$. Find the $f(x)$ My try:let $x=y=0$,then we have $$f(1)-f(0)=[f(0)]^2$$ ...
6
votes
2answers
120 views

Solution of Equation $ 1 + x + \frac{1\cdot3}{2!}x^2 + \frac{1\cdot3\cdot5}{3!}x^3 + \ldots = \sqrt{2} $

I have an equation whose left side is a infinite series . I can solve the equation if I am able to find a close form of the series . The equation is as follows : $$ 1 + x + \frac{1\cdot3}{2!}x^2 + \...
3
votes
1answer
168 views

There is non-trivial function satisfy the given condition?

Let $f:[0,1]\to\Bbb{R}$ to be a function satisfying that $$ f(x)=\begin{cases} \frac{f(2x)}{2} &\text{if }x<1/2 \\ \frac{f(2x-1)}{2}+\frac{1}{2} & \text{if } x\ge1/2\end{cases} \qquad \...
3
votes
1answer
255 views

Minimum period of function such that $f\left(x+\frac{13}{42}\right)+f(x)=f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right) $

Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1,f(x)\neq constant $ and $$f\left(x+\dfrac{13}{42}\right)+f(...
6
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2answers
733 views

Real Analysis Proofs: Additive Functions

I'm new here and could really use some help please: Let $f$ be an additive function. So for all $x,y \in \mathbb{R}$, $f(x+y) = f(x)+f(y)$. Prove that if there are $M>0$ and $a>0$ such that ...
0
votes
1answer
208 views

Functional equation for scale invariant utility functions

Two utility functions $u,v:\mathbb{R}_{>0}\rightarrow\mathbb{R}$ (giving the utility of, say, an amount of money) are considered equivalent if $u(x)$ is given by $m\,v(x)+c$, for some constants $c$ ...
3
votes
2answers
463 views

Alternatives to percentage for measuring relative difference?

If the value of something changes from $\;a\;$ to $\;b\;$, their relative difference can be expressed as a percentage: $$ \newcommand{\upto}{\mathop\nearrow} (D0) \;\;\; a \upto b \;=\; (b-a)/a \color{...
12
votes
3answers
425 views

Is Zorn's lemma necessary to show discontinuous $f\colon {\mathbb R} \to {\mathbb R}$ satisfying $f(x+y) = f(x) + f(y)$?

A UC Berkeley prelim exam problem asked whether an additive function $f\colon {\mathbb R} \to {\mathbb R}$, i.e. satisfying $f(x + y) = f(x) + f(y)$ must be continuous. The counterexample involved ...
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2answers
2k views

how to solve a third degree equation of complex roots and coefficients

It's not a homework it came in one of our exams and I didn't find anything on the internet that is a high-school level. please give me any hint or answer to solve this in a noncomplicated way. solve ...
0
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1answer
37 views

Assist me to obtain an equation please?

I have a plot which contain large number of points. I want to find an equation that calculates the percentage of a certain number of these points $(x,y)$, the ones having $x>5$ and $y>80$. In ...
3
votes
3answers
346 views

A simple but weird functional equation

Let $f$ be a function $f:\mathbb R\to\mathbb R$. Find all functions $f$ that satisfy: $$f(x^2+x+3)+2f(x^2-3x+5)=x^2-x+ \frac{18}{4} + \frac{111}{444} + \frac{222}{333}$$ Maybe the question is ...
2
votes
1answer
74 views

Find all continous functions satistying $ f(x) = f(x^{2} + \frac{x}{3} + \frac{1}{9})$ for all $x \in \mathbb{R}$

The problem I am trying to solve now is to find all continous functions satistying $f(x) = f(x^{2} + \frac{x}{3} + \frac{1}{9})$ for all $x \in \mathbb{R}$ It is the first time for me to face this ...
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vote
2answers
89 views

when does $f(a)^{f(b)}=f(a^b)$?

First $\text{f}\left( 1 \right)=1$ beacause $\text{f}\left( a \right)^{\text{f}\left( 1 \right)}=\text{f}\left( a \cdot 1 \right)$, and $\log_{\text{f}\left( a \right)} \text{f}\left( a \right)^{\...
3
votes
1answer
314 views

Equation for finding maze solvability

I am programming a game where users can edit the state of a maze. The state of each vertical and horizontal wall (present/not present, on/off, 1/0, etc...) is stored in a database and then referenced ...
4
votes
2answers
375 views

Polynomials $P(x)$ satisfying $P(2x-x^2) = (P(x))^2$

I am looking (to answer a question here) for all polynomials $P(x)$ satisfying the functional equation given in the title. It is not hard to notice (given that one instinctively wants to complete the ...
3
votes
1answer
229 views

A Functional Differential Equation

I was having a play with some trig. identities and noticed the following: $\cos{x}=\frac{\sin{2x}}{2\sin{x}}$ Now, $\cos{x} = \frac{d}{dx}\sin{x}$ so I made the following analogous differential ...
4
votes
2answers
232 views

Many other solutions of the Cauchy's Functional Equation

By reading the Cauchy's Functional Equations on the Wiki, it is said that On the other hand, if no further conditions are imposed on f, then (assuming the axiom of choice) there are infinitely ...
1
vote
1answer
120 views

$f(x)=f(x^2+ 1/4)$ , $f$ is continuous from $\mathbb{R}$ to $\mathbb{R}$

Find all continous functions $f\colon\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(x)=f(x^2+ 1/4)$ What I've tried so far: suppose that $f$ is one-one thus $x=x^2+1/4$ ... $x=1/2$ then $f(x)=f(1/...
4
votes
1answer
68 views

A functional equation over a circle

I am interested in the functional equation $$f(r \cos \phi)+f(r\sin \phi)=f(r),\qquad r\geq 0,\ \ \phi\in[0,\pi/2].$$ Let's assume that $f:[0,\infty)\to\mathbb R$ is monotone. Clearly, $f(x)=ax^2$ is ...
2
votes
2answers
49 views

why is it constant?

$f$ is a function from $\Bbb R$ to $\Bbb R$: $\frac{f(x+y)}{(x+y)} - (x+y)^2 = \frac{f(x-y)}{(x-y)} - (x-y)^2$ for all $x$ and $y$ the solution book just says "Thus $\frac{f(x)}{x} - x^2$ is a ...
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0answers
73 views

How to calculate straight line into graph having variety of different results

How to calculate straight line into graph having variety of different results. What I mean for example let say we have this kind of results (measuring persons weight ...
15
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4answers
435 views

Find all functions $f(x+y)=f(x^{2}+y^{2})$ for positive $x,y$

Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}$ such that for any $x,y\in \mathbb{R}^{+}$ the following holds: $$f(x+y)=f(x^{2}+y^{2}).$$
2
votes
2answers
130 views

Functional equation (show that)

Show that there does not exist a function $f:\mathbb N\to \mathbb N$ which satisfy a) $f(2) = 3$ b) $f(mn) = f(m)\cdot f(n)$ for all $m,n \in \mathbb N$ c) $f(m) < f(n)$ whenever $m < n$
10
votes
3answers
302 views

Functional Equation (no. of solutions): $f(x+y) + f(x-y) = 2f(x) + 2f(y)$

Find all the functions $f\colon\mathbb{Q} \to \mathbb{Q}$ such that $f(x+y) + f(x-y) = 2f(x) + 2f(y)$, for all rationals $x,y$.
2
votes
0answers
73 views

Finding every $n$ such that there exists a $n$-th degree polynomial which satisfies $f(x^2+1)={f(x)}^2+1$ [duplicate]

I'm interested in functional equation. I've been thinking about the following functional equation: $$f(x^2+1)={f(x)}^2+1\ \ \ \cdots(\star).$$ I found several functions such as $f(x)=x, x^2+1, (x^2+1)...
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1answer
71 views

Proof read of functional equations

My teacher gave me this functional equation as an excercise $$f(x+f(y))=x+f(f(y))\,\, \forall\,\, x,y \in \mathbb{R}$$ If $f(2)=8$, calculate $f(2005)$ So my solution was For every $y$, let $f(y)=c$...
0
votes
1answer
152 views

Function $f(x)$ such that $f(x-i)+f(x)=\frac{1}{x^2}$

Help me find a function $f(x)$ such that $$f(x-i)+f(x)=\frac{1}{x^2}$$ where $i$ is the imaginary unit.
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vote
2answers
115 views

Determine the polynomials $p(x)$ satisfying $x\cdot p(x-1) = (x-26)\cdot p(x)$

Determine the polynomials $p(x)$ satisfying $x\cdot p(x-1) = (x-26)\cdot p(x)$. My Solution: Put $x=0$, we get $p(0) = 0$, Similarly put $x=26,$ we get $p(26) = 0$. That means $x=0,26$ are two roots ...
16
votes
6answers
901 views

$g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$.

Find all functions $g:\mathbb{R}\to\mathbb{R}$ with $g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$. I think the solutions are $0, 2, x$. If $g(x)$ is not identically $2$, then $g(0)=0$. I'm trying ...
32
votes
7answers
7k views

Find $f(x)$ such that $f(f(x)) = x^2 - 2$

Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$. Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but the ...
1
vote
0answers
48 views

Does one have to check some additional hypothesis to apply the implicit function theorem for infinite dimensional spaces?

Let $L : B_1 \rightarrow B_1$ be an isomorphism of Banach spaces (i.e., L is a bijective bounded linear operator) and $A: B_1 \rightarrow B_2$ a non linear operator. Consider the following equation ...
4
votes
1answer
283 views

Mean density of the nontrivial zeros of the Riemann zeta function

As part of my MSc I am reviewing a paper. The paper is a review on the statistical distribution of the unfolded zeros (see below) of the Reimann functional equation. In the paper there is a sentence: ...
1
vote
0answers
73 views

A functional power series equation

I am interested in solving the following functional equation: $$(1-w-zw)F(z,w)+zw^nF(z,w^2)=z-zw\,.$$ Here $n\geq2$ is a fixed integer and $F(z,w)$ is a power series in two variables with complex ...
0
votes
1answer
55 views

how to solve this optimization problem with functions?

Suppose $\theta\in[0,1]$ and $s(\cdot)$ is strictly increasing in $\theta$ with $0\leq(0)<s(1)\leq1$ and $s(0)+s(1)>1$ How could I solve the program, $\max_{s(\cdot)\in C[0,1]}=\int_0^{s^{-1}(...
5
votes
2answers
279 views

Iterative roots of sine

Is there an analytical function $f(z)$ such that $f(f(z)) = \sin(z)$? More generally, an analytical function such that f applied $n$ times to $z$ gives $\sin(z)$? Is there a general theory for ...