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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Help with a functional equation

I am asked to find all $f : \mathbb{R} \to \mathbb{R}$ such that $f(x - f(y)) = f(f(y)) + 2 x f(y) + f(x) - 1$, which I solved and got that $f(x) = 1 - x^2$ -- a correct solution and everything. What ...
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64 views

On a specific non-linear partial differential equation

Given an $n$-dimensional variable $\mathbf{x}\in\mathbb{R}^n$ and the functions $h_i: \mathbb{R}^n \rightarrow \mathbb{R}$, $i=1,\dots,l$, we would like to find a solution of the following equation: ...
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1answer
532 views

Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.

I've recently encountered this strangely attractive equation (Riemann's functional equation), along with Riemann's original proof. $$\displaystyle\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) ...
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How to prove that $f(f(x))=-x$ implies that $f$ is not continuous? [duplicate]

I am trying to prove that: Given an $f:\mathbb{R} \rightarrow \mathbb{R}$, if $f(f(x))=-x$ then $f$ is not continuous? any help? Thank you!
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87 views

A functional equation for homogeneous functions of degree zero

Let a function $F: \textbf{R}^2_{++} \rightarrow \textbf{R}$ satisfy the relation: if $F(x,y)=F(x',y')$ then $F(x,y)=F(x+x',y+y')$. It is easy to prove that under the additional assumption of ...
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1answer
75 views

If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ and $f'(0)$ exists, then $f$ is differentiable on $\mathbb R$

If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ and $f'(0)$ exists, then $f$ is differentiable on $\mathbb R$. I know how to show it is continuous but no clue how to show ...
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Integral Functional Equation

If $f(x) = 1 + \int_{x/c}^1 f(t) dt$, find $f(x)$ in terms of $c$. This problem was inspired by trying to find the expected value of the longest increasing subsequence $a_1, a_2, a_3, \cdots, a_n$ ...
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Solution of equation with special function using maths

I have the following equation with β∈[0,1] and δ∈[0,1] are 2 parameters and $\mu$, $\sigma$ are the mean and standard deviation of a random variable. Is it possible to use software to get the explicit ...
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Is every function $f : \mathbb N \to \mathbb N$ a composition $f = g\circ g$?

True or wrong: For every function $f: \mathbb N \rightarrow \mathbb N$ there is a function $g: \mathbb N \rightarrow \mathbb N$ with $f=g \circ g$.
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Differential Equation for CDF

Consider the following differential equation $$F(cx) = F(x) + x F'(x)$$ for $c>1$. Does this differential equation belong to a some well known class? Is there a way to find all the solutions ...
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4answers
117 views

Solve $\frac{d}{dx}f(x)=f(x-1)$

I am trying to find a function such that $\dfrac{d}{dx}f(x)=f(x-1)$ Is there such function other than $0$ ?
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1answer
77 views

Convert this scenario into algebra equation

Sales for the month minus the VAT @ 20% = (x). 20% of (x) is profit margin (y). 5% of (y) is commission earned (c). How can I write an equation that demonstrates the above please? I.e x - 20% of y ...
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382 views

Condition for an additive function to be continuous

The problem below is Problem 7 from this year's Miklos Schweitzer competition (contest ended Nov 4th). Suppose that $f: \Bbb{R} \to \Bbb{R}$ is an additive function (that is $f(x+y) = f(x)+f(y)$ ...
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199 views

Functional equation: $f\left(\frac{x-1}{x}\right)+ f\left(\frac{1}{1-x}\right)= 2- 2x$

There is a function given $f\left(\dfrac{x-1}{x}\right)+ f\left(\dfrac{1}{1-x}\right)= 2- 2x ,f\colon \Bbb R\setminus\{0,1\}\to \Bbb R$ How many fuction exist? I have no idea how to start
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is this expansion possible (f+g)(t)= f(t)+g(t)?

Hi if the functions were polynomials is (f+g)(t)= f(t)+g(t) possible? I am trying to integrate a function of that form
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878 views

How to find $f$ if $f(f(x))=\frac{x+1}{x+2}$

let $f:\mathbb R\to \mathbb R$,and such $$f(f(x))=\dfrac{x+1}{x+2}$$ Find the $f(x)$ My try I found $f(x)=\dfrac{1}{x+1}$ because when $f(x)=\dfrac{1}{x+1}$,then ...
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2answers
142 views

Formula for solving for Cx and Cy…

I'm trying to create a formula to find the third point in a triangle based on two known points and three known sides. Known Sides: $AB, BC, AC$ Known Points: $A(x, y), B(x, y)$ Unknown Points: ...
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2answers
99 views

Finding every possible $f(x), a\in\mathbb R$ such that ${\{f(x)\}}^2=a+f(x^2)$

Let $f(x)$ be a rational expression of $x$, and let $a$ be a real number. Then, I'm facing difficulty for finding every possible $a, f(x)$ such that $${\{f(x)\}}^2=a+f(x^2).$$ Here, suppose that ...
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177 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). ...
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207 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 : ...
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211 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 ...
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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 ...
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4answers
596 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 ...
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1answer
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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 ...
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2answers
299 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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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 ...
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2answers
236 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 $$\forall(x;y)\in\mathbb R, f (x+y) + f(x-y) = 2(f(x)+f(y)).$$ ...
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1answer
98 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 ...
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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 $(a,b)$.
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1answer
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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 ...
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2answers
232 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 ...
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1answer
265 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\\ ...
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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 ...
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3answers
779 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 $$f(1)-f(0)=[f(0)]^2$$ ...
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2answers
106 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 + ...
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1answer
156 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 ...
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1answer
234 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 ...
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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
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1answer
148 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
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2answers
269 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 ...
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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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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 ...
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1answer
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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 ...
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3answers
289 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 ...
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1answer
64 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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2answers
86 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 ...
3
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1answer
200 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
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2answers
295 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
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1answer
192 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 ...
3
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2answers
172 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 ...