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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2
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2answers
138 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
3
votes
1answer
97 views

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
27
votes
7answers
721 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 ...
1
vote
2answers
77 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: ...
3
votes
2answers
94 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 ...
3
votes
2answers
150 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
169 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 : ...
0
votes
1answer
81 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
40 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 ...
3
votes
4answers
395 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
42 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 ...
7
votes
1answer
195 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.
5
votes
3answers
101 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
84 views

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

Hi everybody I've found some difficulties in this exercise please could you give me help: let $f$ continuous function in $\mathbb R$ $$\forall(x;y)\in\mathbb R, f (x+y) + f(x-y) = 2(f(x)+f(y))$$ 1 - ...
0
votes
0answers
19 views

Search Space Function:

given a set of integers: ${x_1, x_2, ... x_n}$ Is is possible to construct a generic function $f$ such that there exists $u_1 .... u_n \in R$ where $f(u_k) = x_k$ and: $$f(x+y) = f(x) + f(y)$$ ...
2
votes
1answer
78 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 ...
1
vote
1answer
65 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
vote
0answers
94 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 ...
3
votes
2answers
218 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
232 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
votes
1answer
61 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
votes
3answers
706 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
91 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
123 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
198 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 ...
5
votes
2answers
322 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
90 views

Mathematical Analysis-Continuous Functions on Intervals

Show that if $f:\Bbb{R} \to \Bbb R$ satisfies the equation $f(x+y) = f(x) + f(y)$ for every $x,y$ that exists in $\Bbb R$, and if $f$ is continuous at (at least) one point, then there exists $c$ such ...
0
votes
1answer
83 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 $c$ ...
2
votes
2answers
97 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 ...
0
votes
0answers
67 views

Equation of a conic-IIT JEE Problem

A chord is drawn to a conic section and its mid point lie on the director circle of the conic which of the following can not be its eccentricity (A) sin2θ (B) tan2θ (C) cosec2θ (D) sec2θ – ...
1
vote
2answers
729 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
votes
1answer
35 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
213 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
56 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 ...
1
vote
2answers
79 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
votes
1answer
91 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
206 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
115 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 ...
2
votes
2answers
92 views

Many other solutions of the Cauchy's Functional Equation

By reading the Cauchy's Functional Euqations 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
103 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 ...
4
votes
1answer
44 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
46 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 ...
1
vote
0answers
51 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 ...
10
votes
4answers
342 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
110 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 $m < n$
9
votes
3answers
221 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
63 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, ...
1
vote
1answer
55 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 ...
0
votes
1answer
146 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.
0
votes
0answers
131 views

Solving an equation with 2 unknowns

I've been trying to solve this problem and was wondering if there is a more accurate / efficient way to do it. For the following equation $$y = a \times \left(1 - ...