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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1answer
42 views

Sufficient conditions for this function being linear [duplicate]

Let $f$ be a real-valued function for which, for every real $x,y$: $$f(x+y) = f(x)+f(y)$$ Does this imply that $f$ is a linear function ($f(x)=a\cdot x$)? If $f$ is differentiable, I think the ...
6
votes
1answer
182 views

Transformation of the functional equation $f(x+y)=f(x+1)f(y)$

Is there a way to reduce the following functional equation $$ f(x+y)=f(x+1)f(y),\qquad x,y>0, $$ to the equation $$ f(x+y)=f(x)f(y),\qquad x,y>0, $$ whose solutions are known? Thanks in ...
0
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0answers
30 views

A problem on analysis specifically on functions

Let $f(x)$ be a function from reals to reals obeying the following: $f(x)$ is continuous, $f(0)=1$, and $f(m+n+1)=f(m)+f(n)$. Show that $f(x) =1 +x$ for all real numbers $x$. I am a bit confused on ...
1
vote
1answer
25 views

Understanding of solution for a functional equation.

Problem For all $x,y \in \mathbb{R}$ which is $x^2 \not = y^2$, a function $f$ satisfies the following. $$(x-y)f(x+y) - (x+y)f(x-y) = 4xy(x^2 -y^2)$$ Find the function $f$. Solution Divide ...
10
votes
4answers
212 views

Find all functions f such that $f(f(x))=f(x)+x$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $f(f(x))=f(x)+x, \forall x\in\mathbb{R}$. Find all such functions $f$. Clearly, $f$ is an "one-to-one function". I have tried setting ...
2
votes
3answers
74 views

Prove the equation has unique class of solutions

Find the solutions of equation: $$ x^y + y^x = 1 + xy \quad x,y \in \mathbb{R} \quad x,y >0 $$ My quest First, $x=1$ or $y=1$ gives us obvious solutions, so let's suppose $x \not =1$ and $y \not= ...
1
vote
2answers
40 views

Expressing Math Equations

I'm confused how to express the following expressions in math equations for publication: $x =$ integer part of $y$ $x =$ fraction part of $y$ image $x =$ shifted version of image $y$ left with $z$ ...
0
votes
0answers
90 views

Riemann zeta function, functional equation, what completes this analogy?

What completes this analogy? This: $$\zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s)\;\;\;\;\;\;\;\;\;\;(1)$$ is to: $$\chi(s)=\pi ^{-\frac{s}{2}} \Gamma ...
0
votes
0answers
26 views

$\int_0^t f(x) - x dx = f^{[t]}(0) - t - 1$

Let $t,x$ be nonnegative reals. Let $* ^{[k]}$ denote k th iteration. Find real-analytic $f(x)$ such that $\int_0^t f(x) - x dx = f^{[t]}(0) - t - 1$ Holds. We require analytic iterations. ( $ ...
0
votes
1answer
32 views

modelling trough polynomial equation

Set up a polynomial equation that models each problem below.Then solve the equation, and state the answer to each problem. 1.One dimension of a cube is increased by 1 inch to form a rectangular ...
2
votes
2answers
44 views

Contest problem in functional equations.

Let n be a positive integer with $f(n)= 1! +2! +3!+... +n!$ and P(x), Q(x) be polynomials in $x$ such that $f(n+2)=P(n)f(n+1)+Q(n)f(n)$ for all $n \geq 1$, then which of the options is/are ...
3
votes
1answer
56 views

Is the functional equation for $\zeta (s) \left(1-\frac{1}{3^{s-1}}\right)$ known?

It says in wikipedia that Hardy gave a simple proof of the functional equation for: $$\eta(s)=\zeta (s) \left(1-\frac{1}{2^{s-1}}\right)$$ and that it is: $$\eta(-s) = 2 ...
0
votes
0answers
35 views

Can this relation be made into a functional equation?

I am trying to find the functional equation for this: $$\zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ Therefore I let: $$x_1=\left(1-\frac{1}{n^{s-1}}\right)$$ which I substitute with ...
2
votes
1answer
68 views

Is $f(\sqrt{x}) = \sqrt{f(x)}$ true, where $f(xy) = f(x) + f(y)$ for all positives $x, y \in \mathbb{R}$?

Is $f(\sqrt{x}) = \sqrt{f(x)}$ true, where $f(xy) = f(x) + f(y)$ for all positives $x, y \in \mathbb{R}$? It's obviously false. But the point is that "can it be proved without using the fact that the ...
0
votes
0answers
10 views

Real-valued function with kind of additivity

I am looking for (family of) real-valued functions $f:\mathbb{N}^k \rightarrow \mathbb{R}$ such that for any $i$ one can compute $f(n_1, n_2, \dotsc,n_i+1, \dotsc, n_k)$ knowing only $f(n_1, n_2, ...
3
votes
1answer
48 views

functional with two functions

Find all functions $g:R\rightarrow R$ with the prorepty: There exists a strictly monotonic function $f:R\rightarrow R$ such that $$f(x+y)=f(x)g(y)+f(y),\forall x,y \in R$$ We have one official ...
3
votes
1answer
71 views

Existence of a solution to $f(x) = \int_0^1 k(x,y) f(y) dy$

Let $X = (0,1)\times (0,1)$ with the Lebesgue measure, and $k\colon X \to \mathbb{R}$ be a measurable non-negative function such that $$ \int_0^1 k(x,y) dy = 1$$ for every $x \in (0,1)$. My question ...
3
votes
1answer
86 views

Determine all functions $f:\mathbb{Q}\to\mathbb{Q}$ satisfying the functional equation $f(2f(x) + f(y)) = 2x + y$

Determine all functions $f$ defined on the set of rational numbers that take rational values for which $$f(2f(x) + f(y)) = 2x + y \tag{1}$$ for each x and y. This question is from the 2008 ...
1
vote
1answer
29 views

Finding a Recurrence Relation.

This is from AMC 2015 . For each positive integer n, let S(n) be the number of sequences of length n consisting solely of the letters A and B, with no more than three As in a row and no more than ...
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votes
2answers
17 views

Unit price equation.

having some trouble with an equation.. 50x + 40y = 375 75x + 70y = 621 I know x=2.82 and y=5.85 (I looked it up in the answers), but I have literally no idea how to get there... Thanks in advance.
11
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3answers
164 views

Determine all functions satisfying $f\left ( f(x)^{2}y \right )=x^{3}f(xy)$

Denote by $\mathbb{Q}^{+} $ the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ which satisfy the following equation for all $x,y \in ...
6
votes
0answers
83 views

Pretty Innocuous Functional Equation

If $f$ is a real-valued function with $x,y \in \mathbb{R}$ such that $$f(x+y)=f(x)f(y)$$ then find $f(5)$, given that $f(2)=5$. So, can someone tell me if/where I'm incorrect? This was my approach: ...
5
votes
1answer
63 views

When does $f\left [g\left (x \right ) \right ]=g\left [ f\left ( x \right ) \right ]$ have roots.

I met a interesting equation: $$\sin\left [\cos\left (x \right ) \right ]=\cos\left [\sin\left (x \right ) \right ]$$ (And of course, the equation has no roots). So, Let $f\left ( x \right )$ ...
3
votes
0answers
32 views

What can be said about a function with rotational symmetry of order other than 2?

It is well known that an odd function is a function whose graph has rotational symmetry of order $2$ (about the origin). Suppose the graph of $f:U \to \Bbb{R}$ has rotational symmetry of some higher ...
0
votes
1answer
42 views

Converting equation into Octave / Matlab code and a for loop

I have an array of thousands of values I've only included three groupings as an example below: (amp1=0.2; freq1=3; phase1=1; is an example of one grouping) ...
4
votes
2answers
224 views

Given functional equation $f(x,y)=f(x,y+1)+f(x+1,y)+f(x-1,y)+f(x,y-1)$, show that $f(0,0)=0$

Let $f:\mathbb{Z}\times \mathbb{Z}\to \mathbb{R}$, such that for all $x,y$ $$f(x,y)=f(x,y+1)+f(x+1,y)+f(x-1,y)+f(x,y-1),$$ and if $m,n\in \mathbb{Z},mn\neq 0$,we have $f(2m,2n)=0$. Show that ...
7
votes
1answer
40 views

Find all functions f with the following two properties

Let $f(x): [0, +\infty)\mapsto \mathbb{R}$ be a function such that for one $k\in [0, +\infty)$: $$f^2(x)=k^2+x\cdot f(x+k) \quad \forall x\in \{\;[0, +\infty) : x\geq k\;\}\qquad (1)$$ and ...
1
vote
2answers
47 views

Are there any solutions to $2g(x+y)-g(x-y) =2g(x)g(y)$ with $g(0) \ne 0$?

Are there any solutions to $2g(x+y)-g(x-y) =2g(x)g(y)$ with $g(0) \ne 0$? This came up (by replacing $e^x$ with $g(x)$) in an attempt to generalize this: Solving functional equation gives incorrect ...
2
votes
3answers
47 views

Solving functional equation gives incorrect function

Let $f:\mathbb{R} \to \mathbb{R}$ be a function which satisfies $e^xf(y)+e^yf(x)=2e^{x+y}-e^{x-y}$ for all real x and y. If I place $x=y$, I get $f(x)=e^x-\frac{1}{2}e^{-x}$ which does not satisfy the ...
0
votes
2answers
54 views

Prove that an equation has solution in R

Let $f:\mathbb R\to \mathbb R$, $x\in\mathbb R$ and $$f(x^2 + 3x + 1) = f^2(x) + 3f(x) + 1.$$ Prove that $f(x)=x$ has a solution $\in \mathbb R.$
0
votes
1answer
11 views

rectangular paddock, dimensions, maximise area it encloses

Having trouble trying to work out a question which involves finding a function to graph evidence of the correct answer, any advice would be greatly appreciated. I am struggling with part 'b' a lot, ...
0
votes
1answer
38 views

Solve differential equation

How can we solve (if a closed form expression for f(x) can be found) the following first-order linear differential equation? $$f'(x)=f(x)\cdot (\cos x+\tan x)$$ I have found that one function which ...
5
votes
1answer
61 views

Functional equation: Finding $f(100)$

A polynomial of degree 98 such $f (k)=1/k$ for $k=1,2,3...,98,99$ exists. How to find $f(100)$? What are the possible methods ?
1
vote
0answers
15 views

Criteria when bigger number of functions can be obtained from smaller number

It is known that $$ A_1(x_1, x_2) = \partial \varphi(x_1, x_2)/\partial x_1, $$ $$ A_2(x_1, x_2) = \partial \varphi(x_1, x_2)/\partial x_2 $$ holds if and only if $$ \partial A_1/\partial ...
0
votes
1answer
24 views

Find functions that satisfy this equation.

Give some examples of functions, $F$ and $G$ such that $$x=\sqrt{F(x)+G(x)\sqrt{F(x+n)}}-\sqrt{F(x+n)}.$$ $n$ can be a constant. [Edit]: with $n\gt{0}$
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0answers
23 views

Functional equation + differential equation = way of finding solution?

Question I was wondering about the following: Let's say there is a differential equation whose solution is $f$ And $f$ also satisfies a functional equation. Can anyone construct an (non-trivial) ...
1
vote
1answer
39 views

A functional equation that is equal to 7x

I wish to find all of the functions $f:\mathbb R \to \mathbb R$ such that $$ f(x) + 3 f\left( \frac {x-1}{x} \right) = 7x $$ for all nonzero $x$. I have tried plugging in $\frac{x-1}{x}$, but that ...
1
vote
0answers
41 views

Example of a well defined functional integral?

So I was playing around with the notion of a functional integral. Basically given a set $S$ of functions we can define $$ \int_{f \in S} L(f) $$ As the sum of of every function $f$ evaluated by ...
2
votes
4answers
93 views

Prove that $\lim_\limits{x\to 0}{f(x)}=0$

Let $f:\mathbb{R}\mapsto \mathbb{R}$ be a function such that: $$2\cdot f(x)-\sin(f(x))=x, \forall x\in \mathbb{R}$$ Prove that $\lim_\limits{x\to 0}{f(x)}=0$. I think I need to use the sandwich ...
1
vote
3answers
90 views

What is meaning of this question and how to solve it?

I am stuck with understanding the meaning of the question, which states: Show that $\cos(n\theta)=f_n(\cos\theta)$ for polynomials $f_n(x)$ satisfying $$f_{n+1}(x)=2xf_n(x)-f_{n-1}(x) \tag{1}$$ ...
4
votes
3answers
117 views

Determine all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $xf(y)+yf(x)=(x+y)f(x^2+y^2)$ for all $x,y\in\mathbb{N}$ (contest question)

The question below is from the 2002 Canada National Olympiad. I have found one family of functions but need help in finding (or proving the non-existence) of others. Suggestions on how to improve the ...
5
votes
4answers
402 views

Find a polynomial from an equality

Find all polynomials for which What I have done so far: for $x=8$ we get $p(8)=0$ for $x=1$ we get $p(2)=0$ So there exists a polynomial $p(x) = (x-2)(x-8)q(x)$ This is where I get stuck. How do I ...
5
votes
2answers
127 views

given $2f(x) + f(1-x) = x^2$ find $f(-5)$

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. A function $f$ has property that $2f(x)+ f(1-x) = x^2$ ...
6
votes
2answers
113 views

Finding a function $h$ that satisfies $h \left ( \frac{x}{x^2+h(x)} \right )=1$

Someone gave me a random maths problem to solve: Given that $h \left ( \dfrac{x}{x^2+h(x)} \right )=1$, what is $h(x)$ The restrictions given were: $h(x) \neq constant$ $\exists \frac{dh}{dx}$ ...
0
votes
1answer
24 views

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$ subject to $\int_\mathbb{R} f(x)\,dx = 1$.

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$, where $\alpha$ is a real number, subject to $\int_\mathbb{R} f(x)\,dx = 1$. I have no idea ...
1
vote
2answers
34 views

A smooth function satisfying these functional constraints

I am looking for any function on a square $$f:[-1,1]\times [-1,1] \rightarrow [0,1]$$ with the following properties: The function $f$ is as smooth as possible, e.g. differentiable almost everywhere. ...
2
votes
1answer
41 views

Substitution with functional equations

I've found this nice introduction worksheet that I started to work through with the goal to get a better understanding of functions and finding them in equations. I've gotten so far but in this one ...
2
votes
2answers
108 views

Is a function $f:\mathbb R\to\mathbb R$ such that $f(x+y)=f(x)+f(y)$ always continuous? [duplicate]

Is there a function $f:\mathbb R\to\mathbb R$ such that $f(x+y)=f(x)+f(y)$ which is not continuous? I have proved that if it's continuous in one point $a\in\mathbb R$ then it's continuous on all ...
4
votes
1answer
63 views

If $h(x)=f(g(f(x)))$ is bijective, what do we know about $f,g$?

Question: If $h(x)=f(g(f(x)))$ as a function $\mathbb R \rightarrow \mathbb R$ is bijective, what do we know about $f,g$, which are also functions $\mathbb R \rightarrow \mathbb R$? Is my proof ...
0
votes
2answers
41 views

Solve the functional equation $ q \, \frac{f(x+1)}{f(x)}=\frac{h(x+1)}{h(x)}. $

Let $f(x),h(x)$ be two differentiate on $\mathbb{R}$ functions, $f(0)=h(0)=1$. Solve the functional equation $$ q \, \frac{f(x+1)}{f(x)}=\frac{h(x+1)}{h(x)}, $$ here $q$ is a constant. For ...