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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Sequence of *compact* operators that converges to a bounded linear operator $K_{\lambda}$

Proposition: Fix $1\leq p\leq \infty$ and a bounded sequence of real number $\lambda=(\lambda_i)_{i\in\mathbb{N}}$ and $e_i=\delta_{ij}\in l^p$. Defined the bounded linear operator \begin{equation} ...
5
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1answer
106 views

Find all solutions to $f\left(x^2+xf(y)\right)=xf(x+y)$

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^2+xf(y)\right)=xf(x+y)$$ for all $x,y\in\mathbb{R}$. This is somewhat related to this question, but with an $xf(y)$ term instead ...
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62 views

The Bi-linear Functional Equation $\sum_{i=1}^{n}f_i(x)g_i(y)=0$

Consider the following so-called bi-linear functional equation $$\sum_{i=1}^{n}f_i(x)g_i(y)=f_1(x)g_1(y)+f_2(x)g_2(y)+\cdot\cdot\cdot+f_n(x)g_n(y)=0 \tag{1}$$ where $f_i(x)$ and $g_i(y)$ are ...
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2answers
65 views

How can I know how many real roots this equation has?

How many real roots does $2 \sin x-x=0$ have?
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52 views

Does there exist a multiplicative $f:\mathbb{Q}^+\to\mathbb{Q}^+$ such that $f\neq x\mapsto x^a$ for all $a$?

If we consider the functional equation: $f:\mathbb{Q}^+\to\mathbb{R}$ such that $$ f(xy)=f(x)f(y) $$ for all $x,y\in\mathbb{Q}^+$ I think, I have constructed a solution which is not of the form $x\...
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1answer
73 views

What is the value $f(-4)$ in the under function such that $f(x)+f(\frac1x)=\frac{x^2-12x+1}{2x}.$

Let $f$ is a function such that $$f(x)+f(\frac{1}{x})=\dfrac{x^2-12x+1}{2x}.$$ Then what is the value $f(-4)=$?
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17 views

Modify position with variable ratio

I have a line that represents an axis, on the computer's screen, it has a model domain and a screen domain. For example the model domain can be [2, 10], and the screen domain [50, 400]. I need to ...
2
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5answers
138 views

Solve functional equation $f(x^4+y)=x^3f(x)+f(y)$

I need help solving this equation, please: $$f(x^4+y)=x^3f(x)+f(y),$$ such that $ f:\Bbb{R}\rightarrow \Bbb{R},$ and differentiable I've found that $f(0)=0$ and $f(y+1)=f(1)+f(y)$, but I couldn't ...
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5answers
48 views

Inverse Equation of the Given Equation

Having a bit of a problem getting the inverse of the following equation: $$f(x) = \sqrt{9-x^2}$$ I had an answer which was equal to $3-x$ but when I used sites like Mathway and Wolfram to check my ...
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28 views

Finding functions that give nice solutions to a recurrence relation.

In a recent problem I was working through, I came across the following recurrence relation: $$ \text{K}_1(x,\ t;\ g) = 1,\quad x, t\in\Bbb{R}\quad g\in\text{C}^1 \\ \text{K}_{n+1}(x,\ t;\ g) = g'(t)\...
4
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3answers
122 views

Solving for a function

How can I find a general solution to following equation, $$ f\left(\frac{1}{y}\right)=y^2 f(y). $$ I know that $f(y) = \frac{1}{1 + y^2}$ is a solution but are there more? Is there a general technique ...
4
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4answers
119 views

$f(x)$ is a quadratic polynomial with $f(0)\neq 0$ and $f(f(x)+x)=f(x)(x^2+4x-7)$

$f(x)$ is a quadratic polynomial with $f(0) \neq 0$ and $$f(f(x)+x)=f(x)(x^2+4x-7)$$ It is given that the remainder when $f(x)$ is divided by $(x-1)$ is $3$. Find the remainder when $f(x)$ ...
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24 views

Solving for $f_2(x, y)$ in $\int_{\mathbb{R}}f_1(x, y) f_2(x, y) dx = g(y)$

Let me first present the problem in its most general form, and then work down to a specific case of interest and my motivation. Suppose $f_1: \mathbb{R}^2 \rightarrow \mathbb{R}_{\ge 0}$ and $g: \...
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1answer
40 views

How to prove that the following two sets are equal?

Let $M=\{x|f(x)=x\},~N=\{x|f(f(x))=x\}$, and $f(x)=x^2+ax+b$, where $a,b\in R$ are such that $4b=(a-1)^2$. Show that $M=N$.
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31 views

What are some standard methods for solving functional equations?

I have searched the internet for methods on solving functional equations, unfortunately, most of them consist mainly of substituting values for the variables or on guessing solutions. I think those ...
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2answers
71 views

Find $f(x)$ satisfying the functional equation $x^2{f(x)} +f(1- x) =2x -x^4$

A function $f(x)$ satisfies the functional equation $x^2{f(x)} +f(1- x) =2x -x^4$ for all real $x$. Then $f(x)$ is given by. My work $$x^2{f(x)} +f(1- x) =2x -x^4$$ Replacing $x$ $by$ $1- x$ $$(1-...
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1answer
73 views

Functional Equation: $f \left(x+\cos(2017y) \right)=f(x)+2017\cos\left(f(y)\right)$

Find all functions $f: \mathbb R \rightarrow \mathbb R$, such that for all $x,y \in \mathbb R$ satisfies the equation: $$f \left(x+\cos(2017y) \right)=f(x)+2017\cos\left(f(y)\right)$$ My work so ...
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2answers
261 views

Determine all functions $f$ on $\mathbb R$ such that $f(x^2+yf(x))=f(x)f(x+y)$ for all $x,y$

Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that $$f(x^2+yf(x))=f(x)f(x+y). $$ for all $x,y$ real numbers. I think that the only three solutions are: $f(x)=0$, $f(x)=1$ and $f(x)=x$...
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About periodicity of $f(\frac{m}{n})=\frac{3m-1}{2n+1}$ when $\frac{m}{n}$ is reduced form.

Consider a function $f\colon\mathbb{Q}_{>0}\longrightarrow\mathbb{Q}_{>0}$ such that $f(x)=\frac{3m-1}{2n+1}$ where $x=\frac{m}{n}$ and $\frac{m}{n}$ is reduced form. (i.e., $\gcd(n,m)=1$ and $...
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1answer
85 views

The function $f$ has property: $f(x)+f(1/x)=x$ and what is the largest set of real numbers?

Problem. $f(x)+f\left(\frac{1}{x}\right)=x$ and we need to figure out the largest set of real numbers that can be the domain of $f$. My steps: Step 1. I substituted $x$ with $1/x$ to replace the ...
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1answer
185 views

Function that is both midpoint convex and concave

Which functions $f:\mathbb{R} \to \mathbb{R}$ do satisfy $$f\left(\frac{x+y}{2}\right) = \frac{f(x)+f(y)}{2}$$ for all $x,y \in \mathbb{R}$ I think the only ones are of type $f(x) = c$ for some ...
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19 views

functional type equation

Let $f,g$ be two nonconstant positive functions on $I=[0,1]$ and we assume that : $$ \sup_{x\in I}\sqrt{f^2(x)+g^2(x)}=\sqrt{\sup_{x\in I}f^2(x)+\sup_{x\in I}g^2(x)} $$ This implies that $(f,g)$ are ...
0
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0answers
31 views

Can I calculate a fractional sum with functional equations and/or infinite series?

I was wondering if I can calculate fractional sums (non-integer sums) with functional equations in the following manner $$f(x):=\sum_{n=1}^xg(n)$$ $$f(x)=g(x)+f(x-1)\tag1$$ $(1)$ most certainly ...
4
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74 views

I am looking for a mathematical equation to warp an image [closed]

Theoretically, I know that to warp an image, each pixel $(x,y)$ in the source image is transformed to $(x', y')$ using a function f (i.e. $x'=f(x,y)$ & $y'=f(x,y)$ ). But what mathematical ...
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1answer
31 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+...
5
votes
2answers
98 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 $x$?
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3answers
65 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
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1answer
83 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 $$...
2
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1answer
67 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:...
2
votes
2answers
107 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 ...
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1answer
40 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 ...
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1answer
54 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\...
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3answers
114 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 ...
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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
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3answers
55 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(...
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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, \...
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4answers
88 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
90 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 ...
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2answers
39 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 $g$? ...
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1answer
41 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. ...
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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
162 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
3answers
106 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 ...
4
votes
1answer
169 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: ...
4
votes
1answer
50 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, ...
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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 $y$, ...
6
votes
2answers
107 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?
3
votes
0answers
69 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}...
0
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1answer
41 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 ...
0
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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}{...