# Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).

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Let $X$ be a compact metric space, or just $X=\mathbb T$, the unit circle, if it helps. We consider only continuous, complex-valued functions on $X$. Let $\varepsilon >0$. Is there $\delta > ... 0answers 26 views ### Good online references for solving functional equations? [duplicate] I was wondering what are the main tricks one can try when dealing with a functional equation, like trying values, checking injectivity, surjectivity, bijectivity... In particular, how is one sure ... 2answers 82 views ### Functions satisfying$f(x)f(y)=2f(x+yf(x))$over the positive reals From the IMO shortlist: We denote by$\mathbb{R}^+$the set of all positive real numbers. Find all functions$f: \mathbb R^+\rightarrow\mathbb R^+$which have the property: $$f(x)f(y)=2f(x+yf(x))$$ ... 1answer 496 views ### Solve the functional equation$f(x)f(1/x)=f(x+1/x)+1$,$f(1)=2$, where$f(x)$is a polynomial. Solve the functional equation $$f(x)f(1/x)=f(x+1/x)+1,\ f(1)=2,$$ where$f(x)$is a polynomial. It is easy to check that$f(x)=x+1$is a solution. Are there any other solutions? My attempt is ... 2answers 91 views ### How can I determinine all functions or a function f? Determine all functions$f$in$f(x+1)=2f(x)$, for all$x$in real number. So I let$x$be$x+1$. Then I have$f((x+1)+1)=2f(x+1)$. But since there is a new function$f(x+2)$, I couldn't determine the ... 0answers 42 views ###$f : \mathbb{R} \to \mathbb{R}$,$f \in \mathscr{C}^{0}$. Find all$f \not \equiv 0$satisfying a functional equation This problem is from an AoPS thread post.$\blacksquare~$Problem: Let$f: \mathbb{R} \rightarrow \mathbb{R}$be a continuous function that satisfies the functional relation $$f(x+y)=a^{x y} f(x) f(... 1answer 59 views ### Functional equation in \mathbb{Q}^+ Find every function f:\mathbb{Q}^+\to \mathbb{Q}^+ Such that: f(x+1)=f(x)+1, \forall x\in \mathbb{Q}^+ and f(x^2)=f^2(x), \forall x\in \mathbb{Q}^+ All i have managed doing is showing that f(x)=... 1answer 48 views ### How does WolframAlpha solve this recursion? I have the following recursion:$$x_n=\frac{n-1}{n}x_{n-1}+\frac{1}{n}\left\lfloor\frac{n}{2}\right\rfloor.$$WolframAlpha gives a solution to this recursion as$$x_n=\frac{C_1+\left\lceil\frac{1-n}{2}... 1answer 36 views ### Solve a power tower of function compositions Lets begin with a simple functional equation ;) Find all functions$f: \mathbb{N}\to\mathbb{N}$st.$f(x)=x+1$for all$x \in \mathbb{N}$. I know what you are thinking, "math.SE is for ... 0answers 69 views ### What is a solution to the recurrence relation$f(n) = f(n-1) +f\Big(\left\lfloor \frac{n}{2} \right\rfloor\Big)$? Let$\mathbb{N}=\{1,2,3,\ldots\}$. Find a closed form or an asymptotic form of$f: \mathbb{N} \to \mathbb{N}$, where$f$satisfies$f(1) = 1$and $$f(n) = f(n-1) + f\bigg(\left\lfloor \frac{n}{2} \... 1answer 50 views ### Polynomial in two variables such that P(x,y)=P(x+y,x-y) [closed] P:R\times R \to R\times R is a polynomial with real coefficients. It is given that P\left(x,y\right) = P\left(x+y,x-y\right). Find all such P. 1answer 64 views ### Find all functions f:\Bbb R^+\to\Bbb R^+ s.t. for all x\in \Bbb R^+ the following is valid: f\bigg(\frac{1}{f(x)}\bigg)=\frac{1}{x} Find all functions f:\Bbb R^+\to\Bbb R^+ s.t. for all x\in \Bbb R^+ the following is valid:$$f\bigg(\frac{1}{f(x)}\bigg)=\frac{1}{x}$$I tried to substitute \frac{1}{x} for x and compare ... 0answers 19 views ### Cauchy Functional Equation using Laplace transform? I was just curious. is it possible to solve the Cauchy Functional Equation using Laplace transform or other frequency domain transform? f(t_1+t_2) = f(t_1) + f(t_2) ~~~\Rightarrow~~~f(t)=c\cdot t 2answers 76 views ### If f(2n)=\frac1{f(n)+1} and f(2n+1)=f(n)+1 for all n\in\Bbb N, then find n such that f(n)=14/5. [duplicate] The set \mathbb{N} is the set of nonnegative integers. Let f : \Bbb{N} \rightarrow \Bbb{Q} be defined such that 1.) f(2n) = \dfrac{1}{f(n)+1} for all integers n>0, and 2.) f(2n + 1 ) = f(... 2answers 276 views ### Find all function f:\mathbb{R}^+\to \mathbb{R} if xf(xf(x)-4)-1=4x Find all function f:\mathbb{R}^+\to \mathbb{R} such that for all x\in\mathbb{R}^+ the following is valid:$$xf\big(xf(x)-4\big)-1=4x$$All I could do is: f(x)> {4\over x} for all x so f(... 0answers 34 views ### Functional Equation in the rationals [duplicate] Find all the possible functions f:\mathbb{Q}\to\mathbb{Q} such that f(1)=2 and f(xy)=f(x)f(y)-f(x+y)+1 I managed to find the function for the set of natural number, by putting x=1 and y=n ... 2answers 48 views ### Function satisfying some constraints I am in a hunt for a continuous function Q : [0,1] \to \mathbb R that satisfies the following criteria: Q(0) = 0 Q(1) = 1 Q'(x) \geq 0 for all x \in [0,1] \int_0^1 P(x)Q(x)= 0.7, where ... 3answers 58 views ### Show that the following power series satisfies this functional equation f\left(\frac{2x}{1+x^2}\right)=(1+x^2)\,f(x). Show that the following power series satisfies this functional equation$$f\left(\dfrac{2x}{1+x^2}\right)=(1+x^2)f(x)\,,$$where the series given is$$f(x)= 1+\dfrac{1}{3}x^2+\dfrac{1}{5}x^4+\dfrac{1}{... 1answer 28 views ### Solving simple functional relation Satisfying the boundary conditions $$y(0)=1, y(1)=2$$ What general /particular functions obey $$1) \quad y(x) \,y(x+1)= 2,$$ and $$2)\quad \dfrac{y(x)}{y(x+1)}=\dfrac{1}{2}? \;$$ 0answers 63 views ### Example of$f(x)-f(y)=g(x-y)$we know that for linear function$f(x)-f(y)=f(x-y)$for all$x, y\in D$. Now, I am wondering, if we replace the$f$in the right hand side of the equality by some nonlinear$g$, are there any other ... 3answers 129 views ### if$f(\frac{x+y}{2}) =\frac{f(x)+f(y)}{2}$then find$|f(2)|$[closed] if the following functional equation $$f\bigg(\frac{x+y}{2}\bigg) =\frac{f(x)+f(y)}{2} \quad \text{ holds for all real }~ x ~\text{ and }~ y$$ If$f'(0)$exists and equals to$-1$then find$|f(2)|$.... 1answer 87 views ### Let$f:\mathbb{R}\to(0,\infty)$be a differentiable function. For all$x\in\mathbb{R}f'(x)=f(f(x)).$Then show that such function does not exists [duplicate] What i have done is very small. $$f'(x)=f(f(x))\implies f(f'(x))=f(f(f(x)))$$Now $$f(f(f(x)))=f'(f(x))$$Hence$$f(f'(x))=f'(f(x))$$Now i am blank. What to do for the proof 0answers 102 views ### If$f:\mathbb{R}\setminus\{0,1\}\to\mathbb{R}$satisfies$f(x)+2f\left(\frac 1 x\right)+3f\left(\frac x {x-1}\right)=x$, then$8f(4)=?$Let$f: \Bbb{R}\setminus\{0,1\}\to \Bbb{R}$be a function such that $$f(x)+2f\left(\frac 1 x\right)+3f\left(\frac x {x-1}\right)=x\text{ for all }x\neq 0,1\,.$$ Then,$8f(4)=$? My Attempt. Is it ... 2answers 98 views ### Solving functional equation$f(x)+f(1/x)=f(x)\cdot f(1/x)$Let a function$f(x)$, and$x$not equal to zero be such that:$f(x)+f(1/x)=f(x)\cdot f(1/x)$then$f(x)$is ? I tried differentiating it but did not find any useful outcome. answer given at back of ... 1answer 75 views ### Functional equation for$\eta(s)$following Riemann's$2^{nd}$method. Being \begin{equation*} \eta(s)=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{s}}=\frac{1}{1^{s}}-\frac{1}{2^{s}}+\frac{1}{3^{s}}-\frac{1}{4^{s}}+\cdots \end{equation*} and following Riemann's second ... 4answers 369 views ### Dirac delta solutions I am going through some lecture notes on Fourier transforms (here) and it is stated without proof (example 2.16 on page 29) that the general solution to the equation $$x f(x) = a$$ is given by $$f(x) =... 2answers 83 views ### Solving the functional equation f(x)f(y)=c\,f(\sqrt{x^{2}+y^{2}}) Find all probability density function f:\mathbb{R}\to\mathbb{R} such that there exists a constant c\in\mathbb{R} for which$$f(x)f(y)=c\,f(\sqrt{x^{2}+y^{2}})\text{ for all }x,y\in\mathbb{R}\,.$$... 1answer 47 views ### find function f such that f(x)=xf(x-1) and f(1) = 1 Find function f such that f(x)=xf(x-1) and f(1) = 1. I can prove that there is just one function as f (see Proof1). I know that there exists a pi function \Pi(z) = \int_0^\infty e^{-t} t^z\,... 1answer 92 views ### Function f with f(x_1\cdot x_2)=f(x_1)+f(x_2) that is not \log? Is the log-function the only function that enables the transformation of a product to a sum:$$f(x_1\cdot x_2)=f(x_1)+f(x_2)\,?$$Yes, I can approximate the log function by a Taylor Series, but are ... 0answers 41 views ### Finding the barrier height between two local minimums of free energy [closed] Consider the below functional F=\int_0^L dx [d_x f(x)]^2, with boundary conditions \cos 2 f(0)=\cos 2 f(L), \sin 2 f(0)=\sin 2 f(L). The set of functions f(x)=\frac{n \pi x}{L} (with integer ... 1answer 53 views ### Does f(pq)\times f((1-p)(1-q))=f(p(1-q))\times f(q(1-p)) imply f(pq)=f(p)\times f(q) over [0,1]? Let f(x) be a non-negative Lebesgue measurable (or continuous, or differentiable, or strictly monotone, as needed) function defined on [0,1]. Condition A: f(x) satisfies$$ f(pq)\times f((1-p)(1-... 3answers 61 views ### When the function equation$f(x)f(y)=axy+b$is solvable Assume$a,b$are constants. The question is whether there is a continuous function$f$defined on$\mathbb R$or$\mathbb C$so that $$f(x)f(y)=axy+b$$ Of course, such a function$f$exists if$b=0$... 3answers 93 views ### How do I finish solving$f(x)f(2y)=f(x+4y)$? I'm trying to solve this functional equation: $$f(x)f(2y)=f(x+4y)$$ The first thing I tried was to set$x=y=0$; then I get: $$f(0)f(0)=f(0)$$ which means that either$f(0)=0$or we can divide the ... 1answer 70 views ### Did I do something wrong in solving this functional equation or does it have no solutions? I was trying to solve this functional equation that I found in some papers that were given to me by a friend: $$f:\mathbb{R} \rightarrow \mathbb{R}$$ $$2f(x+y)+6y^3=f(x+2y)+x^3$$ for every$x,y \in \...
Find all functions $f : (0, ∞) → (0, ∞)$ such that $f(x +\frac{1}{y})+ f(y +\frac{1}{z})+ f(z + \frac{1}{x})= 1$ for all $x, y, z > 0$ with $xyz = 1$. Alright so my main question is that i first ...