Mathematics Stack Exchange
Mathematics Stack Exchange

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).

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation(s) and possibly other conditions; e.g., the goal can be to find all continuous solutions. Solving the equation means finding all functions satisfying the given equation(s) and any additional conditions.This is different from the more common use of the word "equation", where the solutions are numbers. It is also different from the more common use of the word "functional", referring to a mapping from a space into the reals or complexes. For basic questions about functions use more suitable tags like or .

A common technique used in solving functional equations is finding some properties of satisfying functions by substituting variables for certain values in the equation. Proving properties of satisfying functions is also helpful - finding that a function is injective, surjective, involutive, and so on, is often a key step in finding all possible solutions. Other techniques such as exploiting symmetry, considering fixed points, and even using certain properties of domains (e.g. well-ordering) sometimes help.

Some well-known functional equations are:

More information can be found at Wikipedia.

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Functional equation in one variable: $f(x)=\frac{1}{4}f(2x)+\frac{1}{4}f\left(\frac{x}{3}\right)+\frac{1}{2}f\left(\frac{x}{5}\right)$

Find all continuous functions $f$ from $\mathbb R$ to $\mathbb R$, such that $$f(x)=\frac{1}{4}f(2x)+\frac{1}{4}f\left(\frac{x}{3}\right)+\frac{1}{2}f\left(\frac{x}{5}\right)$$ holds for all $x$. It is easy to see that there is a solution of the…
Analytical Garden
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Find all functions such that $af(x-1)+bf(1-x)=cx$, where $a$,$b$ and $c$ are real and $a^2$ is not equal to $b^2$

I tried to find $f(0)$, by taking $x=1$, $$f(0)=\frac{c}{a+b},$$ Then i deduced that $f(x)$ is linear, Taking $x=(u+1), af(u)+bf(-u)=(u+1)c$ Similarly, $af(-u)+bf(u)=f(1-u)c$, adding them $2c/(a+b)=f(u)+f(-u)$ Now take u+1,…
h-squared
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Show that there exits infinitely many functions $f:\mathbb{N}\to\mathbb{N}$

Show that there exits infinitely many functions $f:\mathbb{N}\to\mathbb{N}$ satisfying (a) $f(2)=4;$ (b) $f(mn)=f(m)f(n)$ for every $m,n\in\mathbb{N};$ (c) $f(m)
Eduline
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Functional equation$ f(k-x)\cdot f(x)=f(k)$ with $k>0$

The functional equation $f(k-x) f(x) = f(k)$ is satisfied by the exponential functions $e^{\lambda x}$. I notice when $k=0$ that it also has the solutions $f(x) = \frac{-x+C}{x+C}$. Does it have other "nice" solutions when $k > 0$?
Barry Smith
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Need explanation on a solution of a functional equation

This is a functional equation I encountered recently. I could understand the whole solution, except the part when they said that if $(2^x-k^y)(2^x-l^y)<0$ then the monotonicity is broken. Why is that? Please help me understand. Thanks in advance!
furfur
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Finding $f(x)$ from the functional equation $f(x)=x f\left(\frac{2x+3}{x-2}\right)+3$

Find $f(x)$ if $$f(x)=\begin{cases} x\cdot f\left(\frac{2x+3}{x-2}\right)+3 \ , &\text{if } x\ne 2\\ 0 \ , &\text{if }x=2 \end{cases} $$ No idea how to begin. I just noticed that $f(0)=3.$
Hrackadont
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"Foldable" functions

Suppose $f:2^X\to X$ satisfies $f(x_1,\dots)=f(f(x_1,x_2),x_3,\dots)$. Min, max and sum are three such examples. I've been calling these functions "foldable" because they bear some similarity to that concept from programming, but is there a real…
Xodarap
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Functional Equation $\frac{g(x)}{g(-x)} = r^{2x}$

Having some problems with this functional equation: $\frac{g(x)}{g(-x)} = r^{2x}$ Given from the assignment is that $x \in\mathbb{R}$ and $r > 0$ ($r \in\mathbb{R}$). We are rather confident that $g(x) = r^x p(x)$, where $p(x)$ is an even function,…
Orre Snorre
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Solving functional equations clarification

how exactly did he apply 1.39 to get $$f(x-1)^2 =f(1-x)^2$$
SuperMage1
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types of solutions of the functional equation $f\left(x^2+y^2\right)=f(x)^2+f(y)^2$

Are functions $0$, $\frac{1}{2}$, $x$ and $|x|$ the only continuous solutions of $$f\left(x^2+y^2\right)=f(x)^2+f(y)^2\text?$$ Does the equation have discontinuous solutions? [Edit: This one is solved by @Gae.S. in a comment below.] (own…
tong_nor
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Functional Equality

If I have the following equations: $$a(r)=\int_0^\infty s\ f(rs)\ g(s(1-r))\ ds\\b(s)=\int_0^1s\ f(rs)\ g(s(1-r))\ dr$$ Where $f,\ g>0$, $s\in (0,\infty)$ and $r\in (0,1).$ Is it possible to write $f$ and $g$ in terms of $a$ and $b$?
user617369
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Proving ideal gas equation from Boyle’s, Charles’ and Gay-Lussac’s laws

Assuming the empirical laws by Boyle, Charles and Gay-Lussac, which respectively say that \begin{align} p&\propto f(T,N)\cdot {1\over V}\\ V&\propto g(p,N)\cdot T\\ p&\propto h(V,N)\cdot T\\ \end{align} Questions: From these how to prove that…
Atom
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Find all functions $f: \mathbb{R} \rightarrow \mathbb {R}$ such that $f(x+y)+f(x-y)=2f(x)\cos y$

Find all functions $f: \mathbb{R} \rightarrow \mathbb {R}$ such that $f(x+y)+f(x-y)=2f(x)\cos y$(*) for any x,y real numbers my attempts: for $x=y=\frac{2\pi}{3}$ in the (*): $f(\frac{2\pi}{3})+f(0)=f(\frac{\pi}{3})$(1) now for $x=0$ and…
user439117
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Functional equation with two inputs: $f(x,y)f(y,x)=1$

Some context, I was trying to derive something and stumbled upon $f(x,y)f(y,x)=1$, when $x , y \ne 0$. Could someone suggest some tricks or books that I could apply to these types of questions? By the way, I'm trying to solve for $f(x,y)$ and I'm…
BonInSossusvlei
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Find all functions for $f:\Bbb{N}\to\Bbb{N}$ such that $f\left(m^2+f(n)\right)=f\left(m^2\right) +n$

I would have given my approach but i didnt get anywhere. I just substituted zeroes and got $f(f(n)) =n$ and I'm just lost. Any help would be appreciated
Deveshi Singh
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