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.

3976 questions
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Is $f(x) = Cx\log x$ the only solution to $f(xy) = xf(y) + yf(x)$?

I was studying $L(x) = x \log x$ function and found that it satisfies the following functional equation for positive $x, y$: $$ f: \mathbb R^+ \to \mathbb R\\ f(xy) = x f(y) + y f(x) $$ I have a feeling that $L(x)$ is the only (up to multiplying by…
uranix
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Functions $f:\mathbb Z\to\mathbb Q$ satisfying $f\left(\frac{x+y}3\right)=\frac{f(x)+f(y)}2$ knowing that $\frac{x+y}3\in\mathbb Z$

What is the solution of the following functional equation? (I must confess it is a headache for me.) Find all the functions $ f : \mathbb Z \to \mathbb Q $ such that $$ f \left( \frac { x + y } 3 \right) = \frac { f ( x ) + f ( y ) } 2 $$ fora ll…
Prof. Julian Cordero Arroyo
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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 the…
John Zimmerman
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What functions satisfy one of these equalities: $x \cdot \Phi(x) = (\Phi \ast \Phi)(x)= \int_{- \infty}^{x} \Phi(t) dt$?

When reading about the unit step or Heaviside function and it's derivative, the ramp function I encountered the following characterisations of the Heaviside function $\mathscr{H}$: $x \cdot \mathscr{H}(x) = \int_{- \infty}^{x} \mathscr{H}(t)…
ViktorStein
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Find all the functions which satisfy the functional equation $f(y+f(x))=f(x)f(y)+f(f(x))+f(y)-xy$.

I was given by a friend of mine the following problem. Find all the functions $f:\mathbb R\to\mathbb R$ which satisfy the following functional equation $$f(y+f(x))=f(x)f(y)+f(f(x))+f(y)-xy.$$ We spent together quite a lot of time trying to solve it.…
guido giuliani
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Finding functions satisfying $f(f(x))+f(x)+x=0$

Find the number of continous functions $f:\Bbb R\to\Bbb R$ which satisfy the following functional equation $$f(f(x))+f(x)+x=0$$
RITIK007
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Funcional equation $f(xyf(x+y))=f(x)+f(y)$

Find all functions $f$ defined in the set of Real Numbers without zero, satysfying equation $$f(xyf(x+y))=f(x)+f(y)$$ For all $x\neq 0, y\neq 0$ and $x+y\neq0$ Thanks Edit: I found out that function $\frac{1}{x}$ is a solution but I dont know how…
TheRlee
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Prove there are no other solutions of functional equation $f(x+y) = \frac{f(x)+f(y)}{1-f(x)f(y)}$

I have the following functional equation. Find all continuous functions $f:(-1,1) \to \mathbb R$ such that $$ f(x+y)=\frac{f(x) + f(y)}{1 - f(x)f(y)} $$ The first obvious solution is $f(x) \equiv 0$. Another one I guessed, it is $f(x) = \pm \tan x$.…
Yauhen Yakimenka
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All functions satisfying $f(x+(f(y))^2)=(f(x+y))^2$.

Let $f: \mathbb{R} \to \mathbb{R} $ be a function such that $f(0)$ is rational and for real numbers $x$ and $y$ $$f \big(x+(f(y))^2 \big)=(f(x+y))^2$$ Find all functions satisfying above conditions. My try: Let $a \in \mathbb{R}$ be a number…
Ghartal
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Functional Equation with Inverse

How do I solve the following functional equation: $$f(x)+12f^{-1}(x)=\frac{1}{x}f(x)$$ I've been doing a lot of functional equations, but I haven't done one yet that has the function and its inverse together. All I've done so far is figure out that…
Franklin Pezzuti Dyer
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$f(x+y) + f( f(x) + f(y) ) = f( f( x+f(y) ) + f( y+f(x) ) )$

Suppose $f\colon \mathbb R\to\mathbb R$ is a strictly decreasing function which satisfies the relation $$f(x+y) + f( f(x) + f(y) ) = f( f( x+f(y) ) + f( y+f(x) ) ) , \quad \forall x , y \in\mathbb R $$ Find a relation between $f( f(x))$ and $x$.
Souvik Dey
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Function with derivative-like property: $f(ab) = af(b) + bf(a)$

Let $f$ be a function $(0;+\infty)\to\mathbb{R}$ with following property: $f(ab) = af(b) + bf(a)$. What can $f$ be? It can be seen that functions $\delta_p(x) = px\space ln(x)$ work. Now, let $f_0$ be a solution and $f_0(x_0)=y_0$, $x_0\neq1$.…
Serge Seredenko
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$f(550)$ if $f(11)=1$ and $f(a)f(b)=f(a+b)+f(a-b)$

$f:\Bbb Z\to\Bbb Z$ is such that $f(11)=1$ and $f(a)f(b)=f(a+b)+f(a-b)$ for all integers $a,b$. What is $f(550)$? I've first set $f(11)f(0)=2f(11)$, which yields $f(0)=0,2$. Yet I know that 0 won't work because that was a hint. I've also heard…
John Rawls
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Find all functions $f(f(f(...(f(x_1,x_2),x_3),...),x_{2016}))=x_1+x_2+...+x_{2016}$

I am trying to solve the functional equation: Find all functions $f:\mathbb R^2\rightarrow \mathbb R$ such that for all $\left \{x_1,x_2,...,x_{2016} \right \}\subset \mathbb R$: …
Roman83
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Find all continuous functions over reals such that $f(x)+f(y) = f(x+y)-xy-1$ for all $x,y \in \mathbb{R}$

Find all continuous functions over reals such that $f(x)+f(y) = f(x+y)-xy-1$ for all $x,y \in \mathbb{R}$. I saw first that $f(0) = -1$ but then I am struggling to see how to get a formula for $f(x)$. If I do $x = 0$ we get $f(0) + f(x) = f(x)-1$…
user19405892
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