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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Do there exist functions satisfying $f(x+y)=f(x)+f(y)$ that aren't linear?

Do there exist functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x+y)=f(x)+f(y),$ but which aren't linear? I bet you they exist, but I can't think of any examples. Furthermore, what hypotheses do we need to put on $f$ before no such…
goblin GONE
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$f(x^2) = xf(x)$ implies that $ f(x) = mx$?

Suppose a function $f : \mathbb{R} \to \mathbb{R} $ satisfies the relation $$f(x^2) = xf(x) \ \ \forall x$$ Does this imply $f$ must be a straight line, $f(x) = mx$? If so, why? If not, are there other such functions?
Simon S
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Functional Equation $f(x+y)=f(x)+f(y)+f(x)f(y)$

I need to find all the continuous functions from $\mathbb R\rightarrow \mathbb R$ such that $f(x+y)=f(x)+f(y)+f(x)f(y)$. I know, what I assume to be, the general way to attempt these problems, but I got stuck and need a bit of help. Here is what I…
user66807
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If $f(1)=1$, then is it true that $f(n)=n$ for all $n \in \mathbb{N}\cup\{0\}$.

Let $f:\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}$ be a function which satisfies $f(x^2+y^2)=f(x)^2+f(y)^2$ for all $x,y \in\mathbb{N}\cup\{0\}$. It' easy to see that $f(0)=0$ and $f(1)=0$ or $f(1)=1$. Suppose let's assume that $f(1)=1$. Then it's…
C.S.
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Solving the Functional Equation $ f \big( x + y f ( x ) \big) = f ( x ) f ( y ) $

I want to find all functions $ f : \mathbb R \to \mathbb R $ satisfying the functional equation $ f \big( x + y f ( x ) \big) = f ( x ) f ( y ) $, for all real numbers $ x $ and $ y $. An interesting fact about the functional equation is a symmetry…
Mohsen Shahriari
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How much does $(f \circ f)(x)=x^2 - x + 1$ determine $f$?

I ran into this little problem somewhere online: If $g(x) = f(f(x)) = x^2 - x + 1$, what is $f(0)$? Plugging first $x=1$ and then $x=0$ into the identity $g(f(x)) = f(g(x))$, it is not hard to see that $f(0) = f(1) = 1$. But that made me wonder:…
svangen
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Solving functional equation using inverses

I am trying to solve $$f(xy)=f\left(\frac{1}{x}\right)+f\left(\frac{1}{y}\right), \text{with }f(0)=0$$ I started by getting $f(1)=2f(1)$ so $f(1)=0$, and then I had $f(x)=f(\frac{1}{x})$. But I do not know how to proceed. How do I solve this?
oiuio
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Find all functions $f$ such that $f(f(x)) - xf(x)=0$

After watching this Michael Penn video, in which Penn found all $f$ such that $f(f(x))+xf(x)=1$, I was inspired to try my hand at solving a similar problem: For $x ∈ ℝ _{≥ 0}$, find all functions $f$ that are continuous in their domain such that…
luftaffe
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Find all the function that satisfy : $f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy$

Find all the function that satisfy : $$f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy$$ I only find $f(0)=0$ but I can't prove $f(x)=2x$
Haruboy15
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A function such that $f(f(n)) = -n$?

This question from somebody's job interview made me puzzled: Design a function f, such that: $f(f(n)) = -n$ , where n is a 32 bit signed integer; you can't use complex numbers arithmetic. If you can't design such a function for the whole range of…
VividD
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2 answers

Solve $f^2(x)=x+f(x+1)$

If the function $f(x)$ is such that $$f^2(x)=x+f(x+1),$$ find a closed-form expression for $f$. I found $$f(x)=\sqrt{x+\sqrt{x+1+\sqrt{x+2+\sqrt{x+3+\cdots}}}}$$ is such an $f$. Does anyone have other solutions? Thank you.
math110
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Functional equation: $R(1/x)/x^2 = R(x) $

The following can be shown without much hassle. Suppose $R$ is a rational function satisfying the following functional equation. \begin{align} \frac{1}{x^2} R\left( \frac{1}{x} \right) = R(x) \qquad \forall \: x \in \mathbb{R} \backslash…
N3buchadnezzar
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2 answers

Factoring x + y

I'm trying to find functions $f(x)$ and $g(y)$ such that $$f(x)\cdot g(y) = x + y$$ I can't seem to find a single solution to this problem. Anything I try becomes of the form $f(x,y) \cdot g(y) or f(x) \cdot g(x,y)$ Here is my work so…
Sidharth Ghoshal
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Characterizing the solutions of the functional equation $ f ( 3 x ) - f ( 2 x ) = f ( 2 x ) - f ( x ) $

Problem: Find all continuous and strictly increasing functions $ f : ( 0 , + \infty ) \to \mathbb R $ with $$ f ( 3 x ) - f ( 2 x ) = f ( 2 x ) - f ( x ) $$ for all $ x > 0 $. A class of solutions is given by $ f ( x ) = a x + b $, where $ a > 0 $…
John
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Is this $f(x) = x+1$ the only solution to this functional equation.

I am considering the problem of finding all functions $f:(0,\infty)\to(0,\infty)$ satisfying the functional equation: $$f\big(xf(y)+f(x)\big) = 2f(x)+xy\text.$$ I have been able to prove the following three results/properties: $f$ is not…
William Crawford
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