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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Solving a functional equation: $f\left(x^{f(y)}\right)=f(x)^{y}$ for all positive $x$ and $y$.

Question: determine all functions $ f : \mathbb R ^ + \to \mathbb R ^ + $, such that: $$ f \left( x ^ { f ( y ) } \right) = f ( x ) ^ y $$ for all positive numbers $ x $ and $ y $. It's easy to see that $ f ( 1 ) = 1 $, because letting $ x = 1 $…
Liuhua Peng
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function $f$ has the properties that $f(1) = 6$

I am not sure what exactly the question wants me to do: The function $f$ has the properties that $f(1) = 6$ and $f(2x + 1) = 3f(x)$ for every integer $x$. What is the value of $f(63)$? What exactly does it mean when $f(1) = 6$? and what is its…
RudyGoburt
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Indeterminate equation and functional equation

I was wondering what differences and relations are between indeterminate equation and functional equation? Are they the same concept? Thanks and regards!
Tim
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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 $$2)\quad \dfrac{y(x)}{y(x+1)}=\dfrac{1}{2}? \;$$
Narasimham
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General solutions for a functional equation

I wonder if we can show the existence of solutions other than $f(x)=x$ for the functional equation $$f(x+1)=f(x)+1,\text{ }f(x^3)=(f(x))^3$$ where f is a real-valued function on the real line.
Taro
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Cauchy's Problem

I am looking at the Cauchy's functional equation here: http://en.wikipedia.org/wiki/Cauchy's_functional_equation. Could someone help me on how to generalize the Cauchy's equation to $x \in \mathbb{R}$? I know all the steps leading to the proof…
pseudo2013
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Does a functional equation $f : R → R$ include all solutions of $f : N → N$

If I have a functional equation defined in $f : N → N$ and I have to show that there are no solutions. If I show it for $f : R → R$, does it directly imply there is no solution in $f : N → N$? Question part 2: even if I would substitute 0 for some…
Csaba Daniel Farkaš
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If a function is additive on the interval $(-1/3, 1/3)$ does it follow it is linear?

Suppose that $h:(-\frac{1}{3}, \frac{1}{3})\to \mathbb R$, $h(x+y)=h(x)+h(y)$ for all $x,y\in (-\frac{1}{6},\frac{1}{6})$ and the function is bounded. Does it follow that $h(x)=x\cdot c$? I know that this is true if $h$ would have been defined over…
furfur
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Proving surjectivity from $f\Bigl(\bigl(x+f(x)\bigr)^2\Bigr) = \bigl(x+f(x)\bigr)^2$ in a functional equation problem

Can anyone help me with this? If, in a functional equation problem, I do some substitutions and found out that $f\Bigl(\bigl(x+f(x)\bigr)^2\Bigr) = \bigl(x+f(x)\bigr)^2$, in which $f(x)$ is a function that maps a real number to a real number…
Vann
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Find all $f$ such that $ 2 f\left(m^{2}+n^{2}\right)=f(m)^{2}+f(n)^{2} $

question - Find all $f: \mathbb{N}_{0} \rightarrow \mathbb{N}_{0}$ which obey the functional equation $$ 2 f\left(m^{2}+n^{2}\right)=f(m)^{2}+f(n)^{2} $$ for all nonnegative integers $m, n$ my attempt - i showed that $f(0)=0$ or $1$ by taking…
Ishan
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Functional equations Question from Olympiad book

question - Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function such that (a) $f(m)
Ishan
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Find all continuous functions $ f(x+y)=\frac{f(x)+f(y)+2 f(x) f(y)}{1-f(x) f(y)} $

question - Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy the equation $$ f(x+y)=\frac{f(x)+f(y)+2 f(x) f(y)}{1-f(x) f(y)} $$ for all $x, y$ my try - i proved that f(0)=0 ..then hint says substitute $g(x)=f(x)…
Ishan
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Functional equation with delay: $f(t)\pm f(t-\tau)=g$

Are there known results on functional equations of the type: Given $\tau>0$ and $g$ (real numbers), find a continuous function $f$ such that $f(t)-f(t-\tau)=g$ or $f(t)+f(t-\tau)=g$ (these are distinct equations)? For the second case, the constant…
pluton
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real-valued functional equations on a closed interval

Let $a,b \in \mathbb{R}$ with $a
AB Balbuena
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Solving multivariable functional equation

Solve this functional equation: $$f(s,t)=4f(s,u)f(u,t)-f(s,u)-f(u,t)+\frac{1}{2}, \ \ \mbox{for any} \ \ 0 \leq s
VIVID
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