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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Which functions share a certain property of sinusoids?

Among functions $f$ satisfying $\forall x\in\mathbb R\, f(x+p) = f(x)$ with $p>0$ are sinusoids $f(x) = A\sin(\omega x +\varphi)$ with $p=2\pi/\omega.$ These also satisfy the functional equations $$ \forall n\in\{2,3,\ldots\}\, \forall x\in\mathbb R…
Michael Hardy
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Is a function $f$ satisfying $f(x+1) = f(x) + 1$ and $f(x^2) = (f(x))^2$ odd or even?

The problem statement, all variables and given/known data 1) $f(x+1)=f(x)+1$ 2) $f(x^2) =(f(x))^2$ Let a function $f \colon \mathbb{R} \to \mathbb{R}$ satisfy the above statements. Then prove whether the fuction is odd or even. The attempt at a…
maths lover
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Functional equation $f(x+y)=f(x)f(y)-f(xy)+1$

I'm new to solving functional equations and found the following functional equation from a collection of functional equations. Find all functions $f: \mathbb{Q}\to \mathbb{R}$ for $$f(x+y)=f(x)f(y)-f(xy)+1$$ for all $x,y\in \mathbb{Q}$. By…
110112345
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Find $f:f(xf(x)+f(y))=f(x)^2+y$

Find $f:f(xf(x)+f(y))=f(x)^2+y$ Domain and co-domain is real numbers I did the following: Let $s=f(0)$ Then $f(f(y))=s^2+y$ so $f$ is surjective Also, $f(x)=f(y)\implies f(xf(x)+f(y))=f(xf(x)+f(x))\implies x=y$ so $f$ is injective So, $f$ is…
aman
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The Functional Equation $ ( x + y ) \big( f ( x ) - f ( y ) \big) = ( x - y ) f ( x + y ) $, need solution have answer

Find all functions $ f: \mathbb R \to \mathbb R $ such that for all reals $ x $ and $ y $, $$ ( x + y ) \big( f ( x ) - f ( y ) \big) = ( x - y ) f ( x + y ) \text . $$ I actually got the answer by guessing and checking, $ f ( x ) = a x ^ 2 + b x $,…
user829751
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Functions on $ ( 0 , + \infty ) $ satisfying $ f \big( f ( x y ) + 2 x y \big) = 3 x f ( y ) + 3 y f ( x ) $

Find all functions $ f : ( 0 , + \infty ) \to ( 0 , + \infty ) $ such that $$ f \big( f ( x y ) + 2 x y \big) = 3 x f ( y ) + 3 y f ( x ) $$ for all $ x , y \in ( 0 , + \infty ) $. I guess $ f ( x ) = 4 x $, but don't know how to do more. Please,…
prime2357
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functional equation $ 1= f(x)+f \left(\frac x2\right)+\dots+f\left(\frac xN\right) $

Given the equation: $$ 1= f(x)+f\left(\frac x2\right)+f\left(\frac x3\right)+f\left(\frac x4\right) $$ How could I solve it, or the more general equation: $$ 1= f(x)+\left(\frac x2\right)+f\left(\frac x3\right)+f\left(\frac…
Jose Garcia
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$f\circ f$ is linear implies $f$ is linear

I saw a question that asked for the possible values of $f(-1)$ if $(f\circ f)(x)=4x-12$. The solution uses the fact that if the degree of $f$ is $n$, the degree of $f\circ f$ is $n^2$, and concludes that $f$ must be linear i.e. be of the form…
rmdnusr
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The function $f : \mathbb{R} \to \mathbb{R}$ satisfies $f(x) f(y) = f(x + y) + xy$ for all real numbers $x$ and $y.$ Find all possible functions $f.$

I was trying to solve the problem : The function $f : \mathbb{R} \to \mathbb{R}$ satisfies $f(x) f(y) = f(x + y) + xy$ for all real numbers $x$ and $y.$ Find all possible functions $f.$ I started by substituting in $0$ for both, to find that…
Shad0w7
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Finding $f(x)$ of functional equation

I would appreciate if somebody could help me with the following problem: Q: Find all conti-function $f(x)~ (x>0)$ $$xf(x^2)=f(x)$$
Young
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Help needed with the functional equation $f \big(x + y f (x)\big) = f (x) + xf (y)$

Find all functions $f : \mathbb{R} → \mathbb{R}$ such that $$f \big(x + y f (x)\big) = f (x) + xf (y)$$ for all $x, y \in \mathbb{R}.$ Could someone please provide a solution as well as their reasoning and how they reached the solution? I tried…
Aayam Mathur
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An odd function satisfying $g(1-t)+g(1+t)=-t$

I am looking for a continuously diffferentiable odd function $g$ such that $$g(1+t)+g(1-t)=-t$$ for all $t\in\mathbb{R}$. Is this possible?
TCL
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Closed form for functional equation $g(x)=\frac{1}{2}(g(x^2)+g((1-x)^2))$ on $[0,1]$

See bottom for edit to question: Before providing background, my question is if there is a closed-form solution to $$g(x)=\frac{1}{2}(g(x^2)+g((1-x)^2))$$ with $g(0)=g(1)=0$ and $g(1/2)=1$. If this can not be found, can it at least be proven that…
QC_QAOA
  • 11,796
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Continuous Invertible Function Satisfying $ f ( x ) + f ^ { - 1 } ( x ) = x $

While working with inverse functions I come across the following question: Find all functions $ f : \mathbb R \to \mathbb R $, which are continuous and invertible and satisfy the equation $ f ( x ) + f ^ { - 1 } ( x ) = x $, where $ f ^ { - 1 } ( x…
ThomasL
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Functional equation $m(x^y)=m(x)+m(y)$.

Find all functions $m : \mathbb{R}^+ \to \mathbb{R}^+$ such that$$m(x^y)=m(x)+m(y)$$
Max
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