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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Cyclic Function of Order 5

A common example of a cyclic function of order $3$ is $$g(x) = \frac 1{1-x} $$ because $$g^3(x)=x.$$ Question Is there a similar type (i.e. rational function) of cyclic function which is of order $5$ instead?
Hypergeometricx
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Find all fucntions such that $f:\Bbb R \to \Bbb R$ and $f(x^{2}+yf(x))=f(x+y)$

My work : first take $x=y=0$ we get $f(0)=f(0)=c$ , then take $x=x$ and $y=-x$ implies $f(x^{2}-xf(x))=f(0)$ now as there is another function inside the parenthesis I'm asuming there are some entries for which $x^{2}-xf(x)=0$ then we get $f(x)=x$.…
Toneri Otsutsuki
  • 95
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How to find $f(x)$ such that $f(y)-f(x)-(y-x)f'(\frac{x+y}{2})+\frac{(x-y)^2}{4}=0$

Given $f(x)$ to be differentiable such that $f(y)-f(x)-(y-x)f'(\frac{x+y}{2})+\frac{(x-y)^2}{4}=0$, then find $f(x)$ From the given equation, we can't plug in $x=y$ as it just gives $0=0$. For some reason I think $f(x)$ is a quadratic equation ( I…
user635988
  • 858
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find a function that describes win percentage based on skill

I'm searching for a function $C:D\to[0,1]$ (where $D=[0,1]^2\setminus\{(0,0),(1,1)\}$) that can describe the outcome of a match between two players, $p_1$ and $p_2$ with skills of $s_1$ and $s_2$. the function would determine the chance p1 would…
arad ziv
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How to find all functions $f:\mathbb R\to\mathbb R$ such that $\forall a,b\in\mathbb R$: $f(a)+f\big(a+f(b)\big)=b+f\big(f(a)+f^2(b)\big)$

Find all functions $ f : \mathbb R \to \mathbb R $ such that for all $ a , b \in \mathbb R$: $$ f ( a ) + f \big( a + f ( b ) \big) = b + f \big( f ( a ) + f ^ 2 ( b ) \big) \text . $$ Here, for any $ n \in \mathbb N $, $ f ^ n $ denotes the $ n…
mathcactus
  • 247
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Functions which satisfy $f(n)+2f(f(n))=3n+5$

I need to find all functions $f:\mathbb N^*\to\mathbb N^*$ that satisfy the relation: $$f(n)+2f(f(n))=3n+5.$$ Here $\mathbb N^*$ denotes the set of positive integers. I have calculated that $f(1)=2$, $f(2)=3$, $f(3)=4$. Observing from this…
Henry Cai
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linear functional-equation $ \,2f\left(x+1\right)=f\left(x\right)+f\left(2x\right) $

I'm looking for all functions : $ \ R\rightarrow R\ $ satisfying: $ \,2f\left(x+1\right)=f\left(x\right)+f\left(2x\right) $
lebey
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If $f(rx)=r^{\alpha} f(x)$, then which of these options is true?

I am trying questions of previous year real analysis quiz and I was unable to solve this particular question . Let $f$ be a real valued function on $\mathbb R^{3}$ satisfying (for a fixed $\alpha$ belonging to $\mathbb{R}$) , $f(rx)=r^{\alpha}…
user775699
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Functional equations on $ \mathbb Q ^ + $: $ f ( x + 1 ) = f ( x ) + 1 $ and $ f \left( x ^ 2 \right) = f ^ 2 ( x ) $

Find every function $ f : \mathbb Q ^ + \to \mathbb Q ^ + $ such that $$ f ( x + 1 ) = f ( x ) + 1 , \forall x \in \mathbb Q ^ + $$ and $$ f \left( x ^ 2 \right) = f ^ 2 ( x ) , \forall x \in \mathbb Q ^ + \text . $$ Here, $ f ^ 2 ( x ) = f ( x ) ^…
Nikos127
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Is $f(x) = mx + c$ the only set of solutions to $f(x + 1) - f(x) = \text{constant}$ where $m$, $c$, and $x$ are integers?

Is $f(x) = mx + c$ the only set of solutions to $$f(x + 1) - f(x) = \text{constant}$$ where $m$, $c$, and $x$ are integers? I was watching this video in Youtube (https://www.youtube.com/watch?v=uJqbHaFqjmI) and in the video he solves for the…
quantumbracket
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Help working with the functional equation $f(a) f(b) = f(a+b) + ab$

I have looked at some of the other problems on here, but am stuck. Here is the problem: Consider $f: \mathbb{R} \to \mathbb{R} $ such that $f(a) f(b) = f(a+b) + ab$ for all real $a$ and $b$: find all possible $f$. I tried: Let $a=b=0$ to get…
progstudies213
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2 answers

What are the continuous functions $ x f(y)+y f(x)=(x+y) f(x) f(y) ? $

question - What are the continuous functions on $\mathbb{R}$ which are solutions of the equation $$ x f(y)+y f(x)=(x+y) f(x) f(y) ? $$ my try - by putting $y=x$ i get $f(x)=0$ or $1$ for all $x$ not equal to $0$... now my answer is same as mention…
Ishan
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Solving functional equation $f(x)=3f(x+1)-3f(x+2)$

It is given that a function f(x) satisfy: $$f(x)=3f(x+1)-3f(x+2)\quad \text{ and } \quad f(3)=3^{1000}$$ then find value of $f(2019)$. I further wanted to ask that is there some general method to solve such equation. The method that I know to solve…
Smit Patel
  • 324
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Solve this equation: $\frac{f(x)}{\int_{0}^{x_1} f(x)dx}=\frac{g(x)}{\int_{0}^{x_1} g(x)dx}$

Find $f(x)$ and $g(x)$ or some relations between the two functions knowing that $$ \frac{f(x)}{\int_0^{x_1} f(x)dx} = \frac{g(x)}{\int_0^{x_1} g(x)dx}, \quad \forall x \in (0, x_1), \ \forall x_1 \in (0, x_{\max}). $$ One solution is $g(x)=a…
simplex
  • 121
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Easy functional equation: $ f \big( 2 f ( x ) + f ( y ) \big) = 2 x + f ( y ) $

Find all functions $ f : \mathbb R \to \mathbb R $ such that: $$ f \big( 2 f ( x ) + f ( y ) \big) = 2 x + f ( y ) \qquad \forall x , y \in \mathbb R \text . $$ If you put $ x = y = 0 $, you get $ f \big( 3 f ( 0 ) \big) = f ( 0 ) $. What…
John Marty
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