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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Solutions of $f(x+y^{n})=f(x)+[f(y)]^{n}$.

Consider the functional equation $f(x+y^{n})=f(x)+[f(y)]^{n}$ where $f:\mathbb R \to \mathbb R$ and $n$ is given integer $>1$. This equation was discussed yesterday and it was shown that $f$ is necessarily additive. Assuming continuity it was…
Kavi Rama Murthy
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Finding all functions $ f : \mathbb R \to \mathbb R $ satisfying $ f ( x ) f ( y ) + f ( x y ) + f ( x ) + f ( y ) = f ( x + y ) + 2 x y $

Find all functions $ f : \mathbb R \to \mathbb R $ satisfying $$ f ( x ) f ( y ) + f ( x y ) + f ( x ) + f ( y ) = f ( x + y ) + 2 x y $$ for all $ x , y \in \mathbb R $. I tried the standard way: $ x = 0 $, $ x = y $, $ x = 1 $, $ \dots $, but…
user548665
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$f\circ g(x)$ and $g\circ f(x)$ are given; find $f$ and $g$

If $(f\circ g) (x)=(\sin\sqrt{x})^2$ and $(g\circ f) (x)=\lvert\sin x\rvert$, find $f$ and $g$. My progress: By trial and error I got one such pair $f(x) = \sin^2x$, $g(x)=\sqrt{x}$. But I can’t conclude that this is the only such pair or whether…
Makar
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Solving functional equation $f(4x)-f(3x)=2x$

Given that $f(4x)-f(3x)=2x$ and that $f:\mathbb{R}\rightarrow\mathbb{R}$ is an increasing function, find $f(x)$. My thoughts so far: subtituting $\frac{3}{4}x$, $\left(\frac{3}{4}\right)^2x$, $\left(\frac{3}{4}\right)^3x$, $\ldots$, we get…
user263286
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Solve the functional equation $\frac{f(x)}{f(y)}=f\left( \frac{x-y}{f(y)} \right)$

Solve the functional equation $$ \frac{f(x)}{f(y)}=f\left( \frac{x-y}{f(y)} \right), $$ here $f: \mathbb{R} \to \mathbb{R}$ and $f$ is differentiable at $x=0.$ By set $x=y$ we get $f(0)=1$. Differentiate $$ \frac{f'(x)}{f(y)}=f'\left(…
Leox
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Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions...)

My lecturer was talking today (in the context of probability, more specifically Kolmogorov's axioms) about the additive property of functions, namely that: $$f(x+y) = f(x) + f(y)$$ I've been trying to find what functions satisfy this. Intuition says…
tom
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How find this all function $f(x^n+2f(y))=(f(x))^n+y+f(y)$

Question: Given a positive integer $n\ge 2$ . Find all functions $f:R\to R$, such that $$f(x^n+2f(y))=(f(x))^n+y+f(y)$$ let $x=0,y=0,a=f(0)$ then $$f(2f(0))=(f(0))^n+0+f(0)\Longrightarrow f(2a)=a^n+a=[a^{n-1}+1]a$$ if let $y\to…
math110
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a continuous function satisfying $f(f(f(x)))=-x$ other than $f(x)=-x$

Is there a non-trivial solution to the functional equation $f(f(f(x)))=-x$ where $f$ is a continuous function defined on $\mathbb{R}$ ? Also, what about the general one $f^n(x)=-x$ where $f^n$ is $f$ composed with itself $n$ times and $n$ is odd.…
Leyv Zahav
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Solution for exponential function's functional equation by using a definition of derivative

let $f(0)=1$ and $f'(0)=1$. and $f(x+y)=f(x)f(y)$ for $x,y\in R$. How can I found $f(x)$ by using a definition of derivative?
Xiang
  • 677
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Find all real to real function satisfy this functional equation.! $f((x+y)/(x-y))=[f(x)+f(y)]/[f(x)-f(y)]$

Find all real to real function satisfy this functional equation.! $$f\left(\frac {x+y}{x-y}\right)=\frac {f(x)+f(y)}{f(x)-f(y)}$$ I couldn't get to the final answer but I get $f(0) = 0$ and $f(1) = 1$ and this functional is an odd function
maths lover
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Existence of a function

I came across this question Does there exist a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $f(x+y)>f(x)(1+yf(x))$ and $x,y\in \mathbb{R}^+$ and I didn't know how to begin on it.
picakhu
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Functional Equation $f(x + f(x+y)) = x + f(f(x) + y)$

How can I solve the functional equation $$f(x + f(x+y)) = x + f(f(x) + y)$$ where $f : \mathbb{R} \to \mathbb{R}$? I immediately see that the result must be of the form $f(x) = x + C = x + f(0)$ I also noticed that I can use the following: $$y =…
pupli
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find functions f such that $f(f(x))=xf(x)+1$,

let $f:R\longrightarrow R$, and $f$ is continous,and such that $f(f(x))=xf(x)+1$, find all this $f$? follow is my some idea:(but I don't have solution) We have $f(f(0)) = 1$, so there is your $c = f(0)$, such that $f(c) = 1$. Assume there exists…
math110
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Solution(s) to $f(x+f(y))+f(y+f(x))=2f(f(x))f(f(y))$

I would like to know the solutions of the functional equation: $$f(x+f(y))+f(y+f(x))=2f(f(x))f(f(y)), \forall x,y\in\mathbb{R}$$ where $f:\mathbb{R}\rightarrow\mathbb{R}$. I have already determined that $f\equiv 0$ is a solution, and I would not be…
rayradjr
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Prove that $f(x)=8$ for all natural numbers $x\ge{8}$

A function $f$ is such that $$f(a+b)=f(ab)$$ for all natural numbers $a,b\ge{4}$ and $f(8)=8$. Prove that $f(x)=8$ for all natural numbers $x\ge{8}$
Satvik Mashkaria
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