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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Find all function $f:\mathbb{R^+}→\mathbb{R^+}$ that satisfies $f(xy)^{xy} =f(x)^x f(y)^y$

Find all function $f:\mathbb{R^+}→\mathbb{R^+}$ that satisfies the given condition: $$f(xy)^{xy} =f(x)^x f(y)^y$$ If $f:\mathbb{R}→\mathbb{R^+}$ the question would be rather simple, as putting in $y=0 $ yields that $1=f(x)^x$ thus implying…
Chad Shin
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Functions $f:\mathbb R \to \mathbb R$ which satisfy $f(x^2+f(y))=y+(f(x))^2$

Find all the functions $f:\mathbb R \to \mathbb R$ which satisfy $$f(x^2+f(y))=y+(f(x))^2$$ for all $x, y$ in $\mathbb R$. I have the following proof from my math book and want to see if I can get another one. Proof: Let $x=0$. Then we have…
Sierra Res
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How to solve $f\left(\frac{f(x)}{yf(x)+1}\right)=\frac{x}{xf(y)+1}$?

I'm currently working on the following functional equation: Find all $f:\mathbb{R_{>0}}\to\mathbb{R_{>0}}$ such that for all $x,y\in\mathbb{R_{>0}}$: $$ f\left(\frac{f(x)}{yf(x)+1}\right)=\frac{x}{xf(y)+1} $$ I think, i have managed to show that $f$…
Redundant Aunt
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How to solve $f(x+f(x)+2f(y))=f(2x)+f(2y)$?

Another functional equation: Find all surjective functions $f:\mathbb R\to\mathbb R$ such that for all $x,y\in\mathbb R$ it satisfies: $$ f(x+f(x)+2f(y))=f(2x)+f(2y) $$ I couldn't make any progress because I didn't know how to use the surjectivity.…
Redundant Aunt
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What are solutions of the functional equation $f\big(f(x)\big)=-4f(x)+3x$?

How to find all the functions $f:[0,\infty)\rightarrow [0,\infty)$ satisfying the functional equation $$f\big(f(x)\big)=-4f(x)+3x\text?$$ I deduced $f(x) \le \frac 3 4 x.$
user64494
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Functional equation: $f(f(x))=k$

If $k\in\Bbb R$ is fixed, find all $f:\Bbb R\to\Bbb R$ that satisfy $f(f(x))=k$ for all real $x$. If $k\ge 0$, $f(x)=|k+g(x)-g(|x|)|$ is a solution for any $g:\Bbb R\to\Bbb R$.
user263326
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functional equation with two functions: $ f ( x + y ) = f ( x ) g ( y ) + f ( y ) $

Find all functions $ g : \mathbb R \to \mathbb R $ with the property: There exists a strictly monotonic function $ f : \mathbb R \to \mathbb R $ such that $$ f ( x + y ) = f ( x ) g ( y ) + f ( y ) , \forall x , y \in \mathbb R $$ We have one…
Booldy
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Functional equation on integers

Is there a function $f$ such that $$f(x,y,n)=f(x+y,y-x,n+1)$$ $$f(x,y,n)\neq f(x+1,y,n)$$ $$f(x,y,n)\neq f(x,y+1,n)$$ where $x,y$ are integers, $n$ is a positive integer and the range of $f$ is a finite set? If so, what is the least number of values…
Gere
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d'Alembert-like functional equation: $f(x+y)+g(x-y)=\lambda f(x)g(y)$

The D'Alembert functional equation is $f(x+y)+f(x-y)=2f(x)f(y)$. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfy the functional equation for all $x,y\in\mathbb{R}$. It's well known that $f$ is of the form $f(x)=\frac{E(x)+E^∗(x)}{2}$, for some…
user217960
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Functional equation $ f(x)+f(x+1)=x$

What functions satisfy $f(x)+f(x+1)=x$? I tried but I do not know if my answer is correct. $f(x)=y$ $y+f(x+1)=x$ $f(x+1)=x-y$ $f(x)=x-1-y$ $2y=x-1$ $f(x)=(x-1)/2$
user220221
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Find all real functions so that $f(xf(y)+f(x))=f(yf(x))+x$

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ so that $f(xf(y)+f(x))=f(yf(x))+x$ $f(x)=\pm x$ should be the only solution. It's easy to get that $f(f(0))=f(0)$.
CryoDrakon
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Functional equation $f(x)-f(y)=\frac{1}{(x-y)^{2}}$

Please help me to solve the functional equation $$ f(x)-f(y)=\frac{1}{(x-y)^{2}} $$ for all real $x\neq y$. I have reduced it to $$ f(x+h)-f(x)=\frac{1}{h^{2}} $$ for all real $h\neq 0$. But what to do with this equation? Great thanks in advance!
Milena N
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Solving the equation $f(x+t)=f(x)+f(t)+2\sqrt{f(x)}\sqrt{f(t)}$

I am trying to solve the equation $f(x+t)=f(x)+f(t)+2\sqrt{f(x)}\sqrt{f(t)}$ - as in find a function that satisfies this equation. I notice that the RHS is $({\sqrt{f(x)}+\sqrt{f(t)}})^2$ but I am stuck after this.
MathMathCookie
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Solve functional equation $f(f(f(x)))+f(x)=2x$

Please help me solve this functional equation: find $f(x)$ given that $$f(f(f(x)))+f(x)=2x$$ Thanks very much.
user161621
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Find all the functions $f:\mathbb{R}→\mathbb{R}$ such that $f(mx+c)=mf(x)+c$

Find all the functions $f:\mathbb{R}→\mathbb{R}$ such that $f(mx+c)=mf(x)+c$, $m≠1$. I know that $f(x)=x$ and $f(x)=c/(1-m)$ are two solutions. But to completely solve it I have no idea. Can we completely solve it using elementary mathematics…
Bumblebee
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