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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How to find $f$ if $f(f(x))=\frac{x+1}{x+2}$

let $f:\mathbb R\to \mathbb R$,and such $$f(f(x))=\dfrac{x+1}{x+2}$$ Find the $f(x)$ My try I found $f(x)=\dfrac{1}{x+1}$ because when $f(x)=\dfrac{1}{x+1}$,then …
user94270
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3 answers

If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t

Let $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$. I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the same as $g(xy) = g(x) + g(y)$ I looked up the hint and it says let $g(x) = \log…
kuch nahi
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Solution of a function equation $f(x) + f(y) = f(x + y + 2f(xy))$

Find all functions $f:\mathbb{R}_{\geq0}\rightarrow \mathbb{R}_{\geq0}$ which satisfies that for $x,y\in\mathbb{R}_{\geq0}$, $$f(x)+f(y)=f(x+y+2f(xy))$$ I spent quite some time trying to solve it but didn't succeed. It is clear that it has a…
Yoriyuki Yamagata
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$f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$

A function that satisfies both $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$ for all real $x$ is known to be the identity over $\mathbb Q$, but is it also the identity over $\mathbb R$? If not, can you provide me an example of such a function? Thanks.
Leyv Zahav
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27
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A function whose antiderivative equals its inverse.

Does there exist a continuous function $F$ satisfying the property \begin{align} F\left(\int^x_0 F(s)\ ds\right) = x \end{align} If yes, then is the solution unique? As stated, the question is not well-posed since I haven't specified the domain…
Jacky Chong
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Proving that $f(n)=n$ if $f(n+1)>f(f(n))$

How can we prove that if $f:\mathbb{N}\rightarrow\mathbb{N}$ is a function so that $f(n+1)>f(f(n))$ for all $n\in\mathbb{N}$ then $f(n)=n$ for all $n\in\mathbb{N}$? This is problem 6 from the IMO 1977. I found it in this book.
Roiner Segura Cubero
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Find all functions that satisfy $f(\frac{x+4}{1-x}) + f(x) = x$

I found the following task in a book and I would be interested if someone has an idea to solve it: Find all the functions $f$ that satisfy $f(\frac{x+4}{1-x}) + f(x) = x$. My ideas: Assuming that $f$ is a power series or making a substitution. I…
Peter123
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Find all functions $f: \mathbb N \rightarrow \mathbb N$ such that $f(n!)=f(n)!$

Find all functions $f: \mathbb N \rightarrow \mathbb N$ (where $\mathbb N$ is the set of positive integers) such that $f(n!)=f(n)!$ for all positive integers $n$ and such that $m-n$ divides $f(m)-f(n)$ for all distinct positive integers $m,n$. My…
Roman83
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A function that satisfies $f\left({\frac{x+y}3}\right)=\frac{f(x)+f(y)}2$ must be a constant

Prove that a function $f:\mathbb{R}\to\mathbb{R}$ which satisfies $$f\left({\frac{x+y}3}\right)=\frac{f(x)+f(y)}2$$ is a constant function. This is my solution: constant function have derivative $0$ for any number, so I need to prove that $f'$ is…
user164524
19
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4 answers

Solving functional equation $f(x)^2+f(y)^2=f(x+y)(f(f(x))+f(y))$

I was trying to find functions $f:(0,+\infty)\to(0,+\infty)$ satisfying the following functional equation $$ f(x)^2+f(y)^2=f(x+y)(f(f(x))+f(y)) $$ The problem is that I can't find here any reasonable substitution. The only thing I've concluded is…
Norbert
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1 answer

Find continuous functions such that $f(x+y)+f(x-y)=2[f(x)+f(y)]$

Here is the problem: Find all continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy $$f(x+y)+f(x-y)=2[f(x)+f(y)]\;\;\;(1).$$ Here is my attempt: Fix $\delta>0$ and let $C=\int_{0}^{\delta}2f(y)dy.$ Then $$\begin{align*} 2\delta…
Benji
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3 answers

Solution of functional equation $f(x/f(x)) = 1/f(x)$?

I've been trying to add math rigor to a solution of the functional equation in [1], eq. (22). It is: $$ f\left(\frac{x}{f(x)}\right) = \frac{1}{f(x)}\,, $$ where you know that $f(0)=1$ and $f(-x) = f(x)$. I've been trying to fill in the missing…
Ondřej Čertík
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Examples of functions where $f(ab)=f(a)+f(b)$

What are some examples of continuous (on a certain interval) real or complex functions where $f(ab)=f(a)+f(b)$ (like $\ln x$?)
josh
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16
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2 answers

Solve the functional equation $f(xf(y)+yf(x))=yf(x)+xf(y)$

Let $f:\mathbb{R}\to \mathbb{R}$ and such for any real numbers $x,y$ we have $$f(xf(y)+yf(x))=yf(x)+xf(y)$$ Find $f(x)$. I have let $x=y=0$ have $$f(0)=2f(0)\Longrightarrow f(0)=0$$ and I guess the solution is $f(x)=x$ or $f(x)=0$, but I can't…
math110
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15
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3 answers

The function $f (x) = f \left (\frac x2 \right ) + f \left (\frac x2 + \frac 12\right)$

The function $f: [0,1] → \mathbb R $ satisfies the equation $$f (x) = f \left (\frac x2 \right ) + f \left (\frac x2 + \frac 12\right)$$ for every $x$ in $[0,1]$. Can we assert that $f (x) = c (1-2x)$ for some real $c$ if: a) $f$ twice continuously…
Roman83
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