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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Doubt in an alternate solution to an IMO problem from 1987

Consider the following problem from 1987 IMO Show that there is no function $f:\mathbb N→\mathbb N$ Such that $f(f(n))=n+1987$ In the book that I was referring the solution is given as follows: " We show that infact there is no function that…
Aarush Saharan
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Obtaining probability generating function by solving a functional equation

For investigating a queueing model in my research, I need to obtain the probability generating function (PGF), $G(z)$, by solving the following functional equation $$\lambda(z(1-p)+p)G((1-p)z)-(\mu+\lambda p)(1-p)G(z)=(\lambda p-(\mu+\lambda…
גבי חנוכוב
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$f(x) + f(y) = \frac{1}{f(xy)}$

Suppose $f(x) + f(y) = \frac{1}{f(xy)}$ holds for all $x,y$ and $f(x) > 0$ for all $x$. which are real. I understand that you can choose for example, $x=1$ and $y=4$ to get $f(1) + f(4) = \frac{1}{f(4)}$ but that is not very useful. What I did was…
Nav Bhatthal
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Finding values of other functions with the help of given function

To all the viewers , I am really sorry That I didn't used Mathjax as I really don't know how to use it but will learn using it in the future :) So , the problem I am facing it is that I used 2 methods which is cross multiplying , assuming values…
πααρτθ Σαρθι
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Find $f(x)$ that satisfies the equation $f(x^2) + (f(x))^2 = 2x^2 - 2$

We know that $f(x) = x - \frac{1}{x}$ , for $x \neq 0$ , is a particular solution of the equation, but what would be the method to arrive at this solution?
JaberMac
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Prove that the function is injective

Prove that this function is injective. $$f(x + 2xf(y)^2) = yf(x) + f(f(y) + 1)$$ I try to fix $x$ and put two values for $y$: $a$ and $b$, but I can't prove it. Maybe it is better to fix $y$ and choose two values for $x$?
Ilco Daniel
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if we know that $y + f(y)y + 1$ goes over all real numbers once and f is bijective, can we conclude that $f(y)y$ is constant?

yep, that's all I've got to say, we know surely that $y + yf(y)$ goes over all real numbers once as well, and that's about all I got.
11235
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Find $f$ such that $f(af(b))=bf(a)$

Find $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that $$\forall a,b \in \mathbb{R}^+, f(af(b))=bf(a)$$ $f(x)=x$ is a trivial solution, but I'm not sure if this is the only one. I can see that $f(f(x))=x$ and therefore $f$ is bijective, but not anything…
Saturday
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Is every non-zero additive function injective/surjective?

An additive function is a function $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $f(x+y)=f(x)+f(y)$, for all real $x$ and $y$. Aside from the identically $0$ function, is every such function injective/surjective? Certainly, every additive function…
user107952
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How to solve interdependent functional equations with $\max$ constraint

I have three related functions on the closed interval $[0, 1]$: $f(x) = \max(f_S(x), f_C(x))$ $f_S(x) = x+\frac 1 2$ $f_C(x) = (1-x)f(\frac {x+1} 2)$ We are also given that for $0 \leq x \leq 1$, we have $\sup f(x) \leq 2$, that is, that $2$ is an…
joseville
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A functional equation with trigonometric solution

Given a function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x+y)+f(x-y)=2f(x)f(y)$$ for all real $x$ and $y$. Find all possible functions $f(x)$. The optimal solutions are $f(x)=0$, $f(x)=1$, $f(x)=\cos x$, but I can't fully solve the…
xxxx036
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Solve the functional equation $\left(f(x)\right)^2 - f\left(x^2\right)=2x$

$\forall x \in \Bbb R ,$ Find all functions that satisfy: $$\left(f(x)\right)^2 - f\left(x^2\right)=2x$$ My solution: $$f(x) \cdot f(x) - f(x \cdot x)=x+x$$ $$f(x) \cdot f(y) - f(x \cdot y)=x+y$$ $$f(x) \cdot f(0) -f(0)=x$$ $$f(x)=\frac{x}{f(0)}+1…
Rehman
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If $f(x + y) = f(x) + f(y) +xy$ and $f(4) = 10$ then $f(2001) = ?$ Why is my answer wrong?

I saw this question on YouTube: https://www.youtube.com/watch?v=h98AElJPxa8 $f(x+y) = f(x) + f(y) +xy$, and $f(4) = 10$ then $f(2001) = ?$ The following is the way I solved it, and it looks right to me but they have a different solution in the…
Medhaj Patel
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Building a function from 3 given properties

Is there a real function $f$ / algebraic expression, such that $f(-3) = 1$ $f(3) = 1$ $f(x) = - f(x)$ for all $-3 < x < 3$ I was able to construct some $f$ with Fourier analysis and complex arguments but this wasn't what I was looking for. :(
mangolassi
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Analytical solution to $e^{f(x^{\alpha+1})}=e^{f(x)}e^{f(x^\alpha)}$ other than $f(x)=\log{x^\lambda}$

The title is self explanatory... How can I prove (or find a counter example) that $f(x)=\log{x^\lambda}$ is the only analytical solution to this functional equation? Assumptions be made on $\alpha$, $f$ and $x$ belonging to real. Thanks.
gudise
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