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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Find $f(2^{2017})$

The function $f(x)$ has only positive $f(x)$. It is known that $f(1)+f(2)=10$, and $f(a+b)=f(a)+f(b) + 2\sqrt{f(a)\cdot f(b)}$. How can I find $f(2^{2017})$? The second part of the equality resembles $(\sqrt{f(a)}+\sqrt{f(b)})^2$, but I still have…
Student12
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If a function is like $f(f(y))=a^2+y$, does it imply that $f$ is surjective?

If a function is like $f(f(y))=a^2+y$, does it imply that $f$ is surjective? Just for an example, consider this: Find all functions $f:\mathbb{R}\mapsto \mathbb{R}$ such that $$f(xf(x)+f(y))=(f(x))^2+y$$ for all real values of $x,y$. It's…
Mathejunior
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Given that $f(x)+f(y) = f[x \times \sqrt{1-y^2}+ y\times \sqrt{1-x^2}]$ To prove $f(4x^3-3x) + 3f(x) = 0$

Clearly $\arcsin x$ is one such function but how do we prove it in general? I differentiated both sides wrt x but that didn't give me something to get even close to what's required
user481779
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Tripling an involution (Functional equation)

Here's a standard trick which is claimed by Evan Chen in one of his handouts: Introduction to Functional Equations: Tripling an Involution: If you know something about $f(f(x))$, try applying it $f(f(f(x)))$ in different ways. For example, if we…
Mathejunior
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Given $g(x)$, is there a function $f(x)$ so that $f(x) - f(x-1) = g(x)$?

Let $g:\mathbb{R}\to \mathbb{R}$ be a given function. My question is Does there exist a function $f\colon X\subseteq\mathbb{R}\to \mathbb{R}, x\mapsto f(x)$ such that $f(x)-f(x-1)=g(x)?$ for all $x\in X$? I am able to guess some very simple…
Xiangyu Chen
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Linearize the equation $f^2(x)+1=2f(x+1)$

How can I linearize for the continuous function $f$ the functional equation $f^2(x)+1=2f(x+1)$ I' ve been studying recently the book for functional equations by Christopher G. Small and I' ve encountered the term "Linearization" of functional…
Chris Galanis
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Find all functions : $f(x)+f(\frac{x}{2})= \frac{x}{2}$

Find all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ satisfying $$f(x)+f(\frac{x}{2})= \frac{x}{2}$$ $\forall x \in \mathbb{R}^+$. My Attempt : $-\frac{x}{3} + f(x) = \frac{x}{6} - f(\frac{x}{2})$ Let $g(x) = f(x) - \frac{x}{3}$ so…
user403160
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Find all functions for which $\ x \cdot f(xy)+f(-y)=x \cdot f(x)$

Does anyone have idea for solution (for all non-zero numbers)? $$\ x ≠ 0,y ≠ 0$$ $$\ f: R \setminus \{0\} → R$$ $$\ x \cdot f(xy)+f(-y)=x \cdot f(x)$$ Thanks!
dev0experiment
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Find all functions $f(m)[(f(n))^2-1]=f(n)[f(m+n)-f(m-n)]$

Find all functions $f : \mathbb{N} \to \mathbb{N}$ satisfying $$f(m)[(f(n))^2-1]=f(n)[f(m+n)-f(m-n)]$$ $\forall m>n$. Attempted work : Let $P(m,n)$ denote $f(m)[(f(n))^2-1]=f(n)[f(m+n)-f(m-n)], \forall m>n$. Case 1 : $f(n)=1, \forall n \in…
user403160
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Proving that $f(x+y) = f(x)+f(y) $ and $f(xy) = f(x)f(y)$ imply $f(x) = 0$ or $f(x) = x$

Question: If $f$ is a function such that $$f(x+y) = f(x)+f(y) \qquad f(xy) = f(x)f(y)$$ for all $x$, then prove that $f(x) = 0$ or $f(x) = x$ for all $x$. (The fact that every positive number is a square of some number will be important.) My…
user427380
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How to solve the iterated function $f(f(x))=x^2+x$?

Any given setting for $f$ is acceptable. Iterated function
Tongho
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By which assumptions on $g$ could we solve the functional equation $f\circ f=g$?

I'm interested in the solvability of the functional equation (which we would call it $(*)$) $$f\circ f=g$$ where $g$ is some functions to be defined. Certainly, even if $(*)$ has a solution, it would not be expected to be simple. For…
BAI
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$f(n) + f(n+1) + ... + f(n+N-1) = n$

Let $N \in \mathbb{N}$ be constant. Find all non-decreasing functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ satisfying $$f(n) + f(n+1) + ... + f(n+N-1) = n$$ for all integers $n$. Please check my solution. $$ f(n) + f(n+1) + ... + f(n+N-1) = n…
user403160
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Proving a function one to one function from the given function

Here is the function, $$f:\mathbb{N}\rightarrow \mathbb{N}, f(f(m)+f(n))=m+n$$. This is a one to one function. But I cannot proceed further. I have tried to arrange the domains and their respective ranges, like, $f(f(3)+f(1))=4 \land f(2f(2))=4$.…
Chandramauli Chakraborty
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Find the value of $f(\frac14)$

$$f:(-1,1)\mapsto(-1,1)\in C^1\quad f(x)=f(x^2)\quad f(0)=\frac12$$ Then $f(\frac14)$ is? All I could deduce is that $f(x)=f(-x)$. A hint would be great.
Maadhav
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