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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A Problem on Theory Of Equations

Let $f(x) = x^2 + x$, for all real $x$. There exist positive integers $m$ and $n$, and distinct nonzero real numbers $y$ and $z$, such that $f(y) = f(z) = m + \sqrt{n}$ and $f(1/y) + f(1/z) = 1/10$ . Compute $100m + n$.
Ankush Yadav
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How to find $\ f(6)$ given the following functional equation .

Let $f:\Bbb R\to\Bbb R$ be a function such that $\lvert f(x)-f(y)\rvert\le 6\lvert x-y\rvert^2$ for all $x,y\in\Bbb R$. If $f(3)=6$ then $f(6)$ equals: how to calculate this ? facing problem with the inequality up there .
Satyam Kumar
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Functional equation $f(xf(y))=yf(x)$

Find all function $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ satisfying $f(xf(y))=yf(x)$ for all $x,y\in\mathbb{R}^+$. Let $P(x,y)$ denote $f(xf(y))=yf(x)$. Similarly from Functional equation : $f(xf(y))=yf(x)$ , we will get $f(1)=1$ and $f(f(x))=x$,…
Bless
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How can one solve this functional equation $f(u)f(v)=f(u+v)(f(u)+f(v)-(q+q^{-1}))+1$?

Solve for $f$: $$f(u)f(v)=f(u+v)(f(u)+f(v)-(q+q^{-1}))+1$$ where $q$ is a fixed complex value and $f$ is defined on $\mathbb R$. We can assume $f(0)=q$ by symmetry. What I got is $f$ takes values in $\{q,q^{-1}\}$ and two relations…
Jiyuan Zhang
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If $x_1, x_2, \cdots, x_n$ are the roots of $f(x)$, find the roots of $g(x)$

Question : If $x_1, x_2, \cdots, x_n$ are the roots of the equation: $$\mathcal{f}(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots + a_0 ~,$$ find the roots of the following equation : $$\mathcal{g}(x)=a_0x^n-a_1x^{n-1}+\cdots + (-1)^na_n$$ I've stated the…
Mathejunior
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Reducing $2f\left(\frac{x+2y}{2}\right)+2f\left(\frac{x-2y}{2}\right)=f(x)+4f(y)$ to the quadratic functional equation

Consider the functional equation $$2f\left(\frac{x+2y}{2}\right)+2f\left(\frac{x-2y}{2}\right)=f(x)+4f(y)\text.\tag1\label1$$ I noticed that if $f(ax)=a^2f(x)$, then \eqref{1} reduces to the quadratic functional…
mark haokip
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Cauchy's functional equation $f (x+y)=f (x)+f(y)$ in subdomains

Suppose $f:\mathbb{R}\rightarrow\mathbb{R}$ is additive ($f(x+y)=f(x)+f(y)$) and monotonic on a set $D\subset\mathbb{R}$ such that $|D|>1$, $0\in D$ and $-a\in D$ whenever $a\in D$. Assume nothing about the behavior of $f$ in $\mathbb{R}\setminus…
user_xyz
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Find all functions $g$ from the Real numbers itself, satisfying $g(x + y) + g(x)g(y) = g(xy) + g(x) + g(y)$

Find all functions $g$ from the Real numbers to itself, satisfying $g(x + y) + g(x)g(y) = g(xy) + g(x) + g(y) . .(*)$ This is a National Olympiad problem, however, my solution is quite different from the one provided by the author so I need you…
Icosahedron
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Solving a functional equation and general theory of functional equations

$f(x-1)$ + $f(x+1)$ = $kf(x)$ given that $k>1$ then I need to find $f(x)$. I am clueless about solving functional equations please help me this . Also give some general advice about how to solve functional equations.
vishal mishra
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A functional equation: $f\left(x^2-1\right)+2f\left(\frac{2x-1}{(x-1)^2}\right)=2-\frac{4}{x}+\frac{3}{x^2}$

If for all $ x > 1 $ $$ f\left(x^2-1\right)+2f\left(\frac{2x-1}{(x-1)^2}\right)=2-\frac{4}{x}+\frac{3}{x^2} \text, $$ then $f(x)=?$ I don't know how to solve such equations. Help me please. Thank you.
Gordon
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Functional equation $f\left(\frac{x+3f(x)}{4}\right)=x$

If $f$ is a real valued function defined on the set of real numbers and $f$ is strictly increasing on its domain and the following holds: $$f\left(\frac{x+3f(x)}{4}\right)=x$$ for all real $x$, then prove that $f(x)=x$ for all real $x$. I've…
Andreas Ch.
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Functional Equation Problem: Strictly Increasing $ f $ satisfying $ f \big( f ( n ) \big) = n + 2 $

If $ \mathbb N $ denotes all positive integers, then find all functions $ f : \mathbb N \to \mathbb N $ which are strictly increasing and such for all positive integers $ n $, we have: $$ f \big( f ( n ) \big) = n + 2 $$ So far I know that $f(n)$…
hoento
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let $f : \mathbb{R} \setminus \{0\} \to \mathbb{R}$ be differentiable on $\mathbb{R} \setminus \{0\}$

let $f : \mathbb{R} \setminus \{0\} \to \mathbb{R}$ be differentiable on $\mathbb{R} \setminus \{0\}$ And : $$f(xy)=f(x)+f(y)$$ And: $$f'(1)=1$$ prove that : $$f(x)=\int_1^{|x|} \dfrac{dt}{t}$$ Suppose:Not aware of $e^x , \ln x $ . I can…
Almot1960
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Functional equation using Hamel basis

Do you know a functional equation, the solution of which is $f:\mathbb{R}\mapsto \mathbb{R}$, with $f$ a function that is not expressed in function of an additive function and that requires the use of Hamel bases (basis of $\mathbb{R}$ over…
ketherok
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Find the number of functions $f: A \rightarrow A$

Let $A = {1, 2, ..., 2014}$. Find the number of functions $f: A \rightarrow A$ satisfied the following properties. There exists $k \in A$ such that $f$ is non-decreasing function on $\{1, 2, ..., k\}$ and $f$ is non-increasing function on $\{k,…
user403160
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