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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Continuous function $g$ satisfying $g(x + y) = 5g(x)g(y)$

Let $g$ be a continuous function with $g(1) = 1$ such that $$g(x + y) = 5g(x)g(y)$$ for all $x$, $y$. Find $g(x)$.
Argha
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Functional equation: $f(x)=\frac x{\prod_{i|x}^{x-1} f(i)}$

Find an algebraic function $f:\Bbb N\to\Bbb N$ such that $$f(x)=\frac x{\prod_{i|x}^{x-1} f(i)}$$ and $$f(1)=1$$ for all $x\in\Bbb N$ I allready know two things: $f(p^k)=p$ where $p$ is prime and $k\in\Bbb N$ $f(apq)=1$ where $p$ and $q$ is two…
DrainZ
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Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that $f(x)=f(x^y)$

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$f(x)=f(x^y)$$ for all $x,y\in\mathbb{N}$. I'm not intrested in the trivial solution $f(x)=k$, where $k\in\mathbb{N}$.
DrainZ
  • 68
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How to complete this?

Let $f$ a function defined on $]0,+\infty[$ and checking : $$\forall x,y>0,\ f(xy)=f(x)+f(y)$$ Assume that $f$ is bounded on $]1-\eta,1+\eta[$($\subset]0,+\infty[$). Let $\alpha =\underset{x\in ]1,1+\eta[}{\inf}f(x)$ and $\beta =\underset{x\in…
Said
  • 11
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Find the equation of the common part of two objects

How to find the equation of the intersection curve of the ball $ x^2 + y^2 + z^2 = 4a^2 $ (1)and the cylinder $x^2+y^2=2ax(a>0)$(2)? let (1)-(2), we can get $$z^2+2ax-4a^2=0 $$ but this is not the curve equation because $y$ is missing.
canoe
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If $f(xy) = f(x) + f(y)$, show that $f(.)$ can only be a logarithmic function.

As the question states, show that the property exhibited can only be satisfied by a logarithmic function i.e no other family of functions can satisfy the above property.
TheChetan
  • 199
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Sufficiency of the condition $f(x) = f(x^3)$ for $f$ to be even or constant

I've been playing around with some aspects of basic functions, and I reached a function that seemed a bit peculiar. Consider $\forall x \in \mathbb{R}$ a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f(x) =f(x^3).$ I'm wondering if this…
Sandra
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Time delay equation

If $x(t)=\displaystyle\frac{1+(x(t-T))^m}{(x(t-T)))^m}$ for all $t$ where $T$ is constant and $x(t)=x_s$ is the solution to the above equation, why can I write that: $x_s=\displaystyle\frac{1+{x_s}^m}{{x_s}^m}$ when on the RHS $x_s$ is evaluated…
usainlightning
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Solve the functional equation (medium-hard)

Find all real functions $f(x)$ $\mathbb{R} \to \mathbb{R}$ such that $x^2f(yf(x)) = y^2f(x)f(f(x))$ Obviously, let $y=0$ you instantly get, $f(0) = 0$. Also, a relation is: $f(yf(x)) = \frac{y^2 f(x) f(f(x))}{x^2}$ and set $y = f(y)$ to…
Amad27
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Re-arrangement of equation

This type of question might be voted down or frowned upon, but if anyone could point me in the right direction I would be very grateful. I have worked through a question getting the following…
Michael Roberts
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Solve the functional equation $f(x) + f(x^2) = 2$

What are the solutions of the functional equation $f(x) + f(x^2) = 2$? Will they be one to one or many to one? Will they be periodic or not?
Tyrion Lannister
  • 197
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Cauchy-like functional equation $f(h(y)\cdot x+y)= g(y)f(x)+f(y)$

I am looking for the solution to the following two variable functional equation: (*) $f(h(y)\cdot x+y)= g(y)f(x)+f(y)$ where: $h$ is some given continuous function, $f, g,$ unknown functions on some interval $[0,\alpha]$ for some $\alpha>0$, $f$…
mike
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How to rewrite $x-y=\frac{x}{y}$ so that it become $y=$ (something...)?

For example, $x+y=x\times y$ is easy to express as $y=\frac{x}{x-1}$, how about $x-y=\frac{x}{y}$? I tried multiply both sides by $y$ and become $y^2-xy+x=0$ but up to this step I don't know how to continue, can anyone help?
ggrr
  • 149
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A simple two variable functional equation: $f(x+y,y) = f(x,0)+qy$

A real function $f(x,y)$ on $\mathbb R^2$ satisfies $f(x+y,y) = f(x,0)+qy$ for some real number $q$. What form should $f$ assume (without assuming $f$ is continuous)? Is the linear solution $f(x,y)=ax+(q-a)y$ for some constant real number $a$ the…
Hans
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Equation with variable $λ$ that should be $R$ for $x$

I need to find for $λ\in R$ the domain of $f(x)=\sqrt{((λ-2)x^2-2λx+2λ-3)}$ It should be $λ\in[6,+∞ )$ as per my book but I dont understand why. Sorry for my english
Eva
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