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
1
vote
1 answer

functional composition equations $F(F(a,b),c)=F(a,b\cdot c)$ and $F(F(a,b),c)=F(a,b^c)$

(i) Consider a function of two numbers $F(a,b)$ that satisfies the condition $F(F(a,b),c)=F(a,b\cdot c)$. For example $F$ could be one of the following: $F(a,b) = a\cdot b$ $F(a,b) = a^b$ Question: are there any other functions that satisfy this…
user341440
  • 141
1
vote
3 answers

The functional equation $\frac{f(x)}{f(1-x)} = \frac{1-x}{x}$

One set of solutions to this is $f(x) = \frac{c}{x}$ for constant $c$. Are these the only solutions?
user431110
  • 39
1
vote
1 answer

Need hints for solving 2008 A6: $f\left(x+\frac{1}{y}\right)=f\left(y+\frac{1}{x}\right)$

Let $f(x)$ be function that satisfies $f\left(x+\frac{1}{y}\right)=f\left(y+\frac{1}{x}\right) \Big| f:\mathbb{R} \to \mathbb{N} $. Prove that there exists a positive integer that is not in the range of the function. I'm aware that I can easily…
Magic man Wryan Bang
  • 11
1
vote
1 answer

Functional Equation $H(x,y,z)=f(x,y+z) + g(y,z) = h(y,x+z)+ j(x,z)$

For all real numbers $x$, $y$ and $z$ we have: \begin{align} H(x,y,z)=&f(x,y+z) + g(y,z) \\=& h(y,x+z)+ j(x,z) \end{align} All functions $f$, $g$, $h$ and $j$ are continuous (or even continuously differentiable). I originally suspected $H(x,y,z)=…
HRSE
  • 333
1
vote
0 answers

Solving the functional equation f(x+y)=f(x)+f(y)+y√f(x)

The question is to solve the functional equation $$f(x+y)=f(x)+f(y)+y\sqrt{f(x)}$$ $\forall x,y \in \mathbb R$ I tried to put x=y=0,y=x and y=-x in the given functional equation.I ended up getting $$f(2x)=3f(x)+f(-x)$$ and…
user408949
  • 43
  • 5
1
vote
1 answer

The functional equation $f(y) + f\left(\frac{1}{y}\right) = 0$

Let $f : \mathbb{R}\setminus\{0\} \to \mathbb{R}$ be a non-constant function such that $f(y) + f\left(\frac{1}{y}\right) = 0$. I found that $f(y) = h(\log|y|)$ will be a solution , where $h$ is an odd function. Does any other solution exist ? The…
Souvik Dey
  • 8,297
1
vote
2 answers

Find $f(x)$ from $f(f(x))=x^2 - x + 1$

I have this: $$ f\colon \mathbb{R} \to \mathbb{R}, $$ $$ f(f(x)) = x^2 - x + 1 $$ I need to show that $f(1) = 1$ and I need to show that $g(x) = x^2 - xf(x) + 1$ is not an one-to-one fuction. I know how to solve the second problem but I have no idea…
Rrjrjtlokrthjji
  • 506
  • 3
  • 19
1
vote
1 answer

$f(x+y)f(x-y)=[f(x)f(y)]^2$ implies $f(x)=g\left(x^2\right)$ for some $g$

Small's Functional Equations, Exercise 1.3: Let $f$ be a real-valued function such that, for all real $x$ and $y$, $$f(x+y)f(x-y)=[f(x)f(y)]^2.$$ Prove that there exists a function $g$ such that $f(x)=g\left(x^2\right)$. This is from Small's…
Sawyer
  • 580
1
vote
1 answer

Most general solution to the equation $f(x) = f(1/x)$

What is the most general solution of the functional equation $$f(x) = f(1/x)$$ for $x>0$?
user54031
1
vote
3 answers

Equation for curve where $f(x)= f(x-1)+50 x$

I want a curve where $y(x)$ takes the same values as at $y(x-1)$ with $50x$ added on. What equation do you use to calculate this? An easy way would be with a loop, but is there any way to calculate this without a loop? Here is what values you would…
DayDun
  • 13
1
vote
0 answers

Solution to functional equation $f(y,z)g(y+z)=h(y)$?

Consider the functional equation $f(y,z)g(y+z)=h(y)$, where the functions $f,g,h$ are all continuously differentiable and where $h$ is not constant. I have just about convinced myself that any solution to this must take the form $g(x)=e^{cx}$ for…
Ethan Ligon
  • 33
1
vote
5 answers

A function $f(x)$ on the real domain and real constants $a$ and $b\neq 0$ for which $f(x)-f(x-\delta)+a+bx^2=0$ for some real $\delta\neq 0$

Is there a function $f(x)$ on the real domain and real constants $a$ and $b\neq 0$ for which the following is true: $$f(x)-f(x-\delta)+a+bx^2=0$$ for some real $\delta\neq 0$? EDIT: I missed a very important constraint in the original posting of…
M.B.M.
  • 5,406
1
vote
1 answer

find all continuous function $f:\mathbb R^+\to\mathbb R^+$ satisfy: $f(x)=f(\frac{x+1}{x+2})$

Find all continuous function $f:\mathbb R^+\to\mathbb R^+$ satisfy: $f(x)=f(\frac{x+1}{x+2})$ Find all continuous function $f:\mathbb R\to\mathbb R$ satisfy $\forall a
tangkhaihanh
  • 917
1
vote
1 answer

Find all continuous functions $ f:\mathbb{R}\to\mathbb{R}$, that satisfy $f(xy)=f(\frac{x^2+y^2}{2})+(x-y)^2$

Find all continuous functions $ f:\mathbb{R}\to\mathbb{R}$, that satisfy $f(xy)=f(\frac{x^2+y^2}{2})+(x-y)^2$
tangkhaihanh
  • 917
1
vote
2 answers

$f(yf(\frac{x}{y})) = \frac{x^4}{f(y)} $

Solve Functional Equation$$f: \mathbb{R}_+ \to \mathbb{R}_+; \; f(yf(\frac{x}{y})) = \frac{x^4}{f(y)} $$ I'm stuck in the beginning. Any hint will be helpful.
Rezwan Arefin
  • 3,106
  • 1
  • 14
  • 39
1 2 3
…
60 61