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
votes
3 answers

For a function $f:\Bbb N\to\Bbb R$, the following are satisfied, (i) $f(1)=1$ and (ii) $f(1)+2f(2)+3f(3)+....+nf(n)=n(n+1)f(n).$ Find $f(n)$

I found the value of $f(2), f(3),f(4)$ and observed $f(n)=\frac{1}{2n}$ for $n \gt 1$. I am able to prove this by Mathematical induction. Another approach I tried is $f(1)+2f(2)+3f(3)+....+nf(n)=n(n+1)f(n)$. Now subtract $n*f(n)$ from both sides…
Utkarsh Dubey
  • 27
-1
votes
1 answer

$g(x)h(x)=x$ for odd $g(x)$ and even $h(x)$

Question Let $g:\mathbb{R}\to\mathbb{R}$ be an odd function and $h:\mathbb{R}\to\mathbb{R}$ be an even function. Find $g(x)$ and $h(x)$ such that $g(x)h(x)=x$ for all $x\in\mathbb{R}$. Try I can only figure out the trivial cases where $g$ and $h$…
Yuta
  • 1,493
-1
votes
1 answer

Is it possible to solve the functional equation $\frac{\ln\bigl(a(t^*)\bigr)}{\ln\bigl(a(t)\bigr)}=f(t)$?

Is it possible to solve this functional equation? $$\frac{\ln\bigl(a(t^*)\bigr)}{\ln\bigl(a(t)\bigr)}=f(t)$$ where $a(t)$ is the unknown function of the independent variable $t$; $f(t)$ is known, and $t^*$ is a specific value of $t$, also known. If…
Shootforthemoon
  • 1,520
-1
votes
1 answer

functional equation $f\left(x\right)+a=f\left(x+b\right)$

Find the solution of the functional equation $f\colon ℝ\to ℝ\:$ $$f\left(x\right)+a=f\left(x+b\right), \quad\forall x\in\mathbb{R}$$ where $a$ and $b$ are fixed real numbers and $b$ is a positive real number. I suppose that the answer…
shangq_tou
  • 998
-1
votes
1 answer

Golden functions?

If one seeks the functional solutions of $f^2(x)=f(x)+1=f(f(x))$ Are they called 'golden functions' ? Are they always of the form of rational functions ?
QuantumPotatoïd
  • 286
-1
votes
1 answer

The function $f : \mathbb{R} \to \mathbb{R}$ satisfies $f(x) f(y) = f(x + y) + xy$ for all real numbers $x$ and $y$. Find all possible functions $f$.

The function $f : \mathbb{R} \to \mathbb{R}$ satisfies $$f(x) f(y) = f(x + y) + xy$$ for all real numbers $x$ and $y$. Find all possible functions $f$.
littlemix
  • 19
  • 3
-1
votes
2 answers

Find all functions $f$ such that $f(f(x)+yz)=x+f(y)f(z)$

Question Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that $f(f(x)+yz)=x+f(y)f(z).$ My doubt: In the hint of this they write show that $f(0)=0$ or $f(0)^2=2$. I was not able to see why this thing works after a lot of substitution…
IMO 2021 GOLD
  • 416
-1
votes
2 answers

Exponential as solution of a functional equation.

the function $f(t)=2^t$ is a solution of the functional equation: $$f(t+1)=2\cdot f(t),\ f(0)=1$$ Is this unique? Is this unique as continuous funtion? Is this unique as differential funtion? Thank you.
Buschi Sergio
  • 426
-1
votes
2 answers

Find all real-valued functions $f$ satisfying $f(2x+f(y))+f(f(y))=4x+8y$

$ x,y \in\mathbb{R}$ and $f:\mathbb{R} \rightarrow \mathbb{R}$, find all functions satisfying, $$f(2x+f(y))+f(f(y))=4x+8y$$ I couldn't seem to understand it.
Hussaini Said Adam
  • 1
  • 1
-1
votes
1 answer

Is there a function such that $f(x+y) = f(xy)$?

If we define $x, y \in \mathbb{R}$, is there a function that fulfills the condition: $f(x+y) = f(xy)$ for all $x$ and $y$? For now, let's assume there are no stipulations on continuity and differentiability.
dsmalenb
  • 219
  • 1
  • 11
-1
votes
1 answer

Number of solutions of the equation $e^{f(x)}=f(x)+2$

Let $f$ be an everywhere differentiable function, and suppose that $f(x)=0$ has a unique solution, and suppose that $f$ has no local extreme points. What is the number of solutions of the equation $$ e^{f(x)}=f(x)+2. $$ Thanks!
boaz
  • 4,783
-1
votes
1 answer

Solve the functional equation $f(x+1)-f(x)=x*\sin(x) $

Solve $f(x+1)-f(x)=x*\sin(x) $
MorrisonFJ
  • 33
-1
votes
2 answers

Math question related to functional equations

The function $f$ satisfies $f(x) + f(2x + y) + 5xy = f(3x - y) + 2x^2 + 1$for all real numbers $x$, $y$. Determine the value of $f(10)$.
Alex Tang
  • 1
  • 1
-1
votes
2 answers

Solve the functional equation $f(1+xf(y))=yf(x+y)$

Problem Find all functions $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$ such that: $$f(1+xf(y))=yf(x+y)$$ for all $x,y \in \mathbb{R^+}$ Progress I can only prove $f$ is a surjective function. I tried to prove $f$ is an injective function, but I…
Haruboy15
  • 990
-1
votes
1 answer

let $f(x)$ be a real valued function defined for all positive x, satisfying $f(x+y)=f(xy)$ for all positive x, y. Prove that f is a constant function.

I was solving problems from the first chapter of functional Equations and how to solve then and got struck on the second problem. when I looked at the hints section I was not able to understand it completely. Please help me to solve the problem. The…
Ankush Yadav
  • 49
  • 5
1 2 3
…
60 61