The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...

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1answer
877 views

$f(m + f(n)) = f(f(m)) + f(n)$

I found this one in the list of IMO'96 (3) problems and decided to have a go at it, but could not complete the solution. So $m$ and $n$ are non-negative integers and $f$ takes values in the same set: ...
6
votes
2answers
230 views

Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.

Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$. Some solutions I found are $f\equiv0,f\equiv1$, $f(x)=0$ if $x\neq0$ and $f(x)=1$ if $x=0$.
6
votes
2answers
293 views

Proving a function is constant when $f(x)f(y) + f(\frac{a}{x})f(\frac{a}{y}) = 2f(xy)$

I've been working on the following homework problem: Consider a function $f : (0,∞) → \mathbb{R}$ and a real number $a > 0$ such that $f(a) = 1$. Prove that if $f(x)f(y) + ...
6
votes
1answer
96 views

Can we give a definition of the cotangent based on a functional equation?

I've recently learned that the cotangent satisfies the following functional equation: $$\dfrac1{f(z)}=f(z)-2f(2z)$$ (true for $f(z)\neq 0$). Can we solve this equation for real or complex ...
6
votes
1answer
290 views

Iterated polynomial problem

Polynomial $P$ satisfies $P(n)>n$ for all positive integers $n$. Every positive integer $m$ is a factor of some number of the form $P(1),P(P(1)),P(P(P(1))),\ldots $. Prove that $P(x)=x+1$.
6
votes
1answer
145 views

Functional Equation - Am I right?

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$x^2+y^2+2f(xy)=f(x+y)(f(x)+f(y))$$ So here's my solution, If $x=y=0$, $2f(0)=2f(0)^2$ $\implies f(0)=0$ or $f(0)=1$. Case $1$: ...
6
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1answer
140 views

An elementary functional equation.

I am finding this functional equation from a past high school mathematics competition rather tricky. Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}^+$ such that: ...
6
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1answer
80 views

How can find this function by $x\in \mathbb{Q}^+$

Let $f:\mathbb{Q}^+\to \mathbb{Q}^+,f(x)+f(1/x)=1 $ and $f(2x)=2f(f(x)),x\in \mathbb{Q}^+$, prove that $$f(x)=\dfrac{x}{x+1},x\in \mathbb{Q}^+$$ This Problem from my student.
6
votes
1answer
145 views

If $f\circ f$ is smooth, is the monotonic function $f$ smooth?

Let $f:\mathbb R\to\mathbb R$ be continuous and monotonic and assume that $f\circ f$ is analytic. Is $f$ necessarily continuously differentiable/smooth/analytic? My question arose from this: Inspired ...
6
votes
2answers
146 views

Function whose inverse is also its derivative?

What are some good examples of a function $f : \mathbb{R} \to \mathbb{R}$ where its derivative is equal to its inverse? I attempted to find a monomial that satisfied it by starting with $f(x) = ax^b$ ...
6
votes
2answers
402 views

Real Analysis Proofs: Additive Functions

I'm new here and could really use some help please: Let $f$ be an additive function. So for all $x,y \in \mathbb{R}$, $f(x+y) = f(x)+f(y)$. Prove that if there are $M>0$ and $a>0$ such that ...
6
votes
0answers
91 views

Nonlinear inhomogeneous recurrence $f(x)^2=f(x+1)+S(x)$ to find nested radical

I want to solve the recurrence relation $$f(x)^2=f(x+1)+S(x)$$ where $S(x)$ is a given polynomial. The background is to find nested radicals expressions of the form ...
6
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0answers
91 views

How to prove subadditive function?

Let $f: [0, \infty) \to \Bbb R$ be a function satisfying the following conditions: (1) For any $x,y \geq 0, f(x+y) \geq f(x) + f(y)$. (2) For any $x \in [0,2], f(x) \geq x^2 - x$. Prove that, for ...
6
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0answers
216 views

Differentiable functions satisfying $f'(f(x))=f(f'(x))$

I am wondering whether or not there is a reasonable characterization of differentiable functions $f: \mathbb{R}\to \mathbb{R}$ such that $f'(f(x))=f(f'(x))$ for each $x\in\mathbb{R}$. (Or, if you like ...
6
votes
2answers
250 views

Solving the functional equation $f(x) = f(x / \phi) f(x / \phi^2 - 1)$

I'm trying to find a function $f(x)$ such that the spacing between consecutive roots looks like the infinite Fibonacci word: $$1, \phi^{-1}, 1, 1, \phi^{-1}, 1, \phi^{-1}, 1, 1, \phi^{-1}, 1, 1, ...
5
votes
3answers
441 views

Does the functional equation $f(1/r) = rf(r)$ have any nontrivial solutions besides $f(r) = 1/\sqrt{r}$?

Repeating for the sake of TeX rendering: Does the functional equation $f(1/r) = rf(r)$ have any nontrivial solutions besides $f(r) = 1/\sqrt{r}$?
5
votes
6answers
2k views

What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$?

What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$? I'm especially curious if there is both an algebraic and calculus-based derivation of the solution.
5
votes
2answers
189 views

Finding $f(x)$ of functional equation

I would appreciate if somebody could help me with the following problem: Q: Find all conti-function $f(x)~ (x>0)$ $$xf(x^2)=f(x)$$
5
votes
4answers
306 views

3rd iterate of a continuous function equals identity function

If $ f: \mathbb{R} \to \mathbb{R} $ is continuous, and $\forall x \in \mathbb{R} :\;(f \circ f \circ f)(x) = x $, show that $ f(x) = x $. The condition that $f$ is continuous on $\mathbb{R}$ is ...
5
votes
2answers
355 views

Are all multiplicative functions additive?

Suppose $cf(x)=f(cx)$ and $f:\mathbb{R}\to\mathbb{R}$. I believe it follows that $f(x+y)=f(x)+f(y)$. Proof: There is some $c$ such that $y=cx$. Then ...
5
votes
3answers
259 views

Find the value of the function at the given point.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function satisfying the conditions $$\begin{align*} (1)&f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}\\ (2)&f(0)=1\\ (3)&f'(0)=-1 ...
5
votes
1answer
139 views

Does a function exist with the property $f(-n^2+3n+1)=(f(n))^2+1$?

Let $f:\mathbb{R}\to \mathbb{R}$ be a function which fulfills for every $n \in \mathbb{N}$ $$f(-n^2+3n+1)=(f(n))^2+1$$ Is it possible that such a function exists?
5
votes
2answers
974 views

$f(x) + f(1-x) = f(1)$

I was discussing with a friend of mine about her research and I came across this problem. The problem essentially boils down to this. $f(x)$ is a function defined in $[0,1]$ such that $f(x) + f(1-x) ...
5
votes
2answers
355 views

$f: \Bbb N→ \Bbb N$ , $f\big(f(m) +f(n)\big)=m+n$ $\space$ for all $m,n∈\Bbb N$

How to find all functions $f: \Bbb N→ \Bbb N$ which satisfy $f\big(f(m) +f(n)\big)=m+n$ $\space$ for all $m,n∈\Bbb N$ ($\Bbb N$ is the set of all natural numbers, i.e. positive integers) ?
5
votes
1answer
870 views

Examples of functions where $f(ab)=f(a)+f(b)$

What are some examples of continuous (on a certain interval) real or complex functions where $f(ab)=f(a)+f(b)$ (like $\ln x$?)
5
votes
3answers
195 views

Continuous solutions of $f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$

Consider the following functional equation: $$f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$$ where the equation holds for all $x,y,z \in \mathbb{R}$. One solution is $f(x)=cx$ and $g(x)=1$. What are all the ...
5
votes
2answers
254 views

Iterative roots of sine

Is there an analytical function $f(z)$ such that $f(f(z)) = \sin(z)$? More generally, an analytical function such that f applied $n$ times to $z$ gives $\sin(z)$? Is there a general theory for ...
5
votes
3answers
112 views

find $f(x)$ when $3f(x-6)-2f(x-9)=x^2-54$

I can easily show that with the assumption $f$ is a polynomial $f(x)=x^2$. But without that assumption how can I prove that $f(x)=x^2$???. I have tried many change of variables $x=u+k$ but to no ...
5
votes
2answers
175 views

Help with complicated functional equation

Problem: Let $T=\{(p,q,r)\mid p,q,r \in \mathbb{Z}_{\geq0}\}$. Find all functions $f:T\to \mathbb{R}$ such that: $$f(p,q,r)=\\ =\begin{cases} 0, & \text{ if } pqr = 0 \\ 1 + ...
5
votes
1answer
86 views

Functional Equation help

Came across this problem a little while ago but can't seem to get beyond a certain point. Let $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $f(n+1)>f(n)$ and $$f(f(n))=3n$$ for all $n$. ...
5
votes
2answers
312 views

Do there exist functions satisfying $f(x+y)=f(x)+f(y)$ that aren't linear?

Do there exist functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x+y)=f(x)+f(y),$ but which aren't linear? I bet you they exist, but I can't think of any examples. Furthermore, what ...
5
votes
1answer
128 views

Functional Equation with Value

If $f$ is a strictly increasing function from the naturals to the naturals, and $f(f(x))=3x$, what are all values of $f(2012)$? I have only proven that $f(3x)=3f(x)$ but that get's nowhere :(
5
votes
2answers
1k views

How to prove $f(x)=ax$ if $f(x+y)=f(x)+f(y)$ and $f$ is locally integrable

Suppose $f(x)$ is integrable in any bounded interval on $\mathbb R$, and it satisfies the equation $f(x+y)=f(x)+f(y)$ on $\mathbb R$. How to prove $f(x)=ax$?
5
votes
2answers
383 views

Is there a nontrivial solution to $f(f(f(x)))=-8x$?

Let $f:\mathbb R\to\mathbb R$ be a continuous function such that $f(f(f(x)))=-8x$. Must we have $f(x)=-2x$? I can prove this if I assume that $f$ is continuously differentiable everywhere, but is that ...
5
votes
1answer
88 views

Finding a real value of $p$

I am a bit confused about approaching this problem, Let $g(x)$ be a function such that $g(x + 1) + g(x − 1) = g(x)$ for every real $x$. Then for what value of p is the relation $g(x + p) = ...
5
votes
3answers
138 views

Looking for a function $f$ such that $f(i)=2(f(i-1)+f(\lceil i/2\rceil))$

I'm looking for a solution $f$ to the difference equation $$f(i)=2(f(i-1)+f(\lceil i/2\rceil))$$ with $f(2)=4$. Very grateful for any ideas. PS. I've tried plotting the the initial values into ...
5
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2answers
171 views

functional equations for trigonometric functions

It is well known that the following system of functional equations: $\begin{cases} f(x+y)=f(x)f(y)-g(x)g(y) \\ g(x+y)=f(x)g(y)+g(x)f(y) \end{cases}$ admit the solution $(f,g)=(\cos,\sin)$. Are there ...
5
votes
2answers
124 views

Find function $f :\mathbb{R} \to \mathbb{R}$ satisfy that :

Find function $f :\mathbb{R} \to \mathbb{R}$ satisfy that : $$f(1)=1$$ $$f(x+y)=f(x)+f(y)+2xy$$ $$f\left(\frac{1}{x}\right)=\frac{f(x)}{x^4} \hspace{5pt}\forall x \neq 0$$
5
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3answers
393 views

Inverse function of $y=W(e^{ax+b})-W(e^{cx+d})+zx$

I have a simple question for which I am looking for a closed form expression (If there exits one). In other words, given: $$y=W(e^{ax+b})-W(e^{cx+d})+zx$$ where $W$ is the Lambert $W$ function and ...
5
votes
2answers
240 views

Cauchy's functional equation for $\mathbb R^n$

Suppose $f(x+y)=f(x)+f(y)$. If $f:\mathbb R\to \mathbb R$ and is measurable, then $f(x)=cx$. This is referred to as Cauchy's functional equation. Suppose $f:\mathbb R^n\to \mathbb R^n$ instead. Does ...
5
votes
5answers
130 views

Is there analytic solution to $x^y=y^x\land x\neq y$ as $y(x)$?

Equation $x^y=y^x\land x\neq y$ has trivial solution $ y(x) = x$. Is there non trivial solution given say in terms of elementary or special functions as $y(x)$? A solution that would yield $y(2) = 4$ ...
5
votes
3answers
73 views

Simple Functional Equation $\frac{f(a)-f(b)}{a-b}\cdot(-a)+f(a)=-ab$

Compute all real-valued functions $f$ so that the line between any two points on the graph $f$ intersects the $x$-axis at the product of those two points' $x$-coordinates times $-1$. (if we do some ...
5
votes
3answers
108 views

If x and y are different integers , and if $2005 +x =y^2 ; 2005+y =x^2 $ then find xy…

Problem : If $2005 +x =y^2 ; 2005+y =x^2$ then find xy... My approach : Let $2005 +x =y^2 .....(i) ; 2005+y =x^2 ......(ii) $ Now from (i) we get : $ y = \sqrt{x + 2005}$ Now putting this ...
5
votes
1answer
214 views

Generalization of cos: is this function known?

Consider a function $f_1$ defined by $f_1(x)=1-x+o(x)$ and $f_1(2x)=f_1(x)^2 + 0$. It's simple to find that $f_1(x)=e^{-x}$ (for example by writing series near $x=0$). Consider a function $f_2$ ...
5
votes
1answer
92 views

How find this function such $f(2010f(n)+1389)=1+1389+1389^2+1389^3+\cdots+1389^{2010}+n$

Question: Find all function: $f:N\to N$, such that $$f(2010f(n)+1389)=1+1389+1389^2+1389^3+\cdots+1389^{2010}+n,\forall n\in N$$ Maybe this is 2010 Mathematical olympiad problem.But I ...
5
votes
1answer
149 views

How do you solve $f'(x) = f(f(x))$?

A friend told me to solve the following differential equation: $$f'(x)=f(f(x))$$ I have no idea how to solve this! This doesn't seem to be an ordinary differential equation and I can't even solve ...
5
votes
5answers
141 views

$\forall x\in\mathbb R$, $|x|\neq 1$ it is known that $f\left(\frac{x-3}{x+1}\right)+f\left(\frac{3+x}{1-x}\right)=x$. Find $f(x)$.

$\forall x\in\mathbb R$, $|x|\neq 1$ $$f\left(\frac{x-3}{x+1}\right)+f\left(\frac{3+x}{1-x}\right)=x$$Find $f(x)$. Now what I'm actually looking for is an explanation of a solution to this ...
5
votes
1answer
1k views

Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions…)

My lecturer was talking today (in the context of probability, more specifically Kolmogorov's axioms) about the additive property of functions, namely that: $$f(x+y) = f(x) + f(y)$$ I've been trying ...
5
votes
1answer
75 views

Expressing a recursively defined function in terms of factorials or gamma function

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2) $$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = ...
5
votes
2answers
70 views

Problem solve functional equation

Solve functional equation:find all strictly monotone functions $f:(0,+\infty)\to(0,+\infty)$ such that $$(x+1)f(\dfrac{y}{f(x)})=f(x+y),\forall x,y>0$$