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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3
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1answer
510 views

About finding the function such that $f(xy)=f(x)f(y)-f(x+y)+1$

Define a function $f\colon\mathbb{R}\to\mathbb{R}$ which satisfies $$f(xy)=f(x)f(y)-f(x+y)+1$$ for all $x,y\in\mathbb Q$. With a supp condition $f(1)=2$. (I didn't notice that.) How to show that ...
4
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3answers
261 views

Solving the equation $f(x+t)=f(x)+f(t)+2\sqrt{f(x)}\sqrt{f(t)}$

I am trying to solve the equation $f(x+t)=f(x)+f(t)+2\sqrt{f(x)}\sqrt{f(t)}$ - as in find a function that satisfies this equation. I notice that the RHS is $({\sqrt{f(x)}+\sqrt{f(t)}})^2$ but I am ...
2
votes
1answer
4k views

What does it mean for a functional equation to have a unique solution?

It is my thinking that unique conventionally means special or one of its kind. But in the context of solving functional equations*, I am confused what it means to have a unique solution... *e.g. Find ...
1
vote
2answers
263 views

Is there a continuous {permutation,duplicate,translation,stretch}-invariant function on ordered sets of vectors that returns a vector?

Is there any example of such a function $f$, preferably one defined on all $V^n$ and all positive integers $n$ where $V$ is some vector space? It must satisfy the following: $f(T) = f(\sigma(T))$ ...
0
votes
3answers
288 views

What is the solution of $f(x)\cdot f(-x) = 1$

What is the general solution of the equation? $$f(x) \cdot f(-x) = 1$$ I know that $f(x) = A^{k \cdot x}$ is a solution, and I am feeling this is the general solution, but I don't have any proof. ...
9
votes
3answers
2k views

Proving that an additive function $f$ is continuous if it is continuous at a single point

Suppose that $f$ is continuous at $x_0$ and $f$ satisfies $f(x)+f(y)=f(x+y)$. Then how can we prove that $f$ is continuous at $x$ for all $x$? I seems to have problem doing anything with it. Thanks in ...
2
votes
5answers
795 views

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

Is there any function $f$ which would satisfy $f(x)=f(x+1)$ and $f(-1/x)=f(x)$ for every $x$ or at least positive $x$? For the widest possible domains of $x$? If I could turn this functional equation ...
7
votes
1answer
443 views

Entire function with $f(z)=\sin(f(z))$ must be constant?

I'm trying to show why an entire function with the property $f(z)= \sin(f(z))$ everywhere must be constant. Is it sufficient to say that when taking the derivatives, we will get $f'(z)=f'(z) ...
2
votes
1answer
253 views

How to solve polynomial functional equation $P(x,y)=P(\frac{x-y}{2},\frac{y-x}{2})$?

Given $P(x,y)$ which is a polynomial function, satisfying $P(x,y)=\displaystyle P(\frac{x-y}{2},\frac{y-x}{2})$. Then why should $P(x,y)$ be $\displaystyle\sum^n_{i=0}a_i(x-y)^i$? Is it unique?
3
votes
0answers
152 views

Analytic function that provide $f^2(z)=z$

I am trying to solve this problem: Does there exist a function $f(z)$, that is analytic at $E=\{x+iy :x>y\}$ and provides $f^2(z)=z$ for every $z \in \mathbb C$. I have seen a solution that ...
2
votes
1answer
277 views

$f(x+1)=f(x)+1$ and $f(g(x))=g(f(x))$

Let $g_1(x)=x+1$ and $g_2(x)=x^2$ be two real functions. Then it is known that whenever $f$ commutes with $g_1$ and $g_2$, $f$ is the identity function. But in this example we choosed 2 particular ...
7
votes
7answers
381 views

Function satisfying $x = f(f(x))$ and $x \not= f(x)$

Is there a function that would satisfy the following conditions?: $\forall x \in X, x = f(f(x))$ and $x \not= f(x)$, where the set $X$ is the set of all triplets $(x_1,x_2,x_3)$ with $x_i \in ...
0
votes
1answer
229 views

functional equation uniqueness $f(x^{2})= f(x)^{2} $ in $\mathbb{C}$ and $f(x)+f''(x) = 0 $ in $\mathbb{C}$

This is an exercise from a book I tried: One would like to find all holomorphic equations that satisfy:$$i) \ f(z)+f''(z) = 0 \text{ in } \mathbb{C} $$$$ii)\ f(z^{2})=f(z)^{2} \text{ in } ...
5
votes
2answers
417 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 ...
19
votes
6answers
1k views

$f\colon\mathbb{R}\to\mathbb{R}$ such that $f(x)+f(f(x))=x^2$ for all $x$?

A friend came up with this problem, and we and a few others tried to solve it. It turned out to be really hard, so one of us asked his professor. I came with him, and it took me, him and the ...
6
votes
3answers
362 views

Solve $f(f(n))=n!$

What am I doing wrong here: ( n!=factorial ) Find $f(n)$ such that $f(f(n))=n!$ $$f(f(f(n)))=f(n)!=f(n!).$$ So $f(n)=n!$ is a solution, but it does not satisfy the original equation except for ...
6
votes
1answer
157 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 ...
3
votes
2answers
843 views

Trying to solve the equation involving floor function

I am trying to solve the following equation: Floor[logx+1]+x=11 Where Floor function returns the greatest integer smaller than the value in bracket. e.g. Floor[3.3] = 3 And the logarithm is to the ...
1
vote
1answer
91 views

Is this a functional equation of a line?

Let $f$ be a real-valued function satisfying the functional equation $$f(x)=f(x+y)+f(x+z)-f(x+y+z)$$ for all $x,y,z\in\mathbb{R}$. Is it true that $f$ must be the equation of a line, with no ...
9
votes
3answers
521 views

a continuous function satisfying $f(f(f(x)))=-x$ other than $f(x)=-x$

My question is about existence of a non-trivial solution of the functional equation $f(f(f(x)))=-x$ where $f$ is a continuous function defined on $\mathbb{R}$. Also, what about the general one ...
8
votes
3answers
681 views

Help remembering a Putnam Problem

I recall that there was a Putnam problem which went something like this: Find all real functions satisfying $$f(s^2+f(t)) = t+f(s)^2$$ for all $s,t \in \mathbb{R}$. There was a cool trick to ...
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vote
3answers
265 views

Strange functional equation ( hyperfunctions? )

Can we solve this strange functional equation? $$ f(x+i\epsilon)-f(x-i\epsilon) = g(x) $$ I believe that the solution is the Hilbert (finite part) transform of the function g(x) however I do not ...
0
votes
1answer
39 views

designing an equation that compares two values and returns a probability

Given two values, I'm trying to come up with a formula that will return 50% if both values are equal, 25% if the first value is half the second, 75% if the second is half the first. In other words: ...
6
votes
1answer
258 views

$ f(x+f(x+y))=f(x-y)+f(x)^2 \quad \forall x,y\in \mathbb R$

We have to find all functions $f\colon \mathbb R\to\mathbb R$ such that f: $$\forall x,y\in \mathbb R \quad f(x+f(x+y))=f(x-y)+f(x)^2.$$ Could somebody help me solve this problem? Thank you.
2
votes
3answers
349 views

Finding an $f(x)$ that satisfies $f(f(x)) = 4 - 3x$

I need to find $f(f(x)) = 4 - 3x$ In other examples, such as $f(2)$, I can see that the result equates to $-2$ or $f(x^2)$ becomes $-3x^2 + 4$. Do I really just substitute $f(x)$ for $x$ and ...
3
votes
1answer
175 views

Starting out with functional equations

I am thinking of starting learning about various functional equations and ways to solve them, any help as to which books could be of help to me? I have some knowledge about some basic functional ...
3
votes
1answer
284 views

Finding a function that satisfies constraints numerically

I have the following system of equations for function $p(y)$ and I need help debugging my solution: $$\begin{align} 0&=\log(p(y))+1-\lambda-\gamma y^2-\eta ...
11
votes
7answers
2k views

How to find the function $f$ given $f(f(x)) = 2x$?

I was wondered how to find the function in this equality: $f(f(x))=2x$. Also $f$ is continuous. I don't need the answer, how to find it is more important.
19
votes
4answers
938 views

thoughts about $f(f(x))=e^x$

I was thinking, inspired by mathlinks, precisely from this post, if there exists a continuous real function $f:\mathbb R\to\mathbb R$ such that $$f(f(x))=e^x.$$ However I have not still been able to ...
7
votes
1answer
1k views

Solution for exponential function's functional equation by using a definition of derivative

let $f(0)=1$ and $f'(0)=1$. and $f(x+y)=f(x)f(y)$ for $x,y\in R$. How can I found $f(x)$ by using a definition of derivative?
0
votes
1answer
665 views

Solving an equation with a “nested” function

In a little calculation I'm doing for fun, I've come across this equation involving a function of two arguments which is nested on the right side: $$f(t_1 + t_2, K) = f\bigl(t_2, f(t_1, K)\bigr)$$ ...
2
votes
2answers
17k views

Square root of a number squared is equal to the absolute value of that number [duplicate]

Possible Duplicate: Significance of $\displaystyle\sqrt[n]{a^n} $? The square root of a number squared is equal to the absolute value of that number. Why is $\sqrt{x^2} = |x|$? Why not just ...
8
votes
3answers
709 views

How to calculate $f(x)$ in $f(f(x)) = e^x$?

How would I calculate the power series of $f(x)$ if $f(f(x)) = e^x$? Is there a faster-converging method than power series for fractional iteration/functional square roots?
25
votes
2answers
646 views

$f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$

A function that satisfies both $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$ for all real $x$ is known to be the identity over $\mathbb Q$, but is it also the identity over $\mathbb R$? If not, can you provide ...
0
votes
1answer
193 views

Any example of functions are automorphism?

I am looking for functions fulfilling $f(x+y) = f(x) + f(y)$ and $f(x*y) = f(x)*f(y)$. I can only find $f(x)=x$, any more? Any example of functions are automorphism?
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vote
2answers
180 views

If $f(n) + (n+1)^2 = f(n+1)$ then what is $f\phantom{|}$?

Suppose that $$f(n) + (n+1)^2 = f(n+1),$$ How could I find the original (or family of) function(s) that satisfies this property? What is the branch of mathematics that deals with equations like ...
10
votes
3answers
313 views

Existence of a function

I came across this question Does there exist a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $f(x+y)>f(x)(1+yf(x))$ and $x,y\in \mathbb{R}^+$ and I didn't know how to begin on it.
16
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3answers
2k views

The easy(?) part of IMO 2011 Problem 3

Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies $$f(x + y) \leq yf(x) + f(f(x))$$ for all real numbers $x$ and $y$. How can I prove that ...
0
votes
1answer
353 views

Complexity of $T(n)=\sqrt{n}T(\sqrt{n})+n$

I tried to find the complexity of this recursion equation: $T(n)=\sqrt{n}T(\sqrt{n})+n$, by doing couple of iterations and getting a general idea, but I completely got lost. I'd really love your ...
8
votes
1answer
242 views

About $2$-periodic continuous solutions of $f(x)+f(x+1)=f(2x+1)$

Suppose I want to find all the continuous solutions to the functional equation $$f(x)+f(x+1)=f(2x+1),\tag{E1}$$where $f$ is a continuous and $2$-periodic function defined on the dyadic rationals. I ...
5
votes
1answer
90 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) = ...
4
votes
1answer
166 views

Question in solving $\phi(t)=\phi(2t)+\phi(2t-1)$, $\phi\ne0$

Actually one can resort to the two-scale equation in multiresolution analysis. Perform Fourier transformation on both side of $\phi(t)=\phi(2t)+\phi(2t-1)$, it turns out that ...
5
votes
1answer
270 views

Iterative Functional Equation

Find all functions $ f: \mathbb{N} \rightarrow \mathbb{N}\; $ satisfying $$ f(f(f(n))) + 6f(n) = 3f(f(n)) + 4n + 2001 , \forall n \in\mathbb{N} $$ After some trial and error I assumed the ...
4
votes
2answers
195 views

$f(x+f(y))=f(x)+y^n$

Here is the problem: Fix $n\in\mathbb{N}$. Find all monotonic solutions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+f(y))=f(x)+y^n$. I've tried to show that $f(0)=0$ and derive some ...
13
votes
4answers
852 views

Solving for the implicit function $f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$ and $f(1)=1$

How can I find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(1)=1$ and $$f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$$ for all real numbers $x$ and $y$ with $y\neq0$? PS. This is ...
6
votes
1answer
302 views

All functions $\frac{1}{f\left(y^2f(x)\right)} = \big(f(x)\big)^2\left(\frac{1}{f\left(x^2-y^2\right)} + \frac{2x^2}{f(y)}\right)$

How can I find all continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}^+$ such that $$\frac{1}{f\left(y^2f(x)\right)} = \big(f(x)\big)^2\left(\frac{1}{f\left(x^2-y^2\right)} + ...
10
votes
2answers
339 views

Functional equation of irreducible characters

I am preparing to an exam in representations of finite groups. I am trying to tackle a problem regarding a characterization of irreducible characters: Let $f$ be a complex-valued function on a finite ...
3
votes
1answer
230 views

What is known about functions of type $f(n+1) = f(n)^ {f(n)}$?

Update : having looked at Knut's Double Arrow Notation ( Thank you DJC), it seems that this question is nothing more than a frivourless wondering that should be undertaken by whom ever wonders it, it ...
4
votes
2answers
208 views

$f(z_1 z_2) = f(z_1) f(z_2)$ for $z_1,z_2\in \mathbb{C}$ then $f(z) = z^k$ for some $k$

Same as my previous question except domain is complex. I tried assuming that the function was analytic, so for $z_1=z_2=z$ , $f(z^2) = f(z)^2$ $$\sum_{n=0}^\infty a_n z^{2n}=\left(\sum_{n=0}^\infty ...
10
votes
2answers
582 views

Entire functions such that $f(z^{2})=f(z)^{2}$

I'm having trouble solving this one. Could you help me? Characterize the entire functions such that $f(z^{2})=f(z)^{2}$ for all $z\in \mathbb{C}$. Hint: Divide in the cases $f(0)=1$ and $f(0)=0$. ...