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
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
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1answer
249 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.
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3answers
334 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
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1answer
151 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 ...
2
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1answer
231 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
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8answers
1k 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.
13
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4answers
528 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
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1answer
808 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?
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1answer
394 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
7k 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
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3answers
562 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?
24
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2answers
618 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 ...
1
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1answer
175 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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2answers
172 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
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3answers
298 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.
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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
318 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 ...
6
votes
1answer
212 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
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1answer
82 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
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1answer
159 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 ...
4
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1answer
209 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
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1answer
179 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 ...
12
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4answers
733 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
287 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)} + ...
8
votes
2answers
249 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
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0answers
210 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
194 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
516 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 $f(0)=0$. ...
8
votes
3answers
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If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t

Let $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$. I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the same as $g(xy) = g(x) + ...
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vote
2answers
406 views

Functional equation $f(x^2+y)=f(x)+f(y^2)$ from Olympiad

How do i deal with stuff like that? tried to just write many equalities but it just doesnt help... The equation is $f(x^2+y)=f(x)+f(y^2)$. EDIT: the question is to find all functions such that this ...
6
votes
3answers
670 views

Classifying Functions of the form $f(x+y)=f(x)f(y)$ [duplicate]

Possible Duplicate: Is there a name for such kind of function? The question is: is there a nice characterization of all nonnegative functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that ...
4
votes
0answers
170 views

Vector valued contraction

I am familiar with the Banach Fixed Point Theorem and I have used it to prove existence and uniqueness of functional equations in Banach spaces like $C(X)$, the space of bounded continuous function ...
1
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2answers
433 views

Indeterminate equation and functional equation

I was wondering what differences and relations are between indeterminate equation and functional equation? Are they the same concept? Thanks and regards!
1
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1answer
54 views

Transform the sample to make it more similar to a given

$X=\{x_{i}\}$ and $Y=\{y_{i}\}$ are numeric samples: $y_i \ge 0, x_i \ge 0, i \in [0..N]$. I need to find the mapping $F(X)=\{F(x_i)\}$ with fairly simple formula such that: Euclidean distance ...
0
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1answer
85 views

General solution to homogeneous difference equation

With a given example $$ a_{n-1} = ca_{n-2} $$ general solution: $$ a_{n} = c . c . a_{n-2} $$ $$ = c . c . a_{n-3} $$ $$ = c^n a_0 $$ Question: Find the general solution for the ...
4
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1answer
249 views

Solving (and proving) a combinatorial functional recursive equation

I have a sequence of functions $f_k(n)$ defined as follows: $f_1(n)=n^{n-2}$ $f_k(n)=\sum_{i=1}^{n-1}f_{k-1}(i)\cdot(n-i)^{n-i-2}\cdot{n-k \choose n-i-1}$ My goal is to find and prove a closed-form ...
0
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3answers
180 views

Recurrence relation - How to solve this recurrence relation

a person invests 1000 at a bank at 4 percent compound interest compounded annually and every year government and bank charges amounting to C are deducted and if An is the value of the investment at ...
0
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1answer
158 views

Solving general solution of recurrence relation by iteration

$$a_{n-1} = ca_{n-2} $$ Hence $$a_n = c \cdot c \cdot a_{n-2} $$ $$ = c \cdot c \cdot c \cdot a_{n-3} $$ ...... $$ = c^na_0 $$ Why is there a iteration on the constant $c$ ?
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2answers
212 views

Understanding difference equation

I was given an example $$R_n = R_{n-1} + R_{n-2} $$ This equation is given as an second-order equation. Why is it so?
11
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2answers
194 views

Can every real function be represented as two shifted even functions?

I saw the theorem that every function can be represented as the sum of and even and odd function, and this made me wonder: can every function from the reals to the reals, defined on all the reals, be ...
13
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1answer
1k views

$f(x+y)+f(x-y)=2[f(x)+f(y)]$

Here is the problem: Find all continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy $$f(x+y)+f(x-y)=2[f(x)+f(y)]\;\;\;(1).$$ Here is my attempt: Fix $\delta>0$ and let ...
3
votes
1answer
267 views

Functional derivative of $\int \left( \frac{df^2 }{d^2 x} \right)^2 dx$

According to page 7 of the PDF document $$ \frac{\delta}{\delta f} \int \left( \frac{df^2 }{d^2 x} \right)^2 dx = \int \frac{df^4}{d^4 x} dx $$ I would like help proving this statement. Although ...
2
votes
1answer
91 views

All functions with the property $ k \mid f(m+n) \iff k \mid f(m)+f(n)$

Let $\mathbb N$ be the set of all positive integers. How can one find all functions $f: \mathbb N \to \mathbb N$ such that $$ k \mid f(m+n) \iff k \mid f(m)+f(n)$$ For all positive integers $k$.
2
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3answers
89 views

Defining a function with certain properties

I'm a bit rusty in mathematics so I need your help please :) I need a function $y$ that satisfies: $$\begin{align*} y &= ax\\ y &= \left\{\begin{array}{ll} x &\text{if }x\geq 0;\\ 0 ...
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 ...
6
votes
3answers
274 views

Solving $\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$

I recently came across this equation : $$\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$$where $f \in \mathcal{C}^1(\mathbb{R}, \mathbb{R})$. I've done the following, but I'm stuck at ...
5
votes
3answers
428 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 $f(r) = 1/\sqrt{r}$?
3
votes
1answer
169 views

A differential-functional equation: $f'(f^{-1}(x)) = 1/g(x)$

Problem: Given $g(x)$, solve the equation $f'(f^{-1}(x)) = \frac{1}{g(x)}$ for an invertible and differentiable function $f(x)$. So far I have tried setting $y = f^{-1}(x) \Leftrightarrow x ...
1
vote
2answers
1k views

Converting polar equation to cartesian coordinate polar equation and back again?

OK, so I have the following polar equation: $r = Θ/20$ And I would like to translate this a little to the right, and down from the polar origin. Now, I figure since I know cartesian coordinate ...
0
votes
2answers
39 views

Reversing bijections defined via conditional expressions

Let's say that I have a variable $j$ defined by the following formula: $$j=\frac{n(n+2) + m}{2}$$ where $n$ and $m$ are two parameters, both integers, that satisfy the following conditions: $n\in ...