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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0
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2answers
78 views

How can one show that$f(x^n)=nf(x)$

How can one show that if $f(xy)=f(x)+f(y)$ holds for all for real $x$ and $y$ that $$f(x^n)=nf(x).$$ How can i prove that $f(\frac{1}{x})=-f(x)$ To calculate f(1) do i need to pute x=1 ? Do i need ...
1
vote
1answer
18 views

Determine formula to calculate inverse range between two numbers

I've got a measurement for a graphics processing filter that increases intensity as the value decreases. For whatever reason, this filter's default value (minimum) is 1.75, and I've set the maximum ...
0
votes
0answers
41 views

Uniqueness question from functional equation

Let $f(x)$ and $f_2(x)$ be real and continuous on the interval $[0,\infty[$. Let $f(1) = f_2(1) = 1 , f(2) = f_2(2) = e$. Let $g(x)-g(x-1) = f(x-1)$. Let $ f ' (x) = exp(g(x) - g(1)).$ and $f '' ...
6
votes
1answer
96 views

How to prove a function is periodic from a given functional equation?

Given that $f: \mathbb R \to \mathbb R$, and that for some $a \in \mathbb R$, $f(x+a)={1\over 2}+\sqrt{f(x)-f^3(x)}$; prove $f(x)$ is a periodic fucntion. I know that to prove a function is ...
0
votes
1answer
16 views

Re-arrangement of equation

This type of question might be voted down or frowned upon, but if anyone could point me in the right direction I would be very grateful. I have worked through a question getting the following result: ...
0
votes
0answers
13 views

Formulation of a rule

Im new to this forum(hence, formatting issues). I come from CS background. I am trying to figure out a formula to code set of rules. The requirement is as below. Imagine two parameters X and Y ...
3
votes
3answers
59 views

If $g(x) = \frac{x}{x+1}$, and $f\circ g(x) = x^2$, find $f(x)$? [closed]

$$g(x) = \frac{x}{x+1}$$ $$f\circ g(x) = x^2$$ $$f(x) =\text{ ?}$$ Hello, I don't have any clue how to solve that. Do you have any ideas how to solve that? Thanks in advance.
-1
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2answers
69 views

Solve the functional equation $f(x) + f(x^2) = 2$ [closed]

What are the solutions of the functional equation $f(x) + f(x^2) = 2$? Will they be one to one or many to one? Will they be periodic or not?
3
votes
1answer
69 views

Polynimials $P(x)$ and $Q(x)$ satisfying $P(x^2+1)=(Q(x))^2+2x, Q(x^2+1)=(P(x))^2$

I am looking for all polynomials $P(x)$ and $Q(x)$ satisfying the simultaneous functional equations given in the title, i.e. $P(x^2+1)=(Q(x))^2+2x$ $Q(x^2+1)=(P(x))^2$ I've already found one ...
3
votes
2answers
67 views

Can one do anything useful with a functional equation like $g(x^2) = \frac{4x^2-1}{2x^2+1}g(x)$?

I got $$g(x^2) = \frac{4x^2-1}{2x^2+1}g(x)$$ as a functional equation for a generating function. Is there a way to get a closed form or some asymptotic information about the Taylor coefficients ...
3
votes
1answer
81 views

Extremal of a functional $I=\int\limits_0^{x_1} y^2(y')^2dx$

The extremal of the function $$I=\int\limits_0^{x_1} y^2(y')^2dx$$ that passes through $(0,0)$ and $(x_1,y_1)$ is a constant function a linear function of x part of a parabola part of ...
1
vote
2answers
49 views

Can any functions be expressed in one formula? (without conditions)

This is the extension of my previous inquiry: Is it possible to describe the Collatz function in one formula? Can each of all functions be expressed in one formula? That is, can any function ...
3
votes
2answers
93 views

Solving $f(x)=f(\frac{1}{x}),\,\,\,x>0$

What are the non-trivial solutions of the following functional equations: $$f(x)=f(\frac{1}{x}),\,\,\,x>0$$ given $f$ if differentiable on $(0,\infty)$. $\textbf{Edit:}$ Do all the solutions ...
3
votes
2answers
56 views

$f: [0,1] \to [0,1]$ is injective & $ f(2x-f(x))=x$ for $x \in [0,1]$ then $f(x)=x $

Let $f: [0,1] \to [0,1]$ be an injective function such that $ f(2x-f(x))=x$ for $x \in [0,1]$. I have to prove that $f(x)=x, x\in [0,1]$. My attempt: let $$g(x)=2x-f(x)$$ then $$f(g(x))=x \implies ...
0
votes
1answer
30 views

Functional Equation Different Substitution Different Result

Okay Consider this g(x).g(y)=1 for all x,y ∈ ℝ Substitute x=y we get g(x)= 1 or -1 .Now Substitute y=1/x g(x).g(1/x)=1 we get g(x)=x^n where n∈N Different substitutions produced different ...
0
votes
1answer
22 views

Cauchy functional equation three variables

If I have function from $R^3$ to $R$ satisfying $f(x_1,x_2,x_3)+f(y_1,y_2,y_3) = f(x_1+y_1,x_1+y_2,x_3+y_3)$ is it necessarily linear? $f(z_1,z_2,z_3) = \lambda _1 z_1+\lambda _2 z_2+\lambda _3 ...
3
votes
1answer
65 views

How to solve a functional differential equation?

$$(1) \quad \cfrac{d}{dx} (f(x^n))=\cfrac{-f(x^n)^2}{f(n \cdot x^{n-1})}$$ How do I solve this functional differential equation? I need a closed form solution, so approximations won't cut it, I'll ...
2
votes
2answers
58 views

Find the period of a function $\varphi:\mathbb{R}\backslash\{3\}\to \mathbb{R}$ [closed]

Let $\varphi:\mathbb{R}\backslash\{3\}\to \mathbb{R}$ a periodic function so that forall $x\in \mathbb{R}$ $$\varphi(x+4)=\frac{\varphi(x)-5}{\varphi(x)-3}$$ Find the period the $\varphi$.
1
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1answer
26 views

How to state a recurrence equation

I'm working on my homework and I saw this problem: A divide and conquer algorithm $X$ divides any given problem instance into exactly two smaller problem instances and solve them recursively. ...
0
votes
2answers
124 views

why this function is even function

Let $f(x)$ be a continuous differentiable odd function, and for any $x,y\in R$, such $$f(x-y)=f(x)g(y)-g(x)f(y)$$ show that $g(x)$ is an even function. How can I go about solving this? I ...
0
votes
2answers
37 views

Show that $y^n+1/y^n \in \mathbb{N}$

Let $y$ be the solution of the equation $x+\frac{1}{x}=3$ such that $y>1$. Show that $y^n+\frac{1}{y^n} \in \mathbb{N}$ We have $x+\frac{1}{x}=3$ if and only if $ x^2-3x+1=0$. Which has two ...
1
vote
1answer
59 views

Monotonically increasing functions 123

If $f$ is a monotonically increasing function and if $$f\left({x+f(x)\over2}\right)=x$$ for every $x\in\mathbb R$, prove that $$f(x)=x$$
2
votes
1answer
77 views

$F(F(x)+x)^k)=(F(x)+x)^2-x$

I have no idea about this problem. But I feel we have to use chain rule of differentiation here. The Function $F(x)$ is defined by the following identity: $F(F(x)+x)^k)=(F(x)+x)^2-x$ The value of ...
10
votes
1answer
80 views

Vector Space Structures over ($\mathbb{R}$,+)

Consider the abelian group ($\mathbb{R}$,+) of real numbers with the usual addition. Is there a scalar multiplication \begin{equation} \cdot : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}, ...
2
votes
1answer
23 views

Two variable equation, weighting of stock

I am trying to find the weighting of stocks in a portfolio (both variables should be between 0 and 1 but added together should be 1). the equation is: $1.2353 = x(1.2) + y(0.9)$ I have it simplified ...
1
vote
4answers
149 views

How can I find $f'(2x)?$

it might seem a little bit elementary. $f$ is defined on $\Bbb R$ and it is differantiable. and is not equal to zero. if $xf(x)-yf(y)=(x-y)f(x+y)$ then find what is $f'(2x)$ equal to?. ...
2
votes
1answer
85 views

For which polynomials $P$ does there exist a polynomial $Q$ such that $P(x)=Q(x)Q'(x)$?

I am wondering for which polynomials $P$ does there exist a polynomial $Q$ such that $P(x)=Q(x)Q'(x)$? I am quite sure a complete characterisation will be very hard, but I'm looking for partial ...
3
votes
1answer
57 views

What “natural” functions besides $g(x) = x$ and $g(x) =\dfrac{c^{x}}{c^{x}+c^{1/2}} $ satisfy $g(x)+g(1-x) = 1$?

I'm not quite sure how I should state this question. This is one way: What "natural" functions besides $g(x) = x$ and $g(x) =\dfrac{c^{x}}{c^{x}+c^{1/2}} $ satisfy $g(x)+g(1-x) = 1$? By "natural" I ...
1
vote
3answers
44 views

A function that is zero for every integer multiple of $k$

I'm new to this kind of question so it may be a trivial one but I can't find a general solution: I'm searching for a function that has a value of $0$ for every integer multiple of some fixed real ...
5
votes
1answer
39 views

Find the function $f(x)$

Given $f(x)$ is a differentiable function such that $$f(x+y)=e^xf(y)+e^yf(x)$$ $\forall$ $x,y$ $\in$ $\mathbb{R}$ and $f'(0)=1$. Find $f(x)$ if we put $y=0$ we get $$f(x)=e^xf(0)+f(x)$$ $\implies$ ...
0
votes
0answers
41 views

Finding the “hidden values”.

I have this problem but my knowledge of the methods of linear algebra is totally useless if it exists at all (I know what is a matrix at least... maybe). I have a three collections of objects ${\bf ...
5
votes
2answers
133 views

Is there a function satisfying the following properties $ f^{n}(x)=(f(x))^{n+1}$??

Is there a function with the following properties? $$ f(x)=f(x) $$ $$ f'(x)=f(x)^2 $$ $$f^{(n)}(x)=\left(f(x)\right)^{n+1}$$ where $f^{(n)}$ denotes the $n$th derivative, and by convention ...
3
votes
3answers
172 views

if I know $f(x+1) = 2f(x) + 1$, how do I solve f(x)

This is just my thought on run time of a binary search: if you are allowed to make 1 comparison, you can search a sorted list of length 1, but if you are allowed to perform 2 comparisons, you can ...
0
votes
2answers
52 views

Functional equation $f(h(y)x+y)=g(y)f(x)$

(Note: this is a simplified version of my previous question, which was not answered). I am seeking the solution for the functional equation $f(h(y)x+y)=g(y)f(x)$ where $f,g,h$ are continuous. ...
1
vote
2answers
131 views

Let $f:\mathbb R\rightarrow \mathbb R$ be a continuous function such that $f(x)-2f(\frac{x}{2})+f(\frac{x}{4})=x^{2}$ . Then $f(3)$ =? [closed]

Options: (a)$4$, (b)$4f(0)$, (c)$4-f(0)$, (d)$4+f(0)$, (e)$16+f(0)$. CORRECT ANSWER USING REDUCTION Deep thanks to @martini and @A.S. , soo ...
5
votes
1answer
83 views

Solving functional equation $2f(x) = f(2x)$

$f(x)$ is a $\mathbb{R} \rightarrow \mathbb{R}$ differentiable function satisfying the following equation: $$2f(x) = f(2x).$$ Can it be proved that $f(x) = kx$ for some $k$? Note that if $f(x)$ is ...
3
votes
2answers
62 views

Functional equation $f(x+y)=f(x)+2xy+f(y)$

I am interested in classifying solutions $f\,:\,\mathbb R\longrightarrow \mathbb R$ to the functional equation \begin{equation} f(x+y)=f(x)+f(y)+2xy\qquad\qquad(\dagger) \end{equation} and in ...
2
votes
2answers
85 views

What is the vertex of this radical equation?

the question is $$y = \sqrt{-(x -3}) + 4$$ i thought the vertex was (3,4) but i was wrong and that it was supposed to be (3,2). Was i right, can anyone help me with these type of question?
1
vote
1answer
52 views

Solving functional equation $f:Q^+\to R^+$ where $f(xy)=f(x+y)(f(x)+f(y))$

Find all functions $f:\mathbb{Q}^+ \to \mathbb{R}^+$ with the property: $$f(xy)=f(x+y)(f(x)+f(y)),\qquad \forall x, y\in\mathbb{Q}^+ \tag{1}$$ This question is from the 2014 Bulgaria National ...
0
votes
0answers
47 views

Cauchy-like functional equation $f(h(y)\cdot x+y)= g(y)f(x)+f(y)$

I am looking for the solution to the following two variable functional equation: (*) $f(h(y)\cdot x+y)= g(y)f(x)+f(y)$ where: $h$ is some given continuous function, $f, g,$ unknown functions on ...
5
votes
2answers
76 views

Finding a function (?) and computing its definite integral

So I've come across this exercise from one of my old highschool textbooks: $$\text{If}\ 2f\bigg(\frac{x-2}{x+1}\bigg) +f\bigg(\frac{x+1}{x-2}\bigg) = x$$ Considering this, find : ...
1
vote
1answer
27 views

Any chances this can be further reduced?

I've come with the following equation, after a lot of simplification, but can't reduce further. Any chances it can be solved by reducing the $b$ and get the value of $a$? $$a = \frac{1000(1000 - ...
4
votes
2answers
121 views

Prove that $f′(x)=f′(0)f(x)$ derivatives

Let $f:I \to R$ be differrentiable on an open interval $I \subseteq R$ with $$f(a + b) = f(a)f(b) \quad \forall a, b \in R$$ Suppose that $f(0) = 1$ and that $f'(0)$ exists. Show that: $$f'(x) = ...
0
votes
1answer
89 views

A nice form of a given function

First let, $\oplus(a_1,a_2,\ldots,a_n)$ denote the bitwise xor of $a_1,a_2,\ldots,a_n$. Define the function $\Delta(a_1,a_2,\ldots,a_n)$ to be the maximum value of $a_i - ...
3
votes
1answer
103 views

Find continuous functions that satisfy $f(f(x))=x$ over the reals.

I'm looking for a method to solve: $$f(f(x))=x$$ Where $f$ is defined for $x \in R$ So far by inverting both sides I have: $f(x)=f^{-1}(x)$ Which means that my function should be symmetrical over ...
3
votes
1answer
74 views

How to solve this nonlinear functional recurrence

I study two similar nonlinear functional recurrence systems, given by $$P_\pm:\qquad f_n\cdot(1\pm g f_{n-1}) = g\mp(1+2g)f_{n-1} \qquad (n>0)$$ and $$f_0=g$$ Here $f_n$ and $g$ are functions of ...
2
votes
1answer
39 views

How to algebraically prove the following inequation?

Following is the inequation I have been trying to prove for a while. $$\frac{\frac{1}{2}(1-q)}{\frac{1}{2}(1-q) + pq}\neq \frac{\frac{1}{4}(1-q)}{\frac{1}{4}(1-q) + p^2q} + ...
-1
votes
2answers
36 views

Obtaining values from functional equation without solving [closed]

Let $$f(x)=\frac{1}{2}[f(xy)+f(x/y)]$$ for real positive x,y such that f(1)=0 and f'(1)=2. How to find f'(3) and f(e) without explicitly solving the recursion?Any suggestions?
1
vote
1answer
100 views

Find $f:\mathbb{R}\to\mathbb{R}$ such that $f(xy+x+y)=f(xy)+f(x)+f(y)$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function that satisfies $$ f(xy+x+y)=f(xy)+f(x)+f(y) $$ Find $f$ and prove that $$ f(x+y)=f(x)+f(y) $$
0
votes
2answers
32 views

How to rewrite $x-y=\frac{x}{y}$ so that it become $y=$ (something…)?

For example, $x+y=x\times y$ is easy to express as $y=\frac{x}{x-1}$, how about $x-y=\frac{x}{y}$? I tried multiply both sides by $y$ and become $y^2-xy+x=0$ but up to this step I don't know how to ...