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).

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-1
votes
1answer
82 views

What can be $f$ so that $f(f(x)) = -x$? [duplicate]

What can be $f$ so that $f^2(x) = -x$ for all $x\in R$? I know that if $f^2(x) = -x$ then $f(x)$ is injective and $f$ can not be continuous. But I can not find an example of discontinuous function ...
-1
votes
1answer
35 views

Find all possible choices of real-valued functions $f(x)$ and $g(y)$ such that $f (x) + g(y) = log (1 + x + x y + y)$ for all positive $x$ and $y$.

Question: Find all possible choices of real-valued functions $f(x)$ and $g(y)$ such that $$f(x) + g(y) = \log (1 + x + x y + y)$$ for all positive $x$ and $y$. Attempt: Note that $$f(x)+g(y)=\log(...
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votes
5answers
351 views

Midpoint convex functions that are not convex or which are not super-additive but F(0)=0, F injective, and non-negative

When I refer to 'Midpoint Convexity', I mean its form when restricted to the unit interval. I denote this by $(MP)$ . Where, $\text{Dom(F)}=[0,1]$. $$(MP):\forall((x,y)\in [0,1]):\, F\left(\frac{x+y}{...
-1
votes
1answer
69 views

Prove that $f(x) = x$ is the unique solution to the following functional equation [closed]

Let $f : [0 , \infty) \to \mathbb{R},$ continuous at $ x_0 = 0$ satisfying $$f(3x) - 2x = f(x)$$ Prove that $f(x) = x$.
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votes
1answer
18 views

The bivariate function from its marginals

Suppose I have a bivariate function, $f(x,y)$. Fixing $x$, we define $f_x(y)=f(x,y)$. Fixing $y$, we define $f_y(x)=f(x,y)$. Now if we know the form of $f_x(.)$ for all $x$, and the form of $f_y(.)$ ...
-1
votes
1answer
34 views

Searching a formula for scaling/mapping a variable based on 3 known values

I am sending an specc'ed integer (X) between -2048, and 2048 to a synthesizer to control its tuning. When X is 0 = Tuning on Synth is 440 (default) When X is 2048 = Tuning on Synth is 546.42 When X ...
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votes
3answers
68 views

How can one show that $f(n)>n$

Given a function $f\colon \Bbb N^\ast\to \Bbb N^\ast$ such that we have $f(f(n))=f(n+1)-f(n)~\forall~n\in\Bbb N^\ast$. Show that: If $f(n+1)-f(n)>1$ then $f(n)>n$. Since $f(n+1)>f(n)+1$ I ...
-2
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1answer
68 views

How to solve the functional equation $ f(x^2+xf(y))= xf(y)$ [closed]

Hello please how to find all the functions $f:\mathbb{R}\to \mathbb{R}$ such that $$ f(x^2+xf(y))= xf(y)$$ I see that $f(0)=0$ but how to do after
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votes
2answers
129 views

For the given function $f(2)=5$, find $f(3)$

Given that $f(x)$ is differentiable function of $x$ and that $f(x)\cdot f(y) = f(x) + f(y) + f(xy) - 2$ and that $f(2) = 5$. Then find value of $f(3)$. By putting $x=2$ and $y=1$ , I got $f(1)=2$ ...
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votes
3answers
90 views

Prove that $\boldsymbol{\ f}$ is a constant function

if :$$f:[0,1]→\mathbb{R}$$ and for all $x\in[0,1]$ $$f(x)=f\!\left(x^2\right)$$ and for all $b\in[0,1]$ $$ \lim_{ x \to b }f(x)=f(b)$$ prove : for some $c\in\mathbb{R}$, for all $x\in[0,1]$, $f(x)=c$...
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votes
2answers
51 views

Number of functions in $g(g(x))=g(x)$

A function is defined as $g:\{1,2,3,4\}\rightarrow \{1,2,3,4\}$ such that $g(g(x))=g(x)\forall x\{1,2,3,4\}.$ Then number of such functions are Try: from $g(g(x))=g(x)$ implies $g(x)=x$Identity ...
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votes
2answers
55 views

Find all functions $f:\Bbb R \rightarrow \Bbb R$ such that for every $x,y \in \Bbb R$: i) $f(x)\leq x$ ii) $f(x+y)\leq f(x)+f(y) $ [closed]

Find all functions $f:\Bbb R \rightarrow \Bbb R$ such that for every $x,y \in \Bbb R$: i) $f(x)\leq x$ ii) $f(x+y)\leq f(x)+f(y) $ In functional equations, I have trouble when inequality has a ...
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votes
2answers
180 views

Integral equation that's cant solve… Need a hand [closed]

Help me solve this integral equation, I'm having some troubles... I need to use the Fredholm method for second kind integral equations. $$\phi(x)= \sin(x)+ \lambda \int_{0}^{\pi}\cos(2x+y)\phi (y)dy$$...
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votes
2answers
42 views

Function involving real values

Let $f$ be a function satisfying $f(x/2+y/2)=(f(x)+f(y))/2$, for all real $x$ and $y$. If $f'(0)$ exists and equal to $-1$ , then $f(2)$ equals: ? I have tried this question by putting various values ...
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votes
1answer
158 views

Functional differential equation (FDE): $f'(x)=f(x+e)$

I would like to solve the functional differential equation $$f'(x)=f(x-e)$$ I've solved functional equations before, and I've solved differential equations, but I've never solved a functional ...
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votes
1answer
111 views

I am trying to find the function f that satisfies $\cos x=f '(x)+f(-x)$

$\cos x=f '(x)+f(-x)$ I manage to solve $f '(x)+f(x)= \cos x$, starting first by solving $y'=-y$ using $y=\exp(ax)$. But here I get stuck because of $f(-x)$.
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votes
1answer
70 views

Solving for this function

From the Indian National Mathematics Olympiad 1992: Determine all functions $f: \mathbb R -[0,1] \rightarrow \mathbb R$, satisfying the functional relation: $$f(x) + f \left( \frac{1}{1-x} \right) ...
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votes
2answers
50 views

Problem on Definite Integration and functions

Q. The function f is continuous and has the property $f(f(x))=1-x$ for all $x\in [0,1]$ and $$J = \int_{0}^{1} f(x) dx$$ then find $J$. My Attempt- I have no clue to this problem! Instead I tried ...
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votes
1answer
41 views

Find all the polynomials $W (x)$ that meet the following two conditions: [closed]

$W(0)=2$ and $W(x_1+x_2)=W(x_1)+W(x_2)+2x_1x_2-2$ The first condition I know (theorem Bezout). What does the second condition say to us? How to solve this math problem?
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votes
1answer
47 views

Algebra Problem Functions [closed]

Suppose that $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^2$ and $g(f(x)) = x^3$ for all $x \ge 1$. If $g(16) = 16$, then compute $\log_2 g(4)$. (You may assume that $f(x) \ge 1$ and $g(...
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votes
4answers
69 views

let $f( x + \frac{ 1}{ x } ) = x ^ 2 + \frac{ 1}{ x ^ 2} $ then $f(x)$ equals [closed]

let $f( x + \frac{ 1}{ x } ) = x ^ 2 + \frac{ 1}{ x ^ 2} $ then $f(x)$ equals options are $( A ) x ^ 2 - 2$ $( B ) x ^ 2 - 1$ $( C ) x ^ 2$ I don't know how to solve this type of problems
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votes
1answer
43 views

Functional Equations and information theory [duplicate]

What is the solution of the functional equation $f(xy)=f(x)f(y)$ for all $x$ and $y$ in $I$ where $f$ is a real-valued mapping with domain the unit closed interval $I$?
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votes
1answer
131 views

Find a function which satisfies the relation $f(x+1)=(f(x)+1)^{1/2}$

The original question was to find the value this function approaches as $x$ goes to infinity given that the limit exists. This is easy to figure and turns out to be $(1+5^{1/2})/2$. I was though, ...
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votes
1answer
71 views

How to find the derivative of function using given functional equation?

$$f \left(\frac{x+y}{2} \right) = \frac{f(x)+f(y)}{2}, \forall x,y \in \mathbb{R}$$ $$f'(0)= -1,\space f(0)=1$$ $$f'(u)=?$$
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votes
1answer
56 views

Solving a functional equation over the real number line [closed]

Find all functions $f : R \to R$ which satisfy $$f(x)f(y) = f(xy) + xy$$
-3
votes
1answer
48 views

Find all functions $f:\Bbb N\rightarrow \Bbb R $ such that: $f(m+k)=f(mk-n)$ [duplicate]

Find all functions $f:\Bbb N\rightarrow \Bbb R $ such that for a given value $n\in \Bbb N$ , the following identity holds: $$f(m+k)=f(mk-n) ,m,k \in \Bbb N , mk>n$$ This problem has already ...
-3
votes
2answers
1k views

Functional equation $f(ax)=bf(x)$ [duplicate]

What are all the solutions to the functional equation $f(ax)=bf(x)$, where $a,b>0$, and $f$ is continuous, strictly monotone and increasing, and $x$ ranges over the reals? references? proof? ...
-3
votes
2answers
46 views

Functional equation question. [closed]

Suppose that $f(x)$ is a function such that $f(xy) + x = xf(y) + f(x)$ for all real numbers $x$ and $y$. If $f(-1) = 5$ then compute $f(-1001)$.
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votes
1answer
84 views

Is it true that the only solutions are $f(x)=0$ and $f(x)=x$? [closed]

Suppose that $f: \mathbb R \rightarrow \mathbb R$ such that $$f(x^3+y^3)=f(x+y)\left(\left(f(x-y)\right)^2+f(xy)\right),$$ for all $x,y$ real numbers. Is it true that the only solutions are $f(x)=0$ ...
-3
votes
2answers
2k views

$g(x+y)=g(x)g(y)$ for all $x,y \in\mathbb{R}$. If $g$ is continuous at $0$, prove that $g$ is continuous on $\mathbb{R}$ [duplicate]

$g(x+y)=g(x)g(y)$ for all $x,y \in\mathbb{R}$. If $g$ is continuous at $0$, prove that $g$ is continuous on $\mathbb{R}$.
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votes
2answers
335 views

find all function f,g that satisfy

Find all function $f,g$ that satisfy: $$g(x)-g(y)=\frac{1}{6} (x-y)(f(x)+f((x+y)/2)+f(y))$$ For $y=0$ we have an equation in $f$: $$4(x-y)(f(x/2)-f(x/2+y/2))=xf(y)-yf(x)$$ How can i do it?
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votes
2answers
152 views

Nontrivial entire $f(z)$ never equal to $0$ [closed]

I'm looking for nonconstant entire functions $f(z)$ such that $f(z)\neq 0$ for any $z$. More specifically I'm looking for nontrivial cases. So $\exp(z),\exp(z^2),...$ is not what I am looking for. ...
-4
votes
1answer
81 views

Find all the $f:\mathbb{R}\rightarrow\mathbb{R}$,satisfying

$f(f^3(x)+y^3)=x^2+f^3(y)$ where $f^3(x)$ stands for $[f(x)]^3$. I really don't know where to start off$\cdots$
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votes
1answer
156 views

A suggestion on functional equations [closed]

The Help I need I am preparing for a competitive math exam. I need to learn methods of solving questions like this If f$\left(\frac{x+y}{3}\right)= \frac{2+f(x)+f(y)}{3}$ for all real $x$ and $y$ ...