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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3answers
83 views

Is there any functional equation $f(ab+cd)= f(a)+f(b)+f(c)+f(d)$?

I am looking for a real, continuous function that satisfies the functional equation $$ f(ab+cd)= f(a)+f(b)+f(c)+f(d) $$ where $a,b,c,d$ are real. This is equivalent to a function satisfying these two ...
4
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2answers
105 views

find all possible functions : $f(a)f(b)-6ab=\frac{3}{2}f(a+b)$

I'm trying to find all possible functions that satisfy this functional equation: $f(a)f(b)-6ab=\frac{3}{2}f(a+b),$ $f\in \mathbb{R}.$ My attempt : $a=b=0$ then $f(0)=0$ or $1.$ But I don't ...
0
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1answer
41 views

Functional equations in one variable.. [closed]

How do you solve the functional equation involving only one variable...what if and if not given that $f(x)$ is a polynomial... Say for example $f(x)=f(x-1) +2x$
1
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0answers
31 views

$‎\lim_{n\to\infty} (xf(n) + \sum_{k=1}^n f(k) - f(k+x))‎$‎ is‎ ‎convergent for $x\geq 1$. [closed]

‎‎Let $f:[1‎, +‎\infty)\rightarrow\mathbb{R}$ be a function such that ‎$‎\lim_{n\to\infty} (f(n) - f(n+1)) = 0‎$‎‎. ‎Suppose that $\lambda:[1‎, +‎\infty)\rightarrow\mathbb{R}$‎ ‎is an exp-convex ...
2
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1answer
96 views

IMO 1993 b2 proof

"Let $\mathbb{N}=\{ 1,2,3,\ldots \}$. Determine if there exists a strictly increasing function $f:\mathbb{N}\to\mathbb{N}$ such that 1) $f(1)=2$ 2) $f(f(n))=f(n)+n$ for all $n$." Of the solutions ...
2
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2answers
95 views

Find all functions $f : \mathbb{(0,\infty})\to\mathbb{R}$ such that $f(x+y) = xf(y)+ yf(x)$

Find all functions $f : \mathbb{(0,\infty)}\to\mathbb{R}$ such that $f(x+y) = xf(y)+ yf(x)$ if $f$ is continuous at $x=1$ This problem was looking quite easy at first but the domain of positive ...
0
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2answers
139 views

Solve functional equation $f(z)=c+zf(z^2)$ with series expansion?

Let the functional equation $(1)$ be given as $$ f(z)=c+zf(z^2) \tag{1}$$ where $c \in\mathbb R$ and $c \neq 0$. How can this functional equation be solved with series expansion (power, Taylor or ...
2
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1answer
130 views

Find functions $f,g$ such that $f(x+y)=g(x·y)$ for all $x,y$? [closed]

My approach towards this question was that first I putted x=0 and then y=0 which yields f(x)=f(y)and f(x)=g(0), and again for x=1 & y=1 it gives g (x)=f(1+x), and so on. My query is that what ...
4
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0answers
72 views

Solving the equation $f(x)=f^{-1}(x)$. [duplicate]

Exactly under what conditions would the equality $f(x)=f^{-1}(x)$ hold? The proof attached below considers a special case when $f$ is strictly increasing. The theorem then says that the set of ...
4
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2answers
91 views

Functional problems: Find all functions such that $f(x)f(y) = f(xy + 1) + f(x - y) – 2$

Find all functions such that $f(x)f(y) = f(xy + 1) + f(x - y) – 2$ for all $x, y $ are real numbers. I put y=0 into the equation and get $(f(0)-1)f(x)=f(1)-2$. If $f(0)≠1$, then $f(x) = (f(1)-2)/...
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2answers
60 views

Power-like functional equation

I would like to know what are the especifications of a functional equation that give us a power function as a solution. For example, if $f:\Bbb R \to \Bbb R$ is continuous and monotonic, such that $$...
2
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0answers
51 views

Functions satisfying $f(2x)+f(3x)\leq k f(x)$

Given $a<0$ such that $k:=2^{a}+3^{a}\in (0,1)$ and define $f(x):=x^{a}$ for all $x>0$. Then, $f(2x)+f(3x)=x^{a}(2^{a}+3^{a})=kf(x)$. Somebody know another example of a function $f:(0,+\infty)\...
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0answers
27 views

Additive and Bijective function on the real line

I was trying to solve the following functional equation: $f(f(x-y))=f(x)-f(y)$. And I concluded that $f$ must be additive and bijective. The question is: Let $f:\mathbb{R} \to \mathbb{R}$ be an ...
2
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2answers
448 views

$2+ f(x)f(y)=f(x)+f(y) +f(xy) $, if $f(2)=5$ find $f(5)$

This question has been asked before, however I am interested in seeing why my approach to finding a solution does not work. $2+ f(x)f(y)=f(x)+f(y) +f(xy) $, if $f(2)=5$ find $f(5)$ What I have ...
6
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1answer
75 views

If $f(x)f(y)+f(xy)\le -\frac{1}{4},\forall x,y\in[0,1)$, show that $f(x)=-\frac{1}{2}$

Let $f:[0,1) \to \mathbb{R}$ be a function such that $$f(x)f(y)+f(xy)\le -\dfrac{1}{4} \quad \forall\, x,y\in[0,1).$$ Show that $$f(x)=-\dfrac{1}{2} \quad \forall\, x \in[0,1).$$ I have proved that ...
2
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2answers
61 views

Find $x$ where $f(e^{(x+1)})=x-\ln(x)$ approaches one.

Given that $$ x \in [1,\infty) \quad f(e^{(x+1)})=x-\ln(x) $$ and $$ \lim_{x \to a} f(x)=1 $$ Find $a$. I got to the point: $$ \ln(a)-\ln(\ln(a)-1)=2 $$ But from there on I could not get to $a=e^...
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0answers
23 views

How to find a monotone increasing function given its relationship with its inverse?

I am trying to find all functions $$f : [0, 1] \rightarrow [0, 1]$$ continuous monotone strictly increasing such that $$f(v) = v - \frac{v}{2} \cdot f^{-1}(\frac{v}{2}).$$ However I have no idea how ...
6
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1answer
70 views

How to prove $f$ is $C^\infty$

Suppose $f:U \subseteq \mathbb{R}^2 \rightarrow \mathbb{R}$ is continous and $$(x^2+y^4)f(x,y)+(f(x,y))^3=1 \: \text{for all} \: (x,y) \in U. $$ Prove $f$ is $C^\infty$. This kind of exercise ...
22
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3answers
558 views

Find all functions that satisfy $f(\frac{x+4}{1-x}) + f(x) = x$

I found the following task in a book and I would be interested if someone has an idea to solve it: Find all the functions $f$ that satisfy $f(\frac{x+4}{1-x}) + f(x) = x$. My ideas: Assuming that $...
2
votes
2answers
112 views

Find all functions satisfying $f(x+1)=\frac{f(x)-5}{f(x)-3}$

Find all functions satisfying $$f(x+1)=\frac{f(x)-5}{f(x)-3}$$ My try: We have $$f(x+1)=1-\frac{2}{f(x)-3}$$ Letting $g(x) =f(x+1)-3$ We get $$g(x+1)=-2-\frac{2}{g(x)}$$ Any clue here?
4
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3answers
195 views

Find all solutions to the functional equation $f(x+y)-f(y)=\frac{x}{y(x+y)}$

Find all solutions to the functional equation $f(x+y)-f(y)=\cfrac{x}{y(x+y)}$ I've tried the substitution technique but I didn't really get something useful. For $y=1$ I have $F(x+1)-F(1)=\...
2
votes
2answers
163 views

An equation of rational functions

I'm trying to get the set of solutions of the following equation, whose unknowns are the rational functions $f$ and $g$ : $\forall x\in\mathbb{R}$ such that the LHS and RHS are both defined, $$f(x)f(...
4
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3answers
78 views

Math question functions help me?

I have to find find $f(x,y)$ that satisfies \begin{align} f(x+y,x-y) &= xy + y^2 \\ f(x+y, \frac{y}x ) &= x^2 - y^2 \end{align} So I first though about replacing $x+y=X$ and $x-y=Y$ in the ...
6
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2answers
180 views

Multivariate polynomial functional equation

I’m having some difficulties solving the following functional equation: Find all polynomials $P(x,y)\in\mathbb{R}[X,Y]$ for which: $P(x,y)$ is homogeneous (so $\exists n\in\mathbb{N}, \...
0
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2answers
102 views

Find a function $f$ with $xf(x) - 1 = f(3-\frac{1}{x-1})$

I was playing around with a problem and arrived at the following functional equation, $$xf(x)-1 = f\left(3-\frac{1}{x-1}\right),$$ where $x$ is a real number. I know such a function must satisfy $...
1
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1answer
76 views

The functional equation $f(-x+b)=f(x)$

I can solve the (periodic) functional equation $f(x+b)=f(x)$ completely ($x\in \mathbb{R}$ and $b\neq 0$). Indeed, its general solution is $f=\phi o (\; )_b$, where $(\; )_b$ is the $b$-decimal (...
7
votes
3answers
367 views

Help in solving a simple functional equation

I need to find all continuous functions satisfying: $$3f(2x+1)=f(x) + 5x$$ The functional equation looks simple but I am unable to solve it. I tried to convert it into a Cauchy type equation but I ...
2
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3answers
51 views

A functional equation (another)

I would like to find a continuous concave function from $[1/2,1]$ to $[0,1]$ such that $f(1)=1$ and for all $x\in [1/2,1]$ $$f(x)= \frac{1}{2} + \frac{1}{4}f\left(\frac{2x}{1+x}\right).$$ I am ...
0
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1answer
41 views

The condition for the existence of a symmetric form for the reflection formula $f(1-x)= \chi (x) f(x)$

Suppose we have a functional equation in the form $$f(1-x)=\chi (x) f(x)$$ with given function $\chi (x)$. What is the condition on the function $\chi (x)$ so that we can write this reflection ...
3
votes
3answers
123 views

Functional equation: $f(x)f(1/x) = f(x) + f(1/x)$.

If $x \neq 0$ , find $f(x)$ if it satisfies: $f(x)f(1/x) = f(x) + f(1/x)$. I know that the answer is $f(x) = 1 \pm x^n$ where $n \in \mathbb{R}$. I don't know how to show this.
3
votes
1answer
73 views

A particular Functional equation

Bonjour, Find all continuous functions, $f$, such that $f(x)-1999f\big(\frac{2x}{1-x^2}\big)=18$ for $|x|\neq 1$. My try: taking $x=\tan{h}$ leads to: $f(\tan{4h})-f(\tan{2h})=\frac{1}{1999}(f(\...
0
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0answers
21 views

Difficulties with Inner products and polarization identities

I am discussing the general inner product space. Here is what Polarization Identities mean. I denote the inner product by $(x,y)$. I am having a difficult time with the polarization identities. Of ...
3
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1answer
76 views

Find all real functions such that $(x + 1)f(xf(y)) = xf(y(x + 1))$

Find all real functions of real variable such that $$(x + 1)f(xf(y)) = xf(y(x + 1))$$ Let $a=f(0)$. For $y=0$ we get $(x+1)f(ax) = ax$, so if $a\ne 0$ we get $$f(x) = {ax\over x+a}$$ which is actual ...
6
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0answers
162 views

Functional Equation $f(m + f(n)) = f(m) - n$ [duplicate]

Well, similar questions have already been asked. But they are not identical and the solution methods offered there are not the same one as here. Anyway, I want to solve functional equation $f(m + f(n))...
8
votes
2answers
226 views

Solve the functional equation $\frac{f(x)}{f(y)}=f\left( \frac{x-y}{f(y)} \right)$

Solve the functional equation $$ \frac{f(x)}{f(y)}=f\left( \frac{x-y}{f(y)} \right), $$ here $f: \mathbb{R} \to \mathbb{R}$ and $f$ is differentiable at $x=0.$ By set $x=y$ we get $f(0)=1$. ...
6
votes
1answer
148 views

Is it possible to find aproximation of conformal map from sequences of complex points?

I want to find equation of conformal map (= Fatou function $\Psi : z \to u$ ) which: maps some region of complex plane ( attracting petal) to right half of complex plane in u coordinate $Re(u) > ...
1
vote
2answers
81 views

Functional Equation Problems

If $\Bbb N$ denotes all positive integers. Then find all functions $f: \mathbb{N} \to \mathbb{N}$ which are strictly increasing and such for all positive integers $n$, we have: $$f(f(n)) = n+2$$ So ...
2
votes
1answer
60 views

Functional equation in single-variable calculus

Suppose we know that $f(x) \in C^2$ and $f(x)$ defined for all real numbers. Furthermore, $f(x)$ has following property: $$ \forall x,y \in \Bbb R \quad f(x+y) - f(x) = yf'(x + \frac{y}{2})$$ How to ...
-1
votes
2answers
111 views

Functional Equation $f(x)f(f(x)+\frac{1}{x})=1$

I'd like to ask how to find all solutions to the functional equation $f(x)\cdot f(f(x)+\frac{1}{x})=1$, where $f: (0, +\infty)\to\mathbb{R}$ is strictly increasing?
4
votes
3answers
127 views

Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y,z \in \Bbb{R} $ , $f(xf(x)+f(y))=x^2+y$

Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y, \in \Bbb{R} $ , $f(xf(x)+f(y))=x^2+y$ We can easily get a strong condition $f(f(y))=y $ by setting $x=0$ . By this equation we ...
6
votes
2answers
122 views

Find $f(x)$ if $f(x)+f\left(\frac{1-x}{x}\right)=1-x$

If $f: \mathbb{R} \to\mathbb{R} $, $x \ne 0,1$ Find all functions $f(x)$ such that $f(x)+f\left(\frac{1-x}{x}\right)=1-x$ My try: Letting $$g(x)=x+f(x)$$ we get $$g(x)+g\left(\frac{1-x}{x}\right)=\...
1
vote
1answer
177 views

Commutative version of hyperoperators

As I understand it, addition and multiplication are defined on the reals as having identity elements 0 and 1 and being commutative and associative. Multiplication is also distributive over addition. ...
19
votes
4answers
441 views

Solving $f(yf(x)+x/y)=xyf(x^2+y^2)$ over the reals

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that $f(1)=1$ and for all real numbers $x$ and $y$ with $y \neq 0$, $$f\Bigg (yf(x)+\frac{x}{y}\Bigg)=xyf(x^2+y^2)$$ This seems quite hard. $f(x)=...
1
vote
1answer
69 views

Find $f(x)$ where $f(x)(A-\frac{B}{x+B/A})+Cf(x+\frac{B}{A})=0$.

$A, B, C > 0$, $x$ is complex and $Re(x)>0$. My guess is that $f(x)=0$ but I don't know how to prove it.
1
vote
1answer
23 views

Pexider's (/ Cauchy's) functional equation over a bounded domain

I am looking at Pexider's equation $f(x+y)=g(x)+h(y)$, where $f,g,h$ are continuous functions but are defined over bounded domains. Specifically, $f,g,h$ each is defined on a real interval (of length ...
0
votes
0answers
30 views

Cauchy's functional equation from complex to reals

I am looking for the general continuous solutions $f:D \rightarrow \mathbb R$ of the multiplicative Cauchy functional equation $f(x)f(y) = f(xy)$ for the domain $D=\{x \in\mathbb C: |x|<1\}$. (...
2
votes
1answer
43 views

Solving tricky functional equation resembling quadratic equation

I have the following functional equation in hand, I can easily solve it for the case $(a, b ,c)=(1, 1,0)$ which gives $f(x)$ to be $x^2+x$. $\begin{aligned}{g(x)=a\left[f(x)\right]^2+bf(x)+c, \text{...
13
votes
1answer
259 views

Find all pairs of functions $(f,g)$, $\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$

Find all pairs of functions $(f,g)$ : $\mathbb{R} \to \mathbb{R}, g : \mathbb{R} \to \mathbb{R}$ satisfying : $$\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$$ I am really stuck ...
8
votes
1answer
65 views

Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y,z \in \Bbb{R} $ , $f(f(x)+yz)=x+f(y)f(z)$

Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y,z \in \Bbb{R} $ , $f(f(x)+yz)=x+f(y)f(z)$ I was told to do this by proving $f$ is injective and surjective. I have proved it this ...
1
vote
1answer
53 views

Do I need to verify solutions to functional equations?

I am studying the (basics of) solving functional equations. My teacher stipulates that we check any solutions obtained by substitution. Similar guidelines are given in this IMO training material. For ...