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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25
votes
6answers
2k views

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
-2
votes
2answers
206 views

Solve the functional equation $f(1+xf(y))=yf(x+y)$

Problem Find all functions $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$ such that: $$f(1+xf(y))=yf(x+y)$$ for all $x,y \in \mathbb{R^+}$ Progress I can only prove $f$ is a surjective function. I ...
5
votes
1answer
480 views

non-continuous function satisfies $f(x+y)=f(x)+f(y)$

As mentioned in this link, it shows that For any $f$ on the real line $\mathbb{R}^1$, $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable. But how to show there exists such ...
2
votes
1answer
83 views

Functional inequality

Let $S$ be a semigroup such that $S\ne S+S$ and let $f:S\to \Bbb C$ be an unbounded function satisfying $$ |f(s_1)f(s_2)-f(t_1)f(t_2)|\le 1 $$ for all $s_1, s_2, t_1, t_2 \in S$ such that ...
5
votes
3answers
320 views

Find the value of the function at the given point.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function satisfying the conditions $$\begin{align*} (1)&f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}\\ (2)&f(0)=1\\ (3)&f'(0)=-1 ...
5
votes
1answer
157 views

Concerning nonlinear functional equations

There's a problem I've been working on for awhile that involves some hefty functional equations. For example, I may have something along the lines of $$ f(x)f(x) =x+1+f(x+1) $$ I've tried several ...
1
vote
1answer
54 views

Functional inequality

Let $f, g:\Bbb R \to \Bbb R$ be bounded functions satisfying $$ |f(x+y)-f(x)g(y)|\le \frac 1 4 $$ for all $x, y\in \Bbb R$. Prove or disprove $$ |f(x)||1-g(y)|\le \frac 1 4 $$ for all $x, y\in \Bbb ...
3
votes
2answers
175 views

Find all $f:\mathbb{R}^+\to \mathbb{R}^+$ satisfies that:

Find all $f:\mathbb{R}^+\to \mathbb{R}^+$ satisfies that: $$f(x)\cdot f(yf(x))=f(y+f(x))$$ $\forall x,y \in \mathbb{R}^+$
1
vote
0answers
112 views

Periodicity of a function.

If we have two functions $f:\mathbb{R} \to \mathbb{R}$ and $g :\mathbb{R} \to \mathbb{R}$ such that the period of $f$ is 7 and that of $g$ is 11, then the period of $F\left ( x \right ) = f( ...
-1
votes
1answer
507 views

Combine two equation

I have two equations with this format: $$Ds= A+A^2+\alpha_1\tag{1}$$ and $$Ds= M+M^2+\alpha_2 \tag{2}$$ Knowing that $(1)$ explains 72% of $Ds$ and $(2)$ 20%. I want to combine these two equations ...
11
votes
2answers
305 views

When is $f^{-1}=1/f\,$?

I seem to remember a nice article that I read many years ago (perhaps in the American Mathematical Monthly) which investigated the question, "Under what conditions on $f:\mathbb{C}\to\mathbb{C}$ is ...
3
votes
1answer
212 views

Cauchy functional equation

Is there $U\subset \Bbb R^2$ with Lebesgue measure $0$ such that $$f(x+y)=f(x)+f(y)$$ for all $(x, y)\in U$ implies $f(x+y)=f(x)+f(y)$ for all $(x, y)\in\Bbb R^2$ ?
-1
votes
1answer
67 views

Exercise of functions of a real variable

Let $f, g\colon\mathbb{R}\rightarrow\mathbb{R}$ functions so that for all $\,x, y\in\mathbb{R}$, $$f(x+y)+f(x-y)=2f(x)g(y).$$ Prove that if $f$ is not constant zero function and $|f(x)|\leqslant ...
3
votes
3answers
328 views

Find all functions for which $\ x \cdot f(xy)+f(-y)=x \cdot f(x)$

Does anyone have idea for solution (for all non-zero numbers)? $$\ x ≠ 0,y ≠ 0$$ $$\ f: R \setminus \{0\} → R$$ $$\ x \cdot f(xy)+f(-y)=x \cdot f(x)$$ Thanks!
4
votes
1answer
222 views

Solve system of equations with sin, cos, tg

I am trying to solve this system of equations but without any results. How can I solve this system of equations (in real numbers)? $$\sin^2 x + \cos^2 y = \tan^2 z$$ $$\sin^2 y + \cos^2 z = \tan^2 ...
19
votes
2answers
418 views

Proving that $f(n)=n$ if $f(n+1)>f(f(n))$

How can we prove that if $f:\mathbb{N}\rightarrow\mathbb{N}$ is a function so that $f(n+1)>f(f(n))$ for all $n\in\mathbb{N}$ then $f(n)=n$ for all $n\in\mathbb{N}$?
3
votes
1answer
2k views

Solve the recurrence for $T(n) = T(\sqrt n) + 2$. Assume that T(n) is constant for $n\leqslant 2$. [duplicate]

Trying to work out the following question, but I'm stuck.. Can someone direct me please. Using a change of variables $$ \text{Let}\ m = \lg\ n \\ S (m) = T (2m)\\ T (2^{m}) = T ...
6
votes
1answer
108 views

Can we give a definition of the cotangent based on a functional equation?

I've recently learned that the cotangent satisfies the following functional equation: $$\dfrac1{f(z)}=f(z)-2f(2z)$$ (true for $f(z)\neq 0$). Can we solve this equation for real or complex ...
1
vote
0answers
69 views

Functional equation inspired by moment generating function

A user gave the following nice answer http://math.stackexchange.com/a/161584/5031 My question is that although it is clear $\log M_X(t)=Ct^2$ is a form that satisfies the condition ...
-3
votes
2answers
270 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?
6
votes
1answer
124 views

Uniqueness of solution of functional equation

I have a function $f(x_1,x_2) \colon \mathbb{R}^2_{+} \to \mathbb{R}_{+}$ positive homogenous: $$ f(\lambda x_1, \lambda x_2) = \lambda f(x_1,x_2), \; \forall \lambda > 0 $$ and such that ...
2
votes
1answer
276 views

Books on functional equations

Could you help me please with some functional equations? I need some online books with exercises and some explanations. I'm interested in the Cauchy functional equation and Jensen functional equation. ...
2
votes
3answers
127 views

Iterations of $f(x)=\dfrac{ax+b}{cx+d}$

Consider $f(x)=\dfrac{ax+b}{cx+d}$, where $c\neq0$ and $f(x)$ is not equal to a constant. Is it necessarily true that $f^{[n]}(x)=f(x)$ for some natural number $n > 1$?
2
votes
2answers
613 views

Functional Equation. $f(mn)=f(m)f(n)$ and …

I want to prove the following. We have a function $f: \mathbb{Z} \to\mathbb{R}$ s.t. (1) $f(mn) = f(m)f(n)$ (2) $f(m+n) \leq f(m) + f(n)$ (3) $0 \leq f(x) \leq 1$ then $f(m+n) \leq \max\big(f(m), ...
5
votes
2answers
94 views

$f,g:[a,b]\to [a,b]$ are continuous, $f\circ g=g\circ f$ and $f$ is 1-1. Show that $\exists t\in [a,b]$ so that $f(t)=g(t)=t$

Using the Intermediate Value Theorem on $h(x)=f(x)-x$ and $r(x)=g(x)-x$ I can easily show that $\exists t_f,t_g\in[a,b]$ so that $f(t_f)=t_f$ and $g(t_g)=t_g$. It remains to show that $t_f=t_g$. Since ...
1
vote
0answers
130 views

Question about ratios and combinatorics

In this question that I posted yesterday (11/15): I am solving a programming puzzle that consists of finding all the possible ways to build a brick wall of $48$" $\times$ $10$" (width $\times$ height ...
1
vote
1answer
171 views

A functional equation

$$ f(x^2-1)+2f\left(\frac{2x-1}{(x-1)^2}\right)=2-\frac{4}{x}+\frac{3}{x^2}, \ x>1. \ \ f(x) = ? $$ Don't know how to solve such equiations, help me please. Thank you.
3
votes
2answers
381 views

Properties of solutions of the functional equation $f(kx) = kf(x)$

My problem is that I have a function: $f\colon\mathbb R\to\mathbb R$ with the property that $f(kx) = kf(x)$ for all $k,x \in\mathbb R$. a) I shall show that $f(0)=0$ b) If $f$ is not the ...
1
vote
1answer
119 views

Given $2xf(x)+(x-3)f(\frac{1}{1-x})=4x^2-10x-\frac{1}{2}$, find $f(x)$.

Given$$2xf(x)+(x-3)f\left(\frac{1}{1-x}\right)=4x^2-10x-\frac{1}{2}$$ Find $f(x)$. This's the first time I see this kind of question, I have no idea. Please help. Thank you.
5
votes
3answers
114 views

find $f(x)$ when $3f(x-6)-2f(x-9)=x^2-54$

I can easily show that with the assumption $f$ is a polynomial $f(x)=x^2$. But without that assumption how can I prove that $f(x)=x^2$???. I have tried many change of variables $x=u+k$ but to no ...
5
votes
2answers
147 views

Find function $f :\mathbb{R} \to \mathbb{R}$ satisfy that :

Find function $f :\mathbb{R} \to \mathbb{R}$ satisfy that : $$f(1)=1$$ $$f(x+y)=f(x)+f(y)+2xy$$ $$f\left(\frac{1}{x}\right)=\frac{f(x)}{x^4} \hspace{5pt}\forall x \neq 0$$
2
votes
1answer
166 views

A functional equation related to the exponential function

Suppose that $f:\mathbb{R}\rightarrow (0,\infty)$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ are two functions and satisfies the following relation \begin{equation} f(xy)=f(y)^{g(x)} \end{equation} ...
0
votes
3answers
55 views

Prove: we always have at least x>0 is a $x^3+bx^2+cx-d^2=0$ 's root

Prove: we always have at least one x>0 is a $x^3+bx^2+cx-d^2=0$ 's root (b, c, d are real numbers and $d≠0$)
0
votes
3answers
182 views

Find all function $f$ such that $f(x)+f(\frac1x)=\frac1a; a$ is constant

Which function verified that: $f(x)+f(\frac1x)=\frac1a; a$-constant value?
8
votes
1answer
90 views

Are there associative bijections $f\colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$?

This is my first question and I hope this question is not too brief to be acceptable: There are well-known bijections $\mathbb{N \times N \to N}$ and well-known associative compositions on countably ...
0
votes
1answer
178 views

Find all function such that $f(x)-f(y) = (x -y)g(\sqrt{xy})$

Find all functions $f, g$ that satisfy the functional equation $$ f(x)-f(y)= (x -y)g(\sqrt{xy}) \quad \forall\ x,y>0. $$
6
votes
3answers
199 views

Is there a non-constant function $f:\mathbb{R}^2 \to \mathbb{Z}/2\mathbb{Z}$ that sums to 0 on corners of squares?

A problem in the 2009 Putnam asks about functions $f:\mathbb{R}^2 \to \mathbb{R}$ such that whenever $A,B,C,D$ are corners of some square we have $f(A)+f(B)+f(C)+f(D)=0$. Without spoiling the problem ...
22
votes
3answers
1k views

Given $f(f(x))$ can we find $f(x)$?

Given $f(f(x))=x+2$ does it necessarily follow that $f(x)=x+1$? This question comes from a precalculus algebra student.
1
vote
0answers
66 views

What is an affine function?

Consider a functional, what is meant by a minimal sequence consistent of 'piecewise affine functions'?
4
votes
2answers
204 views

A functional equation

Can anything be said about the solutions of the following functional equation? $$ f(x, y + z) = f(x, y) + f(x + y, z) $$ I don't seem to be able to find much in what I think are the standard ...
4
votes
2answers
119 views

How many ways to reach $1$ from $n$ by doing $/13$ or $-7$?

How many ways to reach $1$ from $n$ by doing $/13$ or $-7$ ? (i.e., where $n$ is the starting value (positive integer) and $/13$ means division by $13$ and $-7$ means subtracting 7)? Let the number ...
0
votes
1answer
474 views

Find all positive functions of a positive real such that $ f(xf(y))=yf(x) $ and $\lim_{x\to\infty}f(x)=0$

Find all functions defined on the set of positive reals which take positive real values and satisfy: $$ f(xf(y))=yf(x) $$ for all ; $ f(x)\to0 $ and as $ x\to\infty $
3
votes
2answers
979 views

Given $f(x+1/x) = x^2 +1/x^2$, find $f(x)$

Given $f(x+1/x) = x^2 +1/x^2$, find $f(x)$. Please show me the way you find it. The answer in my textbook is $f(x)=\frac{1+x^2+x^4}{x\cdot \sqrt{1-x^2}}$
1
vote
2answers
85 views

Finding functions $f: \Bbb R_*^+ \to \Bbb R_*^+$ with certain properties

Find the functions $f:\Bbb R_*^+ \to \Bbb R_*^+$ such that: $$f(x)f \left(\frac{1}{x}\right)=1$$
3
votes
3answers
107 views

Find the value of f(343, 56)?

I have got a problem and I am unable to think how to proceed. $a$ and $b$ are natural numbers. Let $f(a, b)$ be the number of cells that the line joining $(a, b)$ to $(0, 0)$ cuts in the region $0 ≤ ...
2
votes
1answer
93 views

How can I simplify this nasty equation between two functions?

I have the following equation: $$ h(n) = n \sum_{i=0}^{\lceil \log_2 n \rceil} \frac{m(2^i)}{2^i} $$ and I'm trying to understand exactly the relationship between the functions $h$ and $m$. The ...
7
votes
1answer
348 views

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

Suppose $f:\mathbb R\to\mathbb R$ is a strictly decreasing function which satisfies the relation $$f(x+y) + f( f(x) + f(y) ) = f( f( x+f(y) ) + f( y+f(x) ) ) , \quad \forall x , y \in\mathbb R $$ ...
3
votes
1answer
218 views

Integral Inequality $|f''(x)/f(x)|$

Let $f$ be a $C^2$ function in $[0,1]$ such that $f(0)=f(1)=0$ and $f(x)\neq 0\,\forall x\in(0,1).$ Prove that $$\int_0^1 \left|\frac{f{''}(x)}{f(x)}\right|dx\ge4$$
1
vote
3answers
83 views

Looking for an equation

This is kind of a reverse question. A few years back I was presented with a functional equation problem, I don't remember it completely, and now I would appreciate the help of the math.SE hivemind to ...
2
votes
1answer
329 views

The functional equation $f(y) + f\left(\frac{1}{y}\right) = 0$

Let $f : \mathbb{R}\setminus\{0\} \to \mathbb{R}$ be a non-constant function such that $f(y) + f\left(\frac{1}{y}\right) = 0$. I found that $f(y) = h(\log|y|)$ will be a solution , where $h$ is an ...