3
votes
0answers
24 views

At most n functions

Some background: I was trying to solve the functional equation f(f(x))=sin(x). I realized that $f(\pi n)$ is a root of f for all integers n, because $f(f(\pi n))=\sin(\pi n)=0$. Thus, we can write f ...
0
votes
0answers
21 views

Find all functions: $f:C\rightarrow C$

Find all functions $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))$ Extension: Find all $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))(x-f(2\pi))(x-f(-2\pi))\cdots ...
0
votes
1answer
40 views

Writing roots of f(x) as f(a) for some a

I was solving a problem when this random thought popped into my head. Suppose you have a function, say $f(x)=x^2-1=(x-1)(x+1)$. The roots of this function are $-1$ and $-1$. We can write these roots ...
1
vote
0answers
47 views

Solve the function from the composition [duplicate]

I have equations as follows $$f(f(x))=x^2+x$$ Then solve for $f(x)$. Can anyone give some hints about this question?
26
votes
5answers
1k views

how to get $ f(x) $ if we know $ f(f(x))=x^2+x $

how to get $ f(x) $ if we know $ f(f(x))=x^2+x $ Is there elementary function of $f(x)$ satisfy the equation?
3
votes
1answer
107 views

Solve for $f(x)$ if $f(f(x))=6x-f(x)$

If $f: [0,\infty) \rightarrow [0,\infty)$ and $f(f(x))=6x-f(x)$ $f(x)>0$ $ \forall x \in (0,\infty) $ Find f(x)
8
votes
1answer
116 views

If the function $f$ satisfies the equation $f(xf(y)+x)=xy+f(x)$, find $f$

Question Let the function $f:\mathbb R\to\mathbb R$,and such $$f(xf(y)+x)=xy+f(x)$$ Find all $f(x)$ Let $x=1,y=1$,then $$f(f(1)+1)=1+f(1)$$ let $f(1)=t$,then $$f(t+1)=1+t$$ So I guess ...
1
vote
2answers
68 views

$f$ is a linear function. If $f(x+3) + f(4x+5) = 10x$,$ f(x)=?$

Question: Suppose $f$ is a linear function. If $f(x+3) + f(4x+5) = 10x$, then $f(x)=?$ My attempts: Okay, this is what I have so far... $$f(x) + f(3) + 5(4x) + f(5)$$ $$f(5x) = 10x - f(8)$$ ...
2
votes
0answers
47 views

If $f(2x-f(x))=x$ . Find all bijective functions.

It is given that $f :[0,1] \rightarrow [0,1] $ and it is bijective. If $f(2x-f(x))=x$ , find all such f. Is my solution correct? My attempt $f(x)$ is bijective. thus there exists g(x) which is the ...
6
votes
2answers
116 views

if the function $f(f(n))+f(n)=2n+2014$,find the $f$

let the function $f:N^{+}\to N^{+}$,and such $$f(f(n))+f(n)=2n+2014$$ Find the $f(n)$ My try: let $n=1$,then we have $$f(f(1))+f(1)=2016$$ let $f(1)=a$,then $$f(a)+a=2016$$ and let ...
18
votes
3answers
1k views

I am searching for an unusual real-valued function.

It needs to fulfill following condition for all positive $x,y ∈ ℝ$: $$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$$ any ideas?
5
votes
2answers
175 views

Help with complicated functional equation

Problem: Let $T=\{(p,q,r)\mid p,q,r \in \mathbb{Z}_{\geq0}\}$. Find all functions $f:T\to \mathbb{R}$ such that: $$f(p,q,r)=\\ =\begin{cases} 0, & \text{ if } pqr = 0 \\ 1 + ...
11
votes
1answer
313 views

Additive functional inequality

The function $f:R_+\to R_+$ is continuously differentiable and increasing. Also, $f(0)=0$ and $f(\infty)=\infty$. Continuity and differentiability of higher orders can be assumed if necessary. ...
7
votes
4answers
169 views

A function satisfying $f(\frac1{x+1})\cdot x=f(x)-1$ and $f(1)=1$?

$f:[0,\infty)\to\mathbb{R}$ is a continuous function which satisfies $f(1)=1$ and: $$f(\frac1{x+1})\cdot x=f(x)-1$$ Does there exist such a function, if they do, are there infinitely many? And is ...
0
votes
0answers
26 views

Can Duhamel's Formula be applied Functional Equations?

Consider an n'th order linear Differential Equation of the form $$y^{[n]} = a_0(x) + a_1(x)y + a_2(x)y' + a_3(x)y'' \ ... \ + a_{n-1}(x)y^{[n-1]}$$ It is well known that to solve this we can create ...
1
vote
0answers
23 views

What is the function $f$ verifying : $f(\frac{x}{2}+\frac{x}{2}\cos(\frac{v\pi}{x}))=\frac{x}{2}\sin(\frac{v\pi}{x})$

What are the solutions to the functional equality (for a constant $v$): $$ \forall\, x > 0, \ \ \ \ ...
0
votes
1answer
55 views

Functions such that $f(\frac{x+y}{2})=\frac{1}{2}f(x)+\frac{1}{2}f(y).$

What are the continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that for every $x,y\in \mathbb{R}$ $$f(\frac{x+y}{2})=\frac{1}{2}f(x)+\frac{1}{2}f(y).$$
0
votes
2answers
59 views

Find all $f:\mathbb R\to\mathbb R$ such that $\forall x,y\in\mathbb R$ the given equality holds: $xf(y)+yf(x)=(x+y)f(x)f(y)$.

Find all $f:\mathbb R\to \mathbb R$ such that $\forall x,y\in\mathbb R$ the given equality holds: $$xf(y)+yf(x)=(x+y)f(x)f(y)$$ My try: whenever $y=0$, we have $$x\cdot f(0)\cdot(1-f(x))=0$$ ...
1
vote
1answer
169 views

A Dangerous Function [closed]

Recently, I have started solving many ques on functional equations. But, this ques for me was tough, $ f(y)^{x^2}+f(x)^{y^2}=f(y^2)^x+f(x^2)^y $ I've started substituting some values, trying to ...
10
votes
4answers
170 views

If $f(3x)=f(x)$ and $f$ is continuous, show that $f(x)$ is a constant function.

If $f(x)$ is a continuous function such that $f(3x)=f(x)$ and the domain of $f$ is all non-negative real numbers. Prove that $f$ is a constant function. What I did: ...
2
votes
2answers
91 views

Functional inequation on $\mathbb{R}$: $f(x+y^2)-f(x)\geq y$

I have the following equation: $$f:\mathbb{R}\rightarrow\mathbb{R}$$ $$\forall (x,y)\in\mathbb{R}^2,\ f(x+y^2)-f(x)\geq y$$ f is not necessarily differentiable/continuous/... (In fact, we can prove ...
7
votes
2answers
126 views

Continuous functions that satisfies $f(x) + f(1-x) + f(\sqrt{x^2+(1-x)}) = 0$ and $f(\frac12)=0$

$f:[0,1]\rightarrow \mathbb{R}$ is a continuous function which satisfies $$ f(x) + f(1-x) + f\left(\sqrt{x^2+(1-x)}\right) = 0 \text{ and } f\left(\tfrac12\right)=0. $$ Can someone give explicit ...
0
votes
0answers
24 views

ODE in case of discontinuity

Consider a function $\varphi(x) : \mathbb{R} \rightarrow \mathbb{R}$ bounded and continuous and $c \in \mathbb{R}$. Question 1 Is there a unique $f : \mathbb{R} \rightarrow \mathbb{R}$, among all ...
1
vote
1answer
42 views

$f(x + y)f(x − y) = ( f(x) + f(y) )^ 2− 4x^ 2 f(y)$ so why $f(2) \neq 2$?

Let $f : \mathbb R → \mathbb R$ be a function that satisfies for all $x,y ∈ \mathbb R $ defined as $f(x + y)f(x − y) = ( f(x) + f(y) )^ 2− 4x^ 2 f(y)$. So, why $f(2) = 2$ is impossible? Why $f(0) = ...
3
votes
2answers
60 views

Determine the smallest number P

I have here a hard problem, which I couldn't solve. Denote $M$ the set of all functions $f:[0,1]\to\mathbb{R}$ with the following properties: $f(x)\ge0, \forall x$ in $[0,1]$, $f(1)=1$, $f(x+y)\ge ...
4
votes
1answer
63 views

submodular-like functions on $\mathbb{R}$

The definition of submodularity led me to define a type of function on numbers (I suppose ordered rings, but consider the reals for now) as $f(x) + f(y) \ge f(x+y) + f(xy)$. Such functions exist: for ...
2
votes
1answer
55 views

Prove That $f(n+f(n))=n$

if $f:\mathbb{N}\rightarrow\mathbb{N}$ is a function such that $f(1)=1$ and $$f(n)=n-f(f(n-1))\,\,\: \forall n \ge 2$$ Prove That $$f(n+f(n))=n$$
2
votes
1answer
62 views

Suppose a function $f : \mathbb{R}\rightarrow \mathbb{R}$ satisfies $f(f(f(x)))=x$ for all $x$ belonging to $\mathbb{R}$.

Suppose a function $f : \mathbb{R}\rightarrow\mathbb{R}$ satisfies $f(f(f(x)))=x$ for all $x$ belonging to $\mathbb{R}$. Show that (a) $f$ is one-to-one. (b) $f$ cannot be strictly decreasing, and ...
5
votes
2answers
70 views

Problem solve functional equation

Solve functional equation:find all strictly monotone functions $f:(0,+\infty)\to(0,+\infty)$ such that $$(x+1)f(\dfrac{y}{f(x)})=f(x+y),\forall x,y>0$$
2
votes
0answers
66 views

Creating a monotonic function

I have $n$ functions $f_i(x) \{i = 1 ,...,n\} $that does not preserve the monotonic mapping order. i.e. if $x_1 < x_2$, then in general, $f_i(x_1)$ is not less than $f_i(x_2)$ (for all $i = 1 ... ...
3
votes
3answers
174 views

if $f(mn)+f(m+n-1)=f(m)f(n)$How find $f(n)$

let $f:N^{+}\to Z$,and $f$ is monotonic nondecreasing,and such $$f(m)f(n)=f(mn)+f(m+n-1),f(4)=5$$ Find all $f(n)$ My try: let $$m=2,n=2\Longrightarrow f^2(2)=f(4)+f(3)$$ ...
7
votes
3answers
320 views

If $f:N→N$ such that $f(f(x))=3x$, then find $f(2013)$

If $f:N→N$ such that $f(f(x))=3x$, then what is the value of $f(2013)$?
3
votes
0answers
75 views

Is this a field of study?

Is there a name for an equation that takes the following form? $$F(f(x),f^{-1}(x),x)=0$$ A nice example being $$f(x)-f^{-1}(x)=0$$ because the solutions of this equation are their own inverses. ...
0
votes
1answer
47 views

Determining quadratic function of this word problem

I have this word problem in my homework: ...
3
votes
1answer
63 views

Solving functional equation 2

Problem: find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x^2+f(y))=(f(x))^2+y^4 +2f(xy),\ \ \ \forall x,y\in\mathbb{R}$$
0
votes
1answer
43 views

Solve a functional equation

Find all functions $f:[0,+\infty)\to [0,+\infty)$ such that $f(x)\geq \frac{3x}{4}$ and $$f(4f(x)-3x)=x,\forall x\in[0,+\infty)$$
1
vote
0answers
29 views

I tried this problem in the following way; is it right?

Let, f(x) be a twice differentiable function defined on (-1, 1) and f(0) = 1. Let, f(x) ≥ 0, f'(x) ≤ 0 and f''(x) ≤ f(x) for all x ≥ 0. Show that, f'(0) ≥ -√2. I am telling you what I did. First, ...
6
votes
2answers
110 views

Show that $f(x) = x$ if $f(f(f(x))) = x$.

If $f: \mathbb{R} \to \mathbb{R}$ and $f$ is strictly increasing, show that $f(x) = x$ if $f(f(f(x))) = x$. So this compulsorily ESTABLISHES that $f(x) = x$ only, and no other solution. So, merely ...
1
vote
0answers
57 views

Another functional equation

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
1
vote
1answer
44 views

Functional equation with $f(1)$ integer

Here is a nice problem: Let $f:R\rightarrow R$ be a function, R is the set of real numbers, satisfying the following properties: $ f(1)$ is an integer and $xf(y)+yf(x)=(f(x+y))^2-f(x^2)-f(y^2)$, for ...
4
votes
2answers
110 views

Functional Equation : If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x).

Problem : If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x). My approach : The given equation can be written as $$(x-y)f(x+y) -(x+y)f(x-y) =4xy(x-y)(x+y)$$ $$\Rightarrow ...
1
vote
1answer
34 views

Solution to a general scaling problem $G(\lambda z)=\frac{G(z)}{\gamma z^n}$

When playing with the scaling problem $$G(4z)=\frac{G(z)}{2z}$$ (see also this question) I discovered, that the general problem $$G(\lambda z)=\frac{G(z)}{\gamma z}$$ with two constants ...
3
votes
1answer
64 views

Functional inequality and one identity

I'm a high school student from Bonn, Germany and I have to solve the following problem: If $g:R \rightarrow R $ is a function with the property $g(ab)-ag(b)\leq bg(a)$, for all real numbers a and b, ...
2
votes
1answer
81 views

Find $g(x)$ if $f(g(x))=f(x)g(x)$ and $g(2)$=37, $f(x)$ and $g(x)$ are polynomials

Suppose $f(x)$ and $g(x)$ are non-zero polynomials with real coefficients, such that $f(g(x))=f(x)\times g(x)$. If $g(2)=37$, find $g(x)$. I tried plugging $f(x)$ and $g(x)$ as $n$ and $m$ ...
4
votes
1answer
61 views

Find all function satisfying a condition with $\min$ and $\max$

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$ \forall (x,y)\in\mathbb{R}^2,x\ne y,\quad \min (f(x),f(y)) \leq \frac{f(x)-f(y)}{x-y} \leq \max(f(x),f(y)) $$ I have started with ...
3
votes
1answer
41 views

Functional equation with function defined on $\mathbb{N}^{*}$

Let $ f:\mathbb{N}^{*} \mapsto \mathbb{N}^{*} $ be a function with the following property: $$ \frac{f(x+1)f(x)-2x}{f(x)}=\frac{2f^2(x)}{x+f(x)}-1$$ Determine all functions with this property. (I'm ...
3
votes
1answer
157 views

Real analytic $f$ satisfying $f(2x)=f(x-1/4)+f(x+1/4)$ on $(-1/2,1/2)$

What are the real analytic functions $f\colon (-\frac{1}{2},\frac{1}{2}) \to \mathbb{C}$ that satisfy the following functional equation, and how are they derived? ...
8
votes
2answers
182 views

Continuous $f$ satisfying $f(2x)=f(x-1/4)+f(x+1/4)$ on $(-1/2,1/2)$

What are the continuous functions $f\colon (-\frac{1}{2},\frac{1}{2}) \to \mathbb{C}$ that satisfy the following functional equation, and how are they derived? ...
2
votes
1answer
94 views

How find this function $f(2^m)=?$

show that:there exists unique function $f:N^{+}\to N^{+}$,such $$f(1)=f(2)=1$$ and $$f(n)=f(f(n-1))+f(n-f(n-1))),n=3,4,\cdots$$ and Find $f(2^m),m>2,m\in N$ My try: let $n=3$ ,then we have ...
5
votes
3answers
73 views

Simple Functional Equation $\frac{f(a)-f(b)}{a-b}\cdot(-a)+f(a)=-ab$

Compute all real-valued functions $f$ so that the line between any two points on the graph $f$ intersects the $x$-axis at the product of those two points' $x$-coordinates times $-1$. (if we do some ...