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 ...

learn more… | top users | synonyms

3
votes
2answers
377 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
113 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
52 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
180 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
177 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
203 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
454 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
947 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
215 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
325 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 ...
39
votes
2answers
739 views

Looking for a function such that…

There was this question on one of the whiteboards at my company, and I found it intriguing. Maybe it's a dumb thing to ask. Maybe there is a simple answer that I couldn't see. Anyway, here it is: ...
0
votes
1answer
54 views

inverse function , asymptotics ..

let be a function given by $ f^{-1} (x) = \sqrt x + G(x) $ with $ G(x)= \sum_{n=0}^{N}a_{n} \sin(nx+ \pi/4) $ finite fourier series with N big my questio is , if for $ x \rightarrow \infty$ the ...
1
vote
0answers
80 views

$f(2^x) = f(4^x + 2^{x+1} + 2) - f(4^x + 1)$

Let $f(x)$ be a non-constant real-analytic function and for real $x$ it satisfies : $f(2^x) = f(4^x + 2^{x+1} + 2) - f(4^x + 1)$ Before you ask if this simplifies by writing $2^x = y$ note that ...
1
vote
0answers
55 views

solving this recurrence equation

is it possibel to solve the equation g$(x)= \sum_{n=1}^{\infty}f(x/n) $ for $ f(x)$ with other methods different from taking the Mellin transform on both sides ?? thanks.
2
votes
2answers
155 views

Find $f(x)$ from $f(f(x))$

I have this: $$ f\colon \mathbb{R} \to \mathbb{R}, $$ $$ f(f(x)) = x^2 - x + 1 $$ I need to show that $f(1) = 1$ and I need to show that $g(x) = x^2 - xf(x) + 1$ is not an one-to-one fuction. I know ...
0
votes
1answer
2k views

How to derive formula from a exponential scattered plot

I am doing some experiments and i got some results which i plot into a graph. The graph at small x values (x<15) gave random scatter plot but in large values more than 1000 it lookes like an ...
0
votes
2answers
92 views

Counting $2010^2$-tuples

Here's a problem i invented myself, but i'm not sure about my solution. I'll show it later, so people can enjoy trying to find one: Consider the function $f:\mathbb{R}^2\to\{1,2,...,2012\}$ that ...
1
vote
1answer
67 views

inverse function and positivity

is there any proof or theorem to say that if the inverse function $ y=f^{-1}(x) $ is POSITIVE in the sense $ f^{-1}(x) >0 $ for $x \ge 0 $ then the function $ f(x) \ge 0 $ will be also positive on ...
0
votes
1answer
32 views

simple substitution on equation giving dev a ride…

Im a developer working on some veterinary calculators, I have found all the required equation substitutions needed for my app, but I left the last one for now and is giving me a bit of a doubt filled ...
7
votes
3answers
258 views

How to find the function $f$ given $f(f(x)) = xf(x)$?

I was wondering if there is a continuous function such that $f(f(x)) = xf(x)$ for every positive number $x$.
0
votes
1answer
131 views

Plotting temperature over time excel

I am doing an uni assignment and have worked out a linear equation which plots temperature over time. I have this in a graph now but that required me to use a lot of calculations in the spreadsheet. ...
5
votes
3answers
427 views

Inverse function of $y=W(e^{ax+b})-W(e^{cx+d})+zx$

I have a simple question for which I am looking for a closed form expression (If there exits one). In other words, given: $$y=W(e^{ax+b})-W(e^{cx+d})+zx$$ where $W$ is the Lambert $W$ function and ...
1
vote
1answer
131 views

Applications of mathematics to some kinds of sporting strategies

I am a rather newbie maths person. Haven't studied maths in a while and so not sure what things are called was hoping to get some information to push me in the right direction so I know what it is I ...
3
votes
1answer
153 views

Find the general solution to $f(z)=f(z/2)f(z-1)$

Find the general solution to $f(z)=f(z/2)f(z-1)$ where $z$ is a complex number.
1
vote
0answers
67 views

Functional equation for the given function

For instance, there is functional equation for Lambert W function $z=W(z) e^{W(z)}$ And moreover, there is differential one: $z(1+W)\frac{dW}{dz}=W$. At the same time, there is no known functional ...
4
votes
1answer
62 views

Solution of $z(t+a) = h(a)z(t)$

I'm reading Howard Georgi online book on The Physics of Waves and found the following argument. Given the functional equation $$ z(t+a) = h(a)z(t) $$ he makes the following derivation (I'm citing ...
2
votes
1answer
136 views

Symmetric homogeneous functions of degree 1

Suppose: $cf(x,y)=f(cx,cy)$ $f(x,y)=f(y,x)$ If $f$ is a polynomial, then $f(x,y)=c(x+y)$ because by Euler's homogeneous function theorem, $f(x,y)=xf_x(x,y)+yf_y(x,y)$ where $f_x,f_y$ are ...
1
vote
5answers
138 views

Help solving a functional equation

Is there a function $f(x)$ on the real domain and real constants $a$ and $b\neq 0$ for which the following is true: $$f(x)-f(x-\delta)+a+bx^2=0$$ for some real $\delta\neq 0$? EDIT: I missed a very ...
1
vote
0answers
90 views

Defining Oblique Lines

Is it correct to classify a line which is neither vertical nor horizontal as oblique. I am trying to classify lines in a plane based on the quadrants through which they pass.
5
votes
2answers
447 views

Are all multiplicative functions additive?

Suppose $cf(x)=f(cx)$ and $f:\mathbb{R}\to\mathbb{R}$. I believe it follows that $f(x+y)=f(x)+f(y)$. Proof: There is some $c$ such that $y=cx$. Then ...
6
votes
2answers
278 views

Solving the functional equation $f(x) = f(x / \phi) f(x / \phi^2 - 1)$

I'm trying to find a function $f(x)$ such that the spacing between consecutive roots looks like the infinite Fibonacci word: $$1, \phi^{-1}, 1, 1, \phi^{-1}, 1, \phi^{-1}, 1, 1, \phi^{-1}, 1, 1, ...
1
vote
1answer
162 views

two functions $ f(x) $ and $ g(x) $

let be two functions $ f(x) $ and $ g(x) $ with an infinite set of roots $ a_{n} $ and $b_{n} $ so $ f(a_{n}) =0= g(b_{n}) $ also they satisfy the same functional equation $ f(1-s)=f(x) $ and $ ...
12
votes
2answers
196 views

About the solution of the infinite recurrence $f(x,f(x,f(x,f(x,f(…))))=a$

On internet I found some recreational problems as $$3=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$$ $$\frac{1}{2}=\frac{1}{x+\frac{1}{x+...}}$$ $$2=x^{x^{x^{...}}}$$ And the trick to solve them was just ...
18
votes
4answers
693 views

Solving functional equation $f(x)^2+f(y)^2=f(x+y)(f(f(x))+f(y))$

I was trying to find functions $f:(0,+\infty)\to(0,+\infty)$ satisfying the following functional equation $$ f(x)^2+f(y)^2=f(x+y)(f(f(x))+f(y)) $$ The problem is that I can't find here any reasonable ...
2
votes
0answers
100 views

Convexity conditions for $f$ and $\dfrac {1} {f}$

Let $f:\mathbb{R} _{+}\rightarrow \mathbb{R} _{+}$ be a real function. Find all conditions on $f$ under which $f$ is convex on $\left( a;b\right) \subset \mathbb{R} _{+}$$\Leftrightarrow$ $\dfrac ...
1
vote
1answer
253 views

find all continuous function $f:\mathbb R^+\to\mathbb R^+$ satisfy: $f(x)=f(\frac{x+1}{x+2})$

Find all continuous function $f:\mathbb R^+\to\mathbb R^+$ satisfy: $f(x)=f(\frac{x+1}{x+2})$ Find all continuous function $f:\mathbb R\to\mathbb R$ satisfy $\forall a<b, \exists c \in (a,b): ...
1
vote
1answer
111 views

Delta function question

Given the functions $$f(x)= \delta (x-a)$$ $$g(x)= \frac{1}{a} \delta \left(x- \frac{1}{a}\right)$$ for a real constant $a\gt0$, is there a relationship between $f$ and $g$? I believe that $ ...