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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2
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1answer
318 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
737 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
78 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
54 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
154 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
1k 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
129 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
422 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
134 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
442 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
275 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 $ ...
5
votes
2answers
259 views

Cauchy's functional equation for $\mathbb R^n$

Suppose $f(x+y)=f(x)+f(y)$. If $f:\mathbb R\to \mathbb R$ and is measurable, then $f(x)=cx$. This is referred to as Cauchy's functional equation. Suppose $f:\mathbb R^n\to \mathbb R^n$ instead. Does ...
10
votes
4answers
636 views

The functional equation $f(x)+f(y)+f(z)+f(x+y+z)=f(x+y)+f(y+z)+f(z+x)$

How to find the all continuous functions $f\colon \mathbb{R}\to\mathbb{R}$ such that $f(x)+f(y)+f(z)+f(x+y+z)=f(x+y)+f(y+z)+f(z+x)$
1
vote
1answer
124 views

Reverse engineer a Bayesian estimate?

My apologies if this is a basic question because I am no mathematician. Struck on my work on this, so came here to get some help.I am working on this bayesian estimate explained here.This is a ...
1
vote
5answers
260 views

$g(x)$ is a function such that $g(x+1)+g(x-1)=g(x)$, $x \in \mathbb{R}$. For what value of $p$, $g(x+p)=g(x)$.

$g(x)$ is a function such that $g(x+1)+g(x-1)=g(x)$, $x \in \mathbb{R}$.For what value of $p$, $g(x+p)=g(x)$. $g(x+2)+g(x)=g(x+1)$
1
vote
1answer
81 views

Show that for this function the stated is true.

For the function $$G(w) = \frac{\sqrt2}{2}-\frac{\sqrt2}{2}e^{iw},$$ show that $$G(w) = -\sqrt2ie^{iw/2} \sin(w/2).$$ Hey everyone, I'm very new to this kind of maths and would really ...
0
votes
1answer
401 views

Find all continuous functions $ f:\mathbb{R}\to\mathbb{R}$, that satisfy $f(xy)=f(\frac{x^2+y^2}{2})+(x-y)^2$

Find all continuous functions $ f:\mathbb{R}\to\mathbb{R}$, that satisfy $f(xy)=f(\frac{x^2+y^2}{2})+(x-y)^2$
4
votes
2answers
183 views

Seeking a contest question on functional equation on $[-1,1]$

I vaguely remember a question going something like Let $f$ be a function on $[-1,1]$ with $f$ satisfying (something like) $$f(x^2-1)=(2x)f(x).$$ Show that $f$ is identically zero on $[-1,1]$. ...
1
vote
1answer
91 views

Determining a function through three equations

I have the following assignment question, and I'm having trouble even getting started: Consider the set of functions $\mathcal{F}=\{f,g\}$, with $f:\mathbb{R}^2\to\mathbb{R}$, and ...
2
votes
0answers
91 views

Indeterminacy in second degree equations of Spencer-Brown's “Laws of Form”

I am trying to follow G. Spencer-Brown's argumentation regarding 'Equations of the Second Degree' in the Laws of Form. He shows how infinitely growing self-referential forms can be created by using a ...
2
votes
2answers
379 views

Correct order of books for a beginner

what should be the order of the books in which a beginner should do the following books in algebra: -1.E.J. Barbeau POLYNOMIALS -2. Polynomials and Polynomial Inequalities (Graduate Texts in ...
3
votes
0answers
52 views

secants, exponentials, quotient structures

Trigonometric functions are . . . . . . somewhat like exponential functions. If $f$ is an exponential function, then $\displaystyle \frac{\prod_i ...
3
votes
1answer
127 views

Solution space to a functional equation

This question comes from my attempts at understanding an example presented by Bill Gasarch on his blog. The example is of a continuous strictly increasing function whose derivative is zero almost ...
5
votes
6answers
2k views

What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$?

What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$? I'm especially curious if there is both an algebraic and calculus-based derivation of the solution.
3
votes
1answer
152 views

What is this topic (competition questions)?

I discovered the following question by accident, and found it interesting, but I only resolved it by brute force: Problem: Let f: $\mathbb{R} \rightarrow \mathbb{R}$ have the property $(\pi)$ iff ...
3
votes
2answers
367 views

How to find all polynomials $P(x)$ with real coefficents satisfying $P^2(x)-1=4P(x^2-4x+1)$

Find all polynomials $P(x)$ with real coefficents satisfying $P^2(x)-1=4P(x^2-4x+1)$. My solution: Let the first term of $P(x)$ be $ax^n$. We see first term of left side is easily $a^2x^{2n}$ ...
3
votes
2answers
167 views

Does anyone recognize this function?

I am looking for a function $f(n)$ that satisfies the following two conditions at the same time $$ \frac{f(n-1)}{f(n)}=(-1)^n\quad ,\quad \frac{f(n+1)}{f(n)}=(+1)^n\equiv 1,\quad \forall ...
3
votes
1answer
116 views

If $g(x) := \int_1^2 f(xt)dt \equiv 0$ then $f \equiv 0$

Let $f \colon \mathbb R \to \mathbb R$ be a continuous function. Let's define $$ g(x) := \int_1^2 f(xt)dt. $$ Prove that $g \equiv 0 \Rightarrow f \equiv 0$. Well, I show you what I have ...
5
votes
1answer
129 views

Functional Equation with Value

If $f$ is a strictly increasing function from the naturals to the naturals, and $f(f(x))=3x$, what are all values of $f(2012)$? I have only proven that $f(3x)=3f(x)$ but that get's nowhere :(
1
vote
2answers
149 views

Continuous function $g$ satisfying $g(x + y) = 5g(x)g(y)$

Let $g$ be a continuous function with $g(1) = 1$ such that $$g(x + y) = 5g(x)g(y)$$ for all $x$, $y$. Find $g(x)$.
0
votes
1answer
94 views

fancy about $f(x+a)=f(x)$ , where $a$ is any non-real complex number

It is well-known that when $a$ is any non-zero real number, the most general solution of $f(x+a)=f(x)$ should be $f(x)=\Theta(x)$, where $\Theta(x)$ is an arbitrary periodic function with period $|a|$ ...
1
vote
3answers
198 views

Solution of functional equation $f(x)=-f(x-a)$

I have a problem with finding solution. I suppose it will be something like $f(x) =G(x)\Re(e^{\frac{x\pi}{a}})$, where $\Re$ is real part of a complex number, $G(x)$ periodic function whith period ...
2
votes
1answer
154 views

Solving $f(2011)=2012$, $f(4xy)=2yf(x+y)+f(x-y)$

How to find the all functions $f$ :$ \mathbb{R}\longrightarrow\mathbb{R}$ such that $f(2011)=2012$,for every $x,y\in\mathbb{R}$ then: $$f(4xy)=2yf(x+y)+f(x-y)$$