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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59 views

If $f(x+y)-f(x-y)=4xy$, then is $f(0)=0$ or $f(0)=1$?

As in the title. It may be very simple, but I'm having difficulty finding the proper substitution.
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1answer
43 views

What are the conditions on this Riemann-Zeta function functional equation?

I am a huge fan of the Riemann Zeta function's functional equation: $$\large{\color\green{\zeta(x)=2^x \Gamma(1-x)\zeta(1-x)\pi^{x-1}\sin\frac{\pi x}{2}}}$$ I am curious as to what conditions on $x$ ...
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2answers
37 views

If,for all $x$,$f(x)=f(2a)^x$ and $f(x+2)=27f(x)$,then find $a$

I am finding this problem confusing : If,for all $x$,$f(x)=f(2a)^x$ and $f(x+2)=27f(x)$,then find $a$. When $x=1$ I have that $f(1)=f(2a)$ using the first identity. Then when $x=2a$ I have by ...
5
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1answer
46 views

Find all solutions to the functional equation $f(x) +f(x+y)=y+2 $

I've started studying functions and I am having trouble with the following question: Find all solutions to the functional equation $f(x) +f(x+y)=y+2 $ Using the substitution technique when $y=0$ ...
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1answer
34 views

How to prove that a function is continuous in functional equation?

I was wondering about different methods or properties to prove that a function in a functional equation is continuous or differentiable. Can somebody give me some examples of such problems or methods, ...
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2answers
104 views

Solve the functional equation $f (2x)=f (x)\cos x$

Find all $f: \mathbb R\longrightarrow \mathbb R $ such that $f $ is a continuous function at $0$ and satisfies $$\;\forall \:x \in \mathbb R,\; f\left(2x\right) = f\left(x\right)\cos x $$ My try: ...
2
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1answer
65 views

Functional equation $f(x + y) = f(x)^m + f(y)^{m + 1}$ [closed]

Let $m \in \mathbb{N}$. How i can find all functions $f$ such that $f(x + y) = f(x)^m + f(y)^{m + 1} \forall x,y \in R$? Thank you in advance.
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1answer
46 views

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

I need to solve the following functional equation:$$f(x+y)^2=f(x)^2+f(y)^2$$ I'm familiar with simpler ones such as $f(x)+f\left(\frac{1}{1-x}\right)=x$ (I use substitutions), but here I cannot find ...
11
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3answers
135 views

If $B(x+y)-B(x)-B(y)\in\mathbb Z$ can we add an integer function to $B$ to make it additive?

Given a function $B:\mathbb R\to\mathbb R$ satisfying $B(x+y)-B(x)-B(y)\in\mathbb Z$ for all real numbers $x$ and $y$, is there a function $Z:\mathbb R\to\mathbb Z$ such that $B+Z$ is an additive ...
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3answers
151 views

Differentiable function satisfying $f(x+a) = bf(x)$ for all $x$

This is an exercise from Apostol Calculus, (Exercise 10 on page 269). What can you conclude about a function which has derivative everywhere and satisfies an equation of the form $$ f(x+a) = ...
11
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2answers
121 views

Finding Symmetry Group $S_3$ in a function

I was considering functions $f: \Bbb{C} \rightarrow \Bbb{C}$ and I defined the following instrument (I call it the Symmetry Group of a function) $$ \text{Sym}(f) = \left< m(x)|f(m(x))=f(x) ...
6
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2answers
81 views

Solving a functional equation $f(x)+f\left(\frac{1}{1-x}\right)=x$

I was given the following homework: list all functions $f:\mathbb{R}\setminus\{0,1\}\rightarrow\mathbb{R}$ such that $f(x)+f\left(\frac{1}{1-x}\right)=x$. And obviously have I no idea what should I do ...
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1answer
36 views

The functional equation $f(f(x) + y) = f(f(x) - y) + 4f(x)y$

I found the following functional equation: $f(f(x) + y) = f(f(x) - y) + 4f(x)y$ Up to now I tried setting $x = 0$ and $f(0) = c$ to get $f(c + y) = f(c - y) + 4cy$ If we define $g(x) = f(x) - x^2 ...
0
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1answer
39 views

Solve $g(x + \frac{1}{x}) = x^y + \frac{1}{x^y}$

Let $y$ be a real number. Find $g$ such that $$g(x + \frac{1}{x}) = x^y + \frac{1}{x^y}$$ Is valid for all real $x$.
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3answers
51 views

Let $f:\mathbb{R}^m \to \mathbb{R}$ be differentiable s.t. $f(x/2)=f(x)/2, \forall x \in \mathbb{R}^m$. Show that $f$ is linear.

Basically, I am not really sure how to start. I thought about going through induction to show for $\mathbb{N}$ and $\mathbb{Q}$, then use the completeness of $\mathbb{R}$, but I think it is a long ...
5
votes
1answer
111 views

About $f(x)= f(\frac{1}{x})$

Consider the equation $$f(x)=f\left(\frac{1}{x}\right)$$ Where we want $f$ to be real-meromorphic. Are all solutions $f$ of the form $$f(x) = g\left(\frac{x}{1+x^2}\right)$$ Where $g$ is a ...
4
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1answer
123 views

How can one show that $f(0)=0$ for $f$ satisfying certain conditions?

Given the functional equation $$f(x+(1+x)f(y))=y+(1+y)f(x)$$ Such that $f:(-1,\infty) \to (-1,\infty)$ and the function $g(x):=\frac{f(x)}{x}$ is strictly increasing in $I=(-1,0)\cup(0,+\infty),$ ...
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1answer
51 views

$3^x-2^y=1$, $x \in \mathbb{N}$ and $y \in \mathbb{N}$

$3^x-2^y=1$ or $y=\log_2{\left(3^x-1\right)}$ $x$ and $y$ must be natural numbers. I know this two solutions: $x=1$ and $y=1$ $x=2$ and $y=3$ Are there more solutions? How can I find them?
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1answer
46 views

Verify proof of $ f(x)=e^x $ if $ f(x+y)=f(x)f(y) $ and $ f'(x)$ exists for all $x$

This is exercise 6.26.8 from Tom Apostol's Calculus I, I'd like to ask someone to verify my proof. I'd be also interested in alternative proofs: If $ f(x+y)=f(x)f(y) $ for all $ x $ and $ y $ and if ...
5
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3answers
119 views

Finding a unique continuous function

Let $f$ be a given continuous function on $[0,1]$. How do you prove that there is a unique continuous function $g$ on $[0,1]$ satisfying $$g(x) = \frac{1}{2}g\left(\frac{x+1}{2}\right) + f(x)$$ for ...
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0answers
21 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 ...
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0answers
53 views

The functional equation $ f(x-f(y))=f(f(y))+xf(y)+f(x)-1$

I came across the functional equation: $f(x-f(y))=f(f(y))+xf(y)+f(x)-1$ So far I tried plugging $x=f(y)$ and got $f(x)=\frac{f(0)-x^2+1}{2}$ which holds for every $x = f(y)$. I suppose that $f(0)=1$ ...
0
votes
1answer
42 views

d'Alembert functioal equation: $f(x+y)+f(x-y)=2f(x)f(y)$

The d'Alembert functioal equation is: $$f(x+y)+f(x-y)=2f(x)f(y)\tag0$$ This equation plays a central role in determining the sum of two vectors in Euclidean and non-Euclidean geometries. Is there a ...
0
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1answer
56 views

The functional equation $f(f(x)+xf(y))=xf(y+1)$

I'm trying to solve the functional equation $f(f(x)+xf(y))=xf(y+1)$. Up to now I found that $f(f(0))=0$ when $x=0$ and that $f(y+1)=f(f(y)+f(1))$ by setting $x=1$. Also $f(x)=x$ is an apparent ...
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4answers
54 views

How can one show that $f(n)>n$

Given a function $f\colon \Bbb N^\ast\to \Bbb N^\ast$ such that we have $f(f(n))=f(n+1)-f(n)~\forall~n\in\Bbb N^\ast$. Show that: If $f(n+1)-f(n)>1$ then $f(n)>n$. Since $f(n+1)>f(n)+1$ I ...
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5answers
135 views

Find all functions $f$ such that $f(x-f(y)) = f(f(x)) - f(y) - 1$

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $f(x-f(y)) = f(f(x)) - f(y) - 1$. So far, I've managed to prove that if $f$ is linear, then either $f(x) = x + 1$ or $f(x) = -1$ must be ...
3
votes
2answers
117 views

Functional equation $f(xy)=f(x)+f(y)$ and continuity

Prove that if $f:(0,\infty)→\mathbb{R}$ satisfying $f(xy)=f(x)+f(y)$, and if $f$ is continuous at $x=1$, then $f$ is continuous for $x>0$. I let $x=1$ and I find that $f(x)=f(x)+f(1)$ which ...
2
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1answer
32 views

discontinuous solutions of functional equation

This is a followup to this question. It's well known that Cauchy's functional equation, $$f(x+y) = f(x) + f(y),$$ has discontinuous solutions. In fact, any discontinuous solution is discontinuous ...
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2answers
47 views

Can you solve the following functional equation?

I found the following functional equation: Find all functions $f : \Bbb R \rightarrow \Bbb R $ such that: $xf(x) - yf(y) = (x - y)f(x + y) $ for all $x, y \in \mathbb R $ Could you please help me? I ...
2
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1answer
46 views

Functional Equation similar to Cauchy's

Find all functions $f:\mathbb{Q}\rightarrow \mathbb{Q}$ such that for any $x,y\in{}\mathbb{Q}$ we have $$\{f(x)\}+\{f(y)\}=\{f(x+y)\}.$$ Note that $\{t\}$ denotes the fractional part of $t$ for ...
2
votes
3answers
99 views

Functional equation $f\left(\frac{1}{x}\right)+(x+1)f(x)=1$

Find all functions $f$ such that $f\left(\frac{1}{x}\right)+(x+1)f(x)=1,\space x\neq0$.
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2answers
37 views

Does induction find all solutions?

Induction shows that an equality holds for all values of $n$. It doesn't show that this is the only equality or formula for the expression that may hold true, correct? For example, say a question asks ...
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0answers
54 views

Is it possible to compute $\Gamma(\rho_k)$ or $\zeta(3+2\omega_k)$, when $\rho_k=\frac{1}{2}+i\omega_k$ is a nontrivial zero of Riemann zeta function?

I believe that I can to use the equation (1) (from reference [1]), a well known equation involving the Gamma function, Riemann zeta function and the theta function $\psi(x)=\sum_{n=1}^\infty ...
3
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0answers
57 views

Functional Equation $f(m + f(n)) = f(m) - n$

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 + ...
2
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1answer
27 views

Functional equation $f(x+y)-f(x)-f(y)=\alpha(f(xy)-f(x)f(y))$ is solvable without regularity conditions

I was reviewing this question and got motivated to solve this general problem: Find all functions $f:\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$, ...
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2answers
47 views

Differentiability of $f(x+y) = f(x)f(y)$ [duplicate]

Let $f$: $\mathbb R$ $\to$ $\mathbb R$ be a function such that $f(x+y)$ = $f(x)f(y)$ for all $x,y$ $\in$ $\mathbb R$. Suppose that $f'(0)$ exists. Prove that $f$ is a differentiable function. This is ...
2
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1answer
52 views

Solution of the functional equation $g(x)g(z) = g(x+z)+g(x-z)$

What is the solution for the following functional equation? $g(x)g(z) = g(x+z)+g(x-z)$ The solution given is: $g(z) = 2\cos(z)$. In the derivation of the result (using Taylor expansion), there is ...
4
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4answers
60 views

Functions satisfying the functional equation $[1-f(x)f(y)]f(x+y)=f(x)+f(y)$

How to prove that there is no real function defined on $\mathbb{R}$, continuous at $0$ and not always vanishing satisfying the functional equation $$[1-f(x)f(y)]f(x+y)=f(x)+f(y) \tag{E}$$
2
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2answers
55 views

Prove that functional equation doesn't have range $\Bbb R.$

Prove that any solution $f: \mathbb{R} \to \mathbb{R}$ of the functional equation $$ f(x + 1)f(x) + f(x + 1) + 1 = 0 $$ cannot have range $\mathbb{R}$. I transformed it into $$ f(x) = \frac ...
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2answers
66 views

Finding $f(n)=f(f(n-1))+f(f(n+1))$

Determine whether a function exists from the positive integers to the positive integers which satisfies the equation: $$f(n)=f(f(n-1))+f(f(n+1))$$. My guess is that this function does not exist, as ...
0
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1answer
38 views

continuous function and functional equation

Let $ g:\mathbb{R} \to \mathbb{R}$ be a continuous function: $g(x) = g(x+1)$. Show that $g$ satisfies the equation: $g(x)=\frac{1}{4} \left(g(\frac{x}{2})+g(\frac{x+1}{2})\right)$ for all $x$. Is it ...
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0answers
23 views

Brute force a formula based on numbers and the result?

I have thought of a very funny thing. In World of Warcraft, all weapons have some Damage Per Second variable specified. I want to know how they calculate that result, based on the result and some ...
1
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2answers
31 views

Does the equation $f(x)+g(y)=x^2+xy+y^2$ have solutions in real functions $f$ and $g$?

Does the equation $$f(x)+g(y)=x^2+xy+y^2 \mbox{ } \forall x,y \in \mathbb{R}$$ have solutions in real functions $f$ and $g$?
1
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2answers
51 views

if $2f\left(\frac{x}{x^2+x+1}\right)=\frac{x^2}{x^4+x^2+1}$ then what is $f(x)$?

assume that: $$2f\left(\frac{x}{x^2+x+1}\right)=\frac{x^2}{x^4+x^2+1}$$ Then what is $f(x)$?
3
votes
4answers
74 views

How to solve the functional equation $f(x+a)=f(x)+a$

Are there any other solutions of the functional equation $f(x+a)=f(x)+a$ ($a=\mathrm{const}$, $a\in\mathbb{R}\setminus\left\{0\right\}$) apart from $f(x)=x+C$ ($C=\mathrm{const}$)? Edit: $a$ is a ...
2
votes
2answers
213 views

Find all the functions satisfying this criterion

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$\left|f(x)-f(y)\right|=2\left|x-y\right|$$
1
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2answers
63 views

Find all functions such that $f: R \longrightarrow R, \forall x \in R, f(x)f(x^2-1)=\sin(x) $.

Find all functions such that $$f: \mathbb{R} \longrightarrow \mathbb{R} \\ f(x)f(x^2-1)=\sin(x), \quad\forall x \in \mathbb{R}$$ That is a difficult problem for me. Help me please.
4
votes
2answers
84 views

Functional equation $x\space f(x^2) = f(x)$

How can I logically lead to the answer from the following conditions? $$ \left\{ \begin{align} & x \, f(x^2) = f(x) \text{ for all } x > 0, \\ & f(x) \text{ is continuous}, \\ &f(1) = ...
3
votes
3answers
138 views

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

Find $f$ if $$ f(x)+f\left(\frac{1}{1-x}\right)=x $$ I think, that I have to find x that $f(x) = f\left(\frac{1}{1-x}\right)$ I've tried to put x which make $x = \frac{1}{1 - x}$, but this equation ...
6
votes
1answer
96 views

Prove $f(x)=0$ when $f(2x^2-1)= f(x)\cdot 2x$

Prove that $f(x)=0$ for $$x\in[-1, 1]$$ $f-continuous$ and for all $x$: $$f(2x^2-1)= f(x)\cdot 2x$$ It is simple for integer numbers. Another fact that I've noticed that $$f(2(-x)^2-1)= f(-x)\cdot ...