Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).

Filter by
Sorted by
Tagged with
24
votes
4answers
787 views
+100

If $f(x)-f^{-1}(x)=e^{x}-1$, what is $f(x)$?

$f(x)$ is an increasing, differentiable function satisfying $f(x)-f^{-1}(x)=e^{x}-1$ for every real number $x$ I couldn't figure it out whether such function $f(x)$ exists or not. And if it exists, I ...
0
votes
1answer
47 views

Problem involving a functional equation.

$\mathbf {The \ Problem \ is}:$Find all possible functions $f \colon \Bbb R→\Bbb R$ such that $f$ is infinitely differentiable on $\Bbb R$ and $f$ satisfies an equation: $$ f(y+x)-f(y-x)= 2xf'(y)\...
4
votes
2answers
71 views

For what continuous $f$ we can write $f(x)+a\cdot f(x+\alpha)$ as $b\cdot f(x+\beta)$?

After reading about harmonic addition theorem, I am interested in: Find all continuous $f:\mathbb R\mapsto\mathbb R$ that admit identities of the form $$f(x)+a\cdot f(x+\alpha)=b\cdot f(x+\beta)$$ ...
3
votes
1answer
31 views

If a random variable $X$ and its mapping $\mathrm{ln}(X+1)$ have the same distribution, what is the distribution of $X$?

Let $f_X(x)$ be the probability density function of a real random variable $X$, $X\geq0$. Define $Y=\mathrm{ln}(X+1)$ and denote the PDF of $Y$ as $f_Y(y)$. I'm curious when will $f_X=f_Y$? I've ...
0
votes
2answers
40 views

find all functions such that $f(x)\geq0$ for all $x$ and $f(x+t) = f(x) +f(t) +2\sqrt{f(x)f(t)}$ [on hold]

I have thought of $f(x)=x^2$, but it’s not always positive and it doesn’t work when $x$ or $t$ is negative.
2
votes
4answers
61 views

solve the functional equation $f(x+t)-f(x-t)=4xt$

I think this question might be related with arbitrary functions, but I’m not sure. I also tried to set $t$ to different values but couldn’t get it to work. I tried to set $t=x$ and end up with $f(2x)=...
3
votes
3answers
152 views

Find all additive real valued functions such that $f(x^{2019})=f(x)^{2019}$

The following is the final problem from this page: Find all the functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $$f(x+y)=f(x)+f(y) \; \; \; \forall \,x,y\in \mathbb{R}$$ and also (this is ...
42
votes
8answers
16k views

Find $f(x)$ such that $f(f(x)) = x^2 - 2$

Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$. Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but the ...
1
vote
0answers
43 views

Help with expressing Partition Numbers as powers of two

Grateful for some help with expressing Partition Numbers as powers of two. I'm not great at maths, but this is as far as I have got: $P(n)=2^{n-1}-(2^{n-2})(2^x)$ where $x$ is an exponential that ...
1
vote
1answer
38 views

Functional Equality

If I have the following equations: $$a(r)=\int_0^\infty s\ f(rs)\ g(s(1-r))\ ds\\b(s)=\int_0^1s\ f(rs)\ g(s(1-r))\ dr$$ Where $f,\ g>0$, $s\in (0,\infty)$ and $r\in (0,1).$ Is it possible to write $...
10
votes
0answers
126 views

$f(f(x)) = 1 + x^2$, then what is f(1)?

I get $f(f(a)) = a^2 + 1 = f(f(-a))$, and so $f(a)^2 + 1 = f(a^2 + 1) = f(-a)^2 + 1$, so $f(a) = f(-a)$ or $f(a) = -f(-a)$, but then I donot know what to do next. Thanks for any help.
2
votes
0answers
59 views

Denesting infinite nested radicals: $\sqrt{1+ \sqrt{2+ \sqrt{4 + \sqrt{8 + \sqrt{\dots}}}}}$ [duplicate]

I tried to denest the infinite nested radical $$\sqrt{1+ \sqrt{2+ \sqrt{4 + \sqrt{8 + \sqrt{\dots}}}}} \; .$$ My first try was to find a functional equation for $$\alpha(x) = \sqrt{1+ \sqrt{x+ \...
2
votes
2answers
31 views

Analytical method to solve the given equations

I require the points of intersection of two curves $y^2=4x$ and $y=e^{-x/2}$ to the find the angle between them. Is there any method to find the points as when I tried to graph them the solution was a ...
1
vote
2answers
40 views

Proving ideal gas equation from Boyle’s, Charles’ and Gay-Lussac’s laws

Assuming the empirical laws by Boyle, Charles and Gay-Lussac, which respectively say that \begin{align} p&\propto f(T,N)\cdot {1\over V}\\ V&\propto g(p,N)\cdot T\\ p&\propto h(V,N)\cdot ...
4
votes
1answer
61 views

Probability of getting a sufficiently long piece from multiple stick breaking

This question asks: Start with a stick of length $1$. Repeatedly remove some fraction $U$ of the remaining stick, where $U$ is uniform in $[0,1]$. What is the probability that at least one of the ...
2
votes
4answers
79 views

Functional equations [Sample paper of Indian Mathematical Olympiad]

Edit- There was information missing (lack of clear printing in my book) in the book through which I referred the question. Confirming with my friend's book I have made a small change. I am really ...
7
votes
2answers
78 views

Classifying Solutions to $f^n(x) = x$

Let $f: \mathbb{R} \to \mathbb{R}$ be continuous everywhere in $\mathbb{R}$ except on some finite set. Suppose we also have $f^n(x) = x$ for all $x$ where $f$ is defined. Note that by $f^n(x)$ I mean ...
3
votes
1answer
104 views

What are all the functions that satisfy $f(x)/f(y) = f(kx)/f(ky)$?

Find all continuous functions defined over real numbers that satisfy $\frac{f(x)}{f(y)} = \frac{f(kx)}{f(ky)}$, for any $x$ and $y$. It is possible to show that the above condition holds for $f(x) = ...
1
vote
0answers
57 views

How to prove that these vectors are linearly independent?

Consider the following vectors: $v_m = (f(ma_1), f(ma_2), \dots, f(ma_n))$ for $m \in \{1,\dots, n\}$, where $f(x) \in \mathbb{R}$ is a continuous, strictly monotonic, and nonlinear function and $...
8
votes
0answers
175 views

Existence of function satisfying $f(f'(x))=x$ almost everywhere

My project is to Study the existence of a continuous function $f : \mathbb{R} \rightarrow \mathbb{R}$ differentiable almost everywhere satisfying $ f\circ f'(x)=x$ almost everywhere $x \in \mathbb{R}$...
0
votes
0answers
11 views

Functional integral in transform space

The integral $ \int D\phi_l (x) \exp (-K(\nabla \phi_l (x) )^2+ c^{-2} (\phi_l (x))^2)) $ can be simplified by the Fourier transformation $\phi (x) = (2\pi)^{-d/2}\int exp(iq.x) \phi_l(q) d^d q$ where ...
19
votes
4answers
1k views

very elementary proof of Maxwell's theorem

Maxwell's theorem (after James Clerk Maxwell) says that if a function $f(x_1,\ldots,x_n)$ of $n$ real variables is a product $f_1(x_1)\cdots f_n(x_n)$ and is rotation-invariant in the sense that the ...
4
votes
1answer
159 views

Questions about the existence of a function

Question 1: Study the existence of $C^1$ function $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfying $\forall x\in\mathbb{R},\mbox{ } f\circ f'(x)=x.$ Question 2: Study the existence of ...
0
votes
0answers
17 views

Concluding existence of a function from functional equations

Suppose each of $X_A, X_B, Y_A, Y_B$ is an implicit function of the other three. Let the same also hold for $X_B, X_C, Y_A, Y_B$ and $X_C, X_A, Y_C, Y_A$. Then we can write for some functions $f_1$ ...
-1
votes
1answer
45 views

Functional inequation [closed]

Is it possible to find all smooth functions $f : \mathbb{R}_+ \to \mathbb{R}$ such that $f(1)=1$ and for all $x,y \in \mathbb{R}_+$ there holds $f(x)f(y)< f(xy)$?
3
votes
1answer
85 views

If $f(xy) = f(x)+f(y)+xy-x-y$ and $f'(1)=4$, evaluate $f$

Let a differentiable function 'f' satisfy the functional rule $$f(xy) = f(x)+f(y)+xy-x-y\quad \forall x,y >0 \text{ and } f'(1)=4.$$ Evaluate 'f' So for this I was taught to start by ...
1
vote
1answer
29 views

Proof of Theorem 1, Calulus of Variation, Gelfand and Fomin

This is theorem 1 on page 12 of Gelfand and Fomin. Why $\phi_1[h]-\phi_2[h]= \epsilon_2 ||h||$ instead of $\epsilon_1 ||h||-\epsilon_2||h||$? Is this a typo and he meant some $\epsilon_3=(\epsilon_1-\...
2
votes
1answer
45 views

How to find the solutions of this functional equation

$$f(\tfrac{1}{2}+x)+f(\tfrac{1}{2}-x)=8xf\big(4(\tfrac{1}{2}+x)(\tfrac{1}{2}-x)\big)\qquad\text{for}\qquad x\in(0,\tfrac{1}{2})$$ I have no idea about how to tackle this equation. The original ...
2
votes
1answer
49 views

Functional equation problem; chain of functions

I have tried more than an hour but couldn't solve it, can somebody please give me a clue? $$f:\mathbb R\rightarrow\mathbb R$$ $$f(f(f(X)))+f(f(X))+X=3f(X).$$ Find $f(X)$ I know that $f(X)=X$ is a ...
0
votes
0answers
45 views

Show that the functional equation $f(f(x))=-x^{3}+g(x)$ has no continuous solution.Here $g$ is a continuous periodic function with positive period.

I want to show $f(f(x))=-x^{3}+g(x)$ has no continuous solution $f:\mathbb{R}\to\mathbb{R}$.Here $g$ is a continuous periodic function with its period $T>0.$ Any help will be thanked.
1
vote
2answers
43 views

$h(x,y)=f(x)+g(y)$

As a followup comment to my answer to the question $\;\;\;\;$Can I split $\frac{1}{a-b}$ into the form $f(a)+f(b)$? "Lord Shark the Unknown" made the following observation: If $h:\mathbb{R}^2\to\...
0
votes
2answers
66 views

Find all such functions $f:R\to R$

It's my last question. Just give me advise how to start. Q: Find all such functions $$f:\mathbb R\to \mathbb R,$$ for all real x, y, the equality $$f(yf(x))=x^2y^4$$
4
votes
0answers
85 views

If $f: \mathbb R^n \to \mathbb R^n$ satisfies $\|f(y)\| = \|f(x + y) - f(x)\|$, is $f$ additive?

Question Main question: Let $\| \cdot \|$ be a norm on a finite-dimensional real vector space $V$. If $f : V \to V$ is a function satisfying $$ \| f(y)\| = \|f(x + y) - f(x)\| $$ for all $x, y ...
0
votes
0answers
37 views

Functional Equations from information theory

I looking for functional equations which are out coming from real life An example from real life to explain and relate to uncertainty is as follows: Example: 3 candidates A, B and C are sitting ...
4
votes
3answers
130 views

Solve for all possible functions f: $|f(x)-f(y)|=2|x-y|$. [closed]

I'm getting $f(x)=2x+f(0)$ and $f(x)=f(0)-2x$ by setting $y=0$, but I'd like to verify. Am I right?
0
votes
0answers
68 views

Find all functions $f\colon\Bbb R\to \Bbb R$ that satisfy $|f(x)-f(y)| =3|x-y|$ for all $x,y \in \mathbb{R}$. [duplicate]

Problem: Find all functions $f\colon\Bbb R\to \Bbb R$ that satisfy $|f(x)-f(y)| =3|x-y|$ for all $x,y \in \mathbb{R}$. My Solution: Take two cases. Case 1: Assume that $f(x)-f(y)=3(x-y)$. Then $f(x)-...
9
votes
1answer
199 views

Finding $f$ such that $f(f(f(f…(x)))) = x$

I would like to find the set of continuous functions $f_n(x)$, where $f_n(x):\mathbb{R}\to \mathbb{R}$ satisfies $$f_n(f_n(f_n(f_n...(x)))) = x$$ where there are $n$ iterations of $f(x)$. For example $...
3
votes
1answer
72 views

System of functional equations $ f(f(x)-f(y))=|f(x)-f(y)|, \; f(1-f(x+2))=1-f(x)$

Find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$\begin{equation} \begin{cases} f(f(x)-f(y))=|f(x)-f(y)|, \\ f(1-f(x+2))=1-f(x) \end{cases} \end{equation}$$ for all $x, \; y \...
10
votes
1answer
104 views

Are $f$ satisfying $|f(y)| = |f(x+y) - f(x)|$ additive?

(This question is inspired by this question, and in particular the comment by Charlie Cunningham.) Let $(V, \|\cdot\|)$ be a normed real vector space. Let $f: V \to V$ be a function satisfying, for ...
0
votes
2answers
40 views

How can I attempt this FE

Prove that there is no function f : ℕ→ℕ such that f (f (n)) = n + 1. Here ℕ is the positive integers {1, 2, 3,...}. I have messed around with the FE but can't seem to produce anything meaningful. I ...
8
votes
0answers
117 views

Uniqueness of solutions of a functional equation

Problem: Find all continuous and strictly increasing functions $f\colon(0,\infty)\to\Bbb R$ with $$f(3x)-f(2x)=f(2x)-f(x)$$ for all $x>0$. A class of solutions is given by $f(x)=ax+b$, where $a>...
2
votes
2answers
70 views

Given $a$ and $c$ positive numbers, is there a function $h$ such that $h(-ax)= -ch(x)$ for all $x$?

Given $a>1$ and $c$ a positive number, is there a (no trivial) function $h$ such that $$h(-ax)= -ch(x)$$ for all $x$?. I know that $h(x)=c^{\log_a|x|}$ satisfies $h(ax)=ch(x)$, however I am not ...
0
votes
1answer
40 views

Problem with existence of discontinuous additive function.

I think I have made a mistake in my justification but I can't see where. Let's assume that $f$ is a discontinuous additive function. From the basic properties of additive functions we know that $f$ ...
0
votes
0answers
19 views

subgroup of additive reals of index 2 [duplicate]

Is there a subgroup of the real numbers under addition of index 2? (and if so, can we classify them somehow?) [I am trying to solve the functional equation $f(x)f(y)=f(x-y)$ for all real $x,y$. If $f$...
1
vote
0answers
25 views

Finding a number theory-oriented solution to the functional equation $f^{f(m)}(n)=n+2f(m)$ over $\mathbb{Z}_{\geq 0}$

Find all functions $f:\mathbb{Z}_\geq0 \to \mathbb{Z}_\geq0$ such that for all non-negative integers $m$ and $n$, where $m\leq n$, $$f^{f(m)}(n)=n+2f(m)$$ (Here, $f^k(x)=\underbrace{(f\circ f\circ \...
1
vote
1answer
84 views

Non-exponential functions satisfying $f(ab) = f(a) + f(b)$

If $f:\mathbb R \to \mathbb R^+$ satisfies $f(a+b) = f(a)f(b)$, and if we write $f(1) = B$, then it is fairly straightforward to prove that for any rational number $x \in \mathbb Q$, $f(x) = B^x$. If ...
1
vote
0answers
53 views

Find all functions $f: \mathbb{R} \rightarrow \mathbb {R}$ such that [duplicate]

Find all functions $f: \mathbb{R} \rightarrow \mathbb {R}$ such that $f(x+y)+f(x-y)=2f(x)\cos y$(*) for any x,y real numbers my attempts: for $x=y=\frac{2\pi}{3}$ in the (*): $f(\frac{2\pi}{3})+f(0)=...
5
votes
1answer
98 views

Is there a continuous function $f$ satisfying $f^{2}(x) = f(x^{2})$, $f(0)=1$ and $ f(1)=0$?

I was wondering if there is a continuous function $f$ satisfying $$[f(x)]^2 = f (x^2)$$ and $f(0)=1$ and $ f(1)=0$. Clearly, some easy functions like polynomials are not satisfied. I guess there is a ...
1
vote
0answers
33 views

Iterative functional equation $f \circ f = f$ [duplicate]

I tried but failed to find some references about the following iterative functional equation: find all functions $f \colon \mathbb{R} \to \mathbb{R}$ such that $f \circ f = f$, that is $f(f(x))=f(x)$ ...
3
votes
2answers
128 views

(Olympiad Question) $f(2x) + 2f(y) = f(f(x+y))$

This was Problem 1 from the 2019 International Mathematics Olympiad. Find all functions $f$ : $\mathbb{Z}$ $\rightarrow$ $\mathbb{Z}$ that satisfy $f(2x) + 2f(y) = f(f(x+y))$ whenever $x , y$ $\...