# 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).

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $x \in \mathbb{R}$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $f_1$ ...
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### 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 $f(x)f(y)< f(xy)$?
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### 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 ...
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### 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 $$f(yf(x))=x^2y^4$$
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### 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 ...
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### 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 ...
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### 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 ...