# Questions tagged [functions]

For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.

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### Find a function such that $f^{-1}=f'$

Let $f:\Bbb{R}^+\rightarrow\Bbb{R}^+$ be a differentiable bijection and let $f$ satisfy: $f'=f^{-1}$ (where $f^{-1}$ denotes the inverse of $f$). Find $f$. This comes from a facebook page "...
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### On injectivity of $F_N$

I am interested in the following problem: Let $F_N:\mathbb{N}\to\mathbb{N}$ be the function that maps a natural number $a$ to the least natural number more than or equal to $a$ for which there exists ...
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### Find the function of separation between two functions

I seriously doubt that is what it is actually called, but I'm not very knowledgeable in this matter. Conceptually, what I am trying to do is calculate the function of a line/curve that shows the ...
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### Compositional “square roots”

Let $A$ be a set and $f\colon A\to A$ a function. The primary and general question is: What conditions are necessary so that there exists $g$ such that for each $x$ in $A$, $g(g(x))=f(x)$? This is a ...
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### Multiplicative Identity analog for absolute value

Is there a standard name for the function: $$f(x) = \begin{cases} x & \text{if |x|≤1;}\\ 1/x & \text{if |x|>1;}\\ \end{cases}$$ And is there a potential application of this ...
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### Approximating the area below average of a concave function

Given a non-decreasing concave function $f:[0,1]\rightarrow \mathbb{R}^+$. Define \begin{align*} F(n)=\displaystyle\sum_{i=1}^n \min\left\{\frac{1}{n},\frac{f\left(\frac{i}{n+1}\right)}{n+1}\right\} ...
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### Understanding the Legendre transform

In physics, I've seen the Legendre transform motivated by "changing the variable $x$ of a function $x \mapsto f(x)$ to the variable $u = \frac{df}{dx}$." I don't quite see what that means and why the ...
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### Olympiad-style question about functions satisfying condition $f(f(f(n))) = f(n+1) + 1$

QN: What functions (from non-negative integers to non-negative integers) satisfy the condition $$f(f(f(n))) = f(n+1) + 1$$ Comment: Evidently $f(n) = n+ 1$ is one solution. Equally evidently no ...
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### When does the “Zetor function” converge?

Let $p_n$ be the n'th non-trivial zero of the Riemann zeta function. We define the Zetor function (acronym of 'zeta' and 'zero') as follows: $$\zeta \rho (s) = \sum_{n=1}^{\infty} \frac{1}{(p_n)^s}.$$...
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### A Ramanujan-type trigonometric identity

At the end of the following article: http://www.ijpam.eu/contents/2013-85-1/15/15.pdf It is asserted that the russian mathematician, Sergey Markelov, in private communication, told them that he ...
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### How to invert this expression involving $\tanh^{-1}$?

I've got the expression: $x = \tanh^{-1}(p) - \sqrt{\frac{2}{3}} \tanh^{-1}\left( \sqrt{\frac{2}{3}} p\right)$ How can I invert this function so I have a function $p(x)$? I thought about using ...
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### Can you add new functions to the set of elementary functions such that every function has an anti-derivative?

Its fairly well known that not every elementary function has an elementary anti-derivative. The common examples of this are $\exp(-x^2)$ and $\sin(x)/x$. The general workaround to this problem is to ...
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### Different Definitions on the Differentiability of Functions on a closed set.

I have encountered three different definitions on the differentiablity of functions on a closed set. In the following, suppose that $\Omega\subset M$ is a (open) domain, where $M$ is a manifold. The ...
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### To determine if$f^{-1}(x)$ is periodic function or not? $f(x)=\int_1^{x} \frac{1}{\sqrt[m]{P(t)}}\;dt$

$$f(x)=\int_1^{x} \frac{1}{\sqrt[m]{P(t)}}\;dt$$ $P(x)$ is polynomial with degree $n$. $m$ is an positive integer and $m>1$ What is the algoritm to determine $f^{-1}(x)$ is periodic function ...
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### Let function $g:A\to A$, $\:\:\:g^{2016}=g\circ g\circ g…\circ g=id_A$ , Prove that $g$ is Inverse function

Let function $g:A\to A$, $\:\:\:g^{2016}=g\circ g\circ g...\circ g=id_A$ , Prove that $g$ is Inverse function My attempted : $g$ is Inverse function if it's Bijection to prove $g$ is one-to-one ...
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### Show that $1^k+2^k+…+n^k$ is $O (n^{k+1})$.

Let $k$ be a positive integer. Show that $1^k+2^k+...+n^k$ is $O (n^{k+1})$. So according to the definition of big-$O$ notation we have: $$1^k+2^k...+n^k ≤ n(n^k) = n^{k+1}$$ whenever $n>1$ Is ...
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### Terminology for functions such that $f(x)\ge x$ for all $x$

Is there a common terminology for a real function $f$ such that $$f(x)\ge x$$ for all $x$? same question for the conditions $\forall x,f(x)>x$; $\forall x,f(x)\le x$; $\forall x,f(x)<x$. (I'm ...
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### Fun functional equation in a group: $af(ab)b=bf(ba)a$

Suppose we have a finite group $G$ and a function $f:G\to G$ that satisfies the following property for all $a,b\in G$: $$af(ab)b=bf(ba)a$$ Under what conditions (or, for what groups $G$) can such a ...