Questions tagged [functions]

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

5,087 questions with no upvoted or accepted answers
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17
votes
0answers
516 views

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 "...
16
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0answers
308 views

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 ...
13
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1answer
2k views

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 ...
13
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0answers
590 views

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 ...
13
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0answers
668 views

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 ...
13
votes
1answer
334 views

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\} ...
12
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0answers
133 views

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 ...
11
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0answers
475 views

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 ...
10
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1answer
230 views

A problem about periodic functions

Suppose that $f(x):\mathbb{R}\rightarrow \mathbb{R}$ is a periodic function with a minimal positive period $T$. Can $g(x)=f(x^2)$ be periodic? I know it is impossible if the condition $\forall x \in (...
10
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0answers
280 views

Number of zeros of the second derivative of composition of sigmoid-like functions

Consider the bounded, positive and monotonic functions $f(x) = \frac{x^k}{a^k+x^k}$ and $g(x) = \frac{b^h}{b^h+x^h}$ with $a,b>0,$ and $k,h > 1$ and defined for $x\in \mathbb{R}^+$. Using ...
9
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431 views

Does this calculation have a name, or a generic formulation?

Background Informatiom I would appreciate help in identifying or explaining this operation: To calculate each of the $n$ values of $f(\Phi)$: Sample from the distribution of each of $i$ parameters, $\...
9
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1answer
238 views

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: $$\zeta \rho (s) = \sum_{n=1}^{\infty} \frac{1}{(p_n)^s}. $$...
8
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303 views

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 ...
8
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0answers
229 views

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 ...
8
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0answers
218 views

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 ...
8
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1answer
231 views

A function is smooth at a point and not smooth in any neighbourhood of it, exist or not?

Suppose that a function $f$ defined in an open set $U \subseteq \mathbb{R}^m$ is smooth at a point $p \in U$. Then we have that there exists an open set $U_n \subseteq U$ $($ say $U_{n+1} \subseteq U_{...
7
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124 views

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that $\alpha\,f(yz)+\beta\,f(zx)+\gamma\,f(xy)\geq f(x+y+z)$ for all $x,y,z\in\mathbb{R}$.

Let $\alpha,\beta,\gamma$ be three real numbers. Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$\alpha\,f(yz)+\beta\,f(zx)+\gamma\,f(xy)\geq f(x+y+z)$$ for all $x,y,z\in\mathbb{R}$. ...
7
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2answers
56 views

Strictly increasing function $f: \mathbb{N} \to \mathbb{R}^+$ such that $\lim\limits_{n \to \infty} \frac{f(n)}{n} = 0$ and $f(n)+f(3n) \ge 2f(2n).$

Does there exist a strictly increasing function $f: \mathbb{N} \to \mathbb{R}^+$ such that $\lim\limits_{n \to \infty} \frac{f(n)}{n} = 0$ and $f(n)+f(3n) \ge 2f(2n)$ for all sufficiently large $n$? ...
7
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1answer
115 views

Does the product rule for differentiation has anything to do with $\sin( \alpha + \beta)$?

When learning about differentation, I came along the product rule: $$D(f \cdot g) = f \cdot Dg + g \cdot Df$$ I immediately thought of this rule from trigonometry: $$ \sin( \alpha + \beta ) = \sin(\...
7
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1answer
123 views

Use of the fact that every function is sum of an odd and an even function.

It is well know that every real variable function $f$ can be written as a sum of an odd and an even function, namely $h$ and $g$ where: $$h(x) = {f(x)-f(-x)\over 2}\;\;\;\;\;\;\;\;\;\;\;\;g(x) = {f(x)+...
7
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0answers
107 views

Is $\pi: \mathcal{C}^\infty (M,N) \to \mathcal{C}^\infty (S,N)$, $\pi(f) = \left. f\right|_{S}$ a quocient map in the $\mathcal C^1$ topology?

Let $M, N$ be smooth connected manifolds (without boundary), where $M$ is a compact manifold, so we can put a topology in the space $\mathcal C^\infty(M, N)$ using $\mathcal{C}^1$ Whitney Topology. ...
7
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326 views

Is there any continuous function that is only differentiable on $\mathbb{Q}$?

I am looking for a continuous function $f: \mathbb R \rightarrow \mathbb R$ so that $f$ is differentiable in $x$, if and only if $x \in \mathbb Q$. I already know there is no function that is ...
7
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0answers
275 views

Why is the unit circle definition of trig functions not rigorous enough?

It has recently come to my attention that the usual unit circle derivation of the elementary trigonometric functions isn't considered rigorous enough. Apparently, this has to do with problems ...
7
votes
1answer
354 views

Approximation of the exponential

Let $c>1,k\in\mathbb{N}$. Let's consider two approximations of the exponential function : The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$ and the second one is $...
7
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0answers
1k views

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 ...
7
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0answers
325 views

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 ...
6
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0answers
76 views

What is a solution to the recurrence relation $f(n) = f(n-1) +f\Big(\left\lfloor \frac{n}{2} \right\rfloor\Big)$?

Let $\mathbb{N}=\{1,2,3,\ldots\}$. Find a closed form or an asymptotic form of $f: \mathbb{N} \to \mathbb{N}$, where $f$ satisfies $f(1) = 1$ and $$f(n) = f(n-1) + f\bigg(\left\lfloor \frac{n}{2} \...
6
votes
1answer
148 views

What properties must $f$ have if $f(x)=f(\sin(\pi x)+x)\iff x\in\Bbb{Z}$?

This is a follow-up from my previous question. I now know that the statement: $$f(x)=f(\sin(\pi x)+x)\iff x\in\Bbb{Z}$$ is not true for all $f$. For example, $f$ can be $x$ to any constant power or ...
6
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0answers
145 views

$\frac{(x-y)^3}{x+y}\neq g(f(x)-f(y))$?

$$h(x,y)=\frac{(x-y)^3}{x+y}$$ Prove that there does not exist 1D real functions $f,g$ such that $h(x,y)=g(f(x)-f(y))$. The problem seems really really easy because it is obvious that $x+y\neq f(x)-...
6
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0answers
90 views

What is the minimum value of $f_\infty=\frac{x}{\sqrt{x-\sqrt[3]{x-\sqrt[4]{x-\cdots}}}}$?

In a similar vein to What is the maximum value of this nested radical?, I'd like to share a similar nested radical, but this time with changing fractional powers. What is the minimum value of $$f_\...
6
votes
0answers
94 views

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 ...
6
votes
0answers
203 views

I am a physicist with a difficult equation (quadratic exponential) I am curious about. No luck with a lit review, sympy or Mathematica.

I have an equation I have been playing with (the variable is $x$ and the constants are positive and real): $$( r_1 - x )^2 \frac{ a r_3 e^{x / r_2} + b r_2 e^{x / r_3} }{\left( a e^{ x / r_2} +...
6
votes
0answers
258 views

Closed form bijection between integers and pairs thereof

I know that it's simple enough to map the integers, $\mathbb{Z}$, to pairs of integers, $\mathbb{Z}^2$, in a bijective way (i.e. a one-to-one mapping). You can wrap the integers around the origin of ...
6
votes
0answers
85 views

Explain approximate lines in graph of this function

Sorry that this is a long question; the crux of it is that I want to know why lines appear in the graph of the function ($\varphi^\infty(x)$) I've defined. Define $\varphi(x)$ as follows: If the ...
6
votes
0answers
152 views

Function with $f(f(n))=f(n-1)f(n+1)-f(n)^2$

Let $\mathbb{N}$ denote the set of positive integers. Does there exist a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $$f(f(n))=f(n-1)f(n+1)-f(n)^2$$ for all $n\geq 2$? If $f$ is linear, ...
6
votes
1answer
519 views

Solution to $xe^{e^x}$

The problem $xe^{e^x}=e$ came up another day and I wondered if it were solvable. My attempt was the following substitution,$$x=W(u)$$$$W(u)e^{e^{W(u)}}=e$$Where I used a Lambert W identity to get $$W(...
6
votes
0answers
6k views

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 ...
6
votes
0answers
111 views

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 ...
6
votes
0answers
474 views

Proving not equicontinuity in $\Bbb R$ but equicontinuity in any other closed subset of $\Bbb R$

Let $F = {f_{n} | n ∈\Bbb N }$ be an infinite collection of functions $f_{n}(x)=e^{−n(x−n)^2} , x ∈ \Bbb R$. Prove that $F$ is not equicontinuous on $\Bbb R$ but equicontinuous on $[−a, a]$ for any $a ...
6
votes
2answers
135 views

Prove function is Fibonacci sequence

Stuck on how to finish a question I'm working on. Have to find the number of bitstrings of length n with no odd length maximal runs of ones. For example, when n=3 there are three such bitsrings: 011, ...
6
votes
0answers
276 views

expansion for $1-|t|$

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $$ I am wondering if one can expand ...
6
votes
1answer
323 views

Closed form for sine graphic rotated by 45 degrees?

Is there a non-parametric closed form for a function looking like a sine rotated 45 degrees? I have encountered also a similar question but it asks for a function resembling the rotated sine, but not ...
6
votes
1answer
92 views

Function for which it is unknown whether it is continuous

Is there any function $f:\mathbb R\rightarrow \mathbb R$ for which at least some values are known but it is unknown whether $f$ is continuous or not? Edit: I am looking for examples from actual ...
6
votes
1answer
143 views

Prove that if $2^x,3^x, 5^x, 7^x, 11^x … $ are all integers then $x$ is an integer as well

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x ... $ are integers then $x$ is an integer as well? I have read the definition of the exponent functions as given in my calculus text, and the ...
5
votes
0answers
75 views

Is there a formula for $f(x)$ where $f(x)=$ the sum all simplest fractions with the numerator$+$denominator equals $x$

Is there a closed-form for $$f(n)=\sum\limits_{\substack{k=1 \\ (k,n)=1}}^{n-1} \frac{k}{n-k}$$ For example, $f(5)=1/4+2/3+3/2+4/1= 6+5/12$; $f(6)=5+1/5$ The list of $f(x)$ from $x=1$ to $x=8$ is: $(0,...
5
votes
0answers
74 views

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 ...
5
votes
0answers
111 views

Need your thoughts on a faster approximation for the signum function I came up with.

I recently was kind of fascinated with the whole signum function out of the blue, and was generally really curious of the already existing approximations of it. I really don't want to lag this down ...
5
votes
1answer
112 views

Convergence of $\sqrt[k]{z+\sqrt[k]{z+\sqrt[k]{z+\cdots}}}$, where $z=(1+x)^k-(1+x)$

If one writes $$1+x=\sqrt{(1+x)^2}=\sqrt{1+2x+x^2}=\sqrt{x+x^2+(1+x)}$$ then one has a recursive definition of the function $1+x$ which can be used to write $1+x$ as the infinite nested radical: $$1+x=...
5
votes
1answer
152 views

A pre-calculus problem about a quadratic function

This is from a test (I'm not a high school student) given to rising high school juniors. The problem was designed to take less than 20 minutes, preferably 5-15. Judging from its source, this is not a ...
5
votes
1answer
60 views

The digit at position $n$ of the number $x$ in base $m$

As a solution to this question, we can define a function $f_b(x, n)$ which finds the digit in the $n$th position of $x$ in base $b$. $$ f_b(x, n) = \left\lfloor \frac{x}{b^n} \right\rfloor \bmod b $$ ...

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