Questions tagged [elementary-functions]

For questions on elementary functions, functions of one variable built from a finite number of polynomials, exponentials and logarithms through composition and combinations using the four elementary operations $(+, –, ×, ÷)$.

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Variants of geometric sum formula

I know there are closed formulas for sums of the form $\sum_{k=0}^n k^sr^k$ and $\sum_{k=0}^n r^{2n}$ or $\sum_{k=0}^n r^{2n+1}$. (See https://en.wikipedia.org/wiki/Geometric_series#Sum) From Sum of ...
Irwin Shure's user avatar
2 votes
2 answers
93 views

Is there a spelling mistake or am I missing something

Here, $[ \cdot]$ is $\lfloor \cdot \rfloor $ floor function. N $ \in N$. Where did $\frac{[Nx]} N + \frac{1}{2N}$ came from and how does $x$ differs by $\frac{1}{2N}$. Shouldn't it be $\frac{1}N$ if ...
Yugant Shewale's user avatar
6 votes
0 answers
115 views

Is there an "elementary irrational number" without a certain digit in its decimal presentation?

In this question, I define an "elementary irrational number" as an irrational number which is built up of a finite combination of integers, field operations (addition, multiplication, ...
Alex-Github-Programmer's user avatar
1 vote
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College Algebra Function Transformations Unique?

Let $f(x)=x^2$ be the parent function for the transformation $g(x)=(2x+12)^2 -1$. Follow the two points $(0,0)$ and $(2,4)$ from the parent function through the transformation. Let's simplify $g(x)$ ...
Watson_Unknown's user avatar
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What is the image of $|z|<\frac{1}{2}$ under the action of $f(z)=\frac{1}{2}\left( z+\frac{1}{z} \right)$?

What is the image of the disc $|z|<\frac{1}{2}$ under the complex function $f(z)=\frac{z^2+1}{2z}=\frac{1}{2}\left( z+\frac{1}{z} \right)$? I guess that a first step is to write: $2f(x+iy) = x+iy+\...
Dr. John's user avatar
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1 answer
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Polynomial evaluation at zero.

Supose we have a polynomial function (what follows is the definition of the book i'm reading) $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $f(x)=\sum_{i=0}^n a_ix^i$ $(0 \leq i \leq n \in \mathbb{...
Batata's user avatar
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Are all multivariate elementary functions $C^\infty$ on any open ball for which they are defined?

Are all multivariate elementary functions $C^\infty$ on any open ball for which they are defined? I believe they are, and can find no counterexample, but do not know how to prove this. (All the ...
SRobertJames's user avatar
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1 answer
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Exact solution to system of first-order coupled nonlinear ODEs

I have recently been trying to find an exact solution to the following system of first-order ODEs: $\begin{cases} \frac{dx}{dt}=y(t)*z(t) \\ \frac{dy}{dt}=x(t)*z(t) \\ \frac{dz}{dt}=x(t)*y(t)\end{...
FabrizzioMuzz's user avatar
2 votes
0 answers
58 views

Constructing polynomials with rational critical values

I have poked around the forum, and haven't been able to find an answer to the following question: Given $n$ integral points $(x_i, y_i)$ with distinct $x_i$, is it possible to construct a polynomial $...
tiral's user avatar
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4 votes
1 answer
187 views

How to determine the monotonicity of this function

Given the definition of the function: $$ f(x) = e^x - e^{-x} + \frac{2x}{(x^2+1)^2} , x∈(-∞,0]$$ What's the monotonicity of this function in the given range? I tried to calculate the derivation to ...
xmh0511's user avatar
  • 205
2 votes
1 answer
117 views

Book Recommendations for Elementary functions

I have been through school. I have had a treatment of (Pre)Calculus (I,II,III). I still feel like I don't know enough Trigonometry, or enough of Exponentials, or other functions. Even through my ...
Kartik Pandey's user avatar
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2 answers
40 views

Intuition about a simple equality involving min and max of some naturals

I am playing for fun with some (natural) numbers and noticing the following: For any $h = 0, ... , \max(a, b)$: $$\min(a, h) - \min(b, h) = \max(0, h - b) - \max(0, h - a)$$ What is this called? Is ...
Jada's user avatar
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1 answer
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Dirichlet series of an elementary function

Is there an example of an elementary function (different from Dirichlet polynomials, i.e. cutoff Dirichlet series) which has a know Dirichlet expansion (known coefficients)? I am aware of the ...
F. Jatpil's user avatar
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1 answer
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Non-elementary antiderivative of rational function: $f(x)= \frac{1}{a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0}$

Is there any function of the form $$f(x)= \frac{1}{a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0}$$ for $n\in\mathbb{N},a_i\in\mathbb{R}$, that has no elementary antiderivative? If there is none, is there a ...
Mattan Feldman's user avatar
3 votes
1 answer
81 views

Inequalities for the solution of $x = (x-a) e^{x+a}$.

Let $a > 0$. The equation $$(x-a) \, e^{x+a} = x $$ must be solved for $x > 0$. Since the solution does not have a closed form, I would like to obtain bounds for the solution. Until now, I was ...
P.S. Dester's user avatar
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Integrals of (Jacobi) elliptic functions

Jacobi elliptic functions create a new trigonometry that we don't teach in high schools. Taking the derivative of composite functions of these functions with elementary functions is easy thanks to ...
Bob Dobbs's user avatar
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11 votes
2 answers
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The rational function $\tan\left(n\arctan x\right)$

While messing around with the formula $$\arctan x+\arctan y=\arctan\frac{x+y}{1-xy},\tag1$$ I decided to iteratively find expressions for $n\arctan x$ by applying the formula $(1)$ $n$ times. I was ...
clathratus's user avatar
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1 vote
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Can we always know whether an infinite series converges or diverges, if we are summing over an elementary expression?

Just like the question stated, can we always determine the convergence of an infinite sum of an elementary function? For example, given some elementary function f(n) can we determine if the sum will ...
Craftinators's user avatar
-2 votes
1 answer
61 views

Solving the equation of the form $a - \frac{1}{x}\ln{(\frac{x}{1-x})} + \frac{1}{x^2}$ [closed]

I got an equation of the form $a - \frac{1}{x}\ln{\frac{x}{1-x}} + \frac{1}{x^2} = 0$. $a$ is a constant and the values of $x \in [0,1]$. Does a closed-form solution exist for this? If not, is there ...
Shew's user avatar
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3 answers
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Inequality involving nested absolute values

I don't understand where I'm wrong in my reasoning about this inequality: $$\big\vert x - \vert 1-x\vert\big\vert < 1$$ Attempts $$\vert 1-x\vert = \begin{cases} 1-x & \text{if}\quad x\leq 1 \\ ...
Martin and Friends's user avatar
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48 views

New Formula for the Payback Period (PBP) of an investment

$\bf1$. Context: Payback Period (or PBP): Time required to recover initial investment in a project. Payback Period = $\bf3.25$ years (to recover the $50,000$) $\bf2$. Unliked Formula: PBP = $\frac{\...
InanimateBeing's user avatar
-2 votes
3 answers
82 views

Is there a simple function mapping -1 to 1 and 1 to 0 [closed]

This is a rather simple question. I'm writing a computer program, and I have a variable for velocity, which takes values $\pm 1$. I also have corresponding sprites, facing both right and left, on ...
Math chiller's user avatar
1 vote
1 answer
95 views

Proof of the equality $\left(\frac{2^{\ln2}}{3^{\ln3}}\right)^{\frac{1}{\ln3-\ln2}}= \frac{1}{6}$ [closed]

I want to show that $$\left(\frac{2^{\ln2}}{3^{\ln3}}\right)^{\frac{1}{\ln3-\ln2}}= \frac{1}{6}$$ I've tried for a while but I am unable to prove it by only manipulating one side. Any help would be ...
TreeGuy's user avatar
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How to find the distance between two funtion inputs given two function outputs

The question: Suppose you have a function given by $f(x)=0.1x^2-x+2.5$ We are given that $f(x_1)=3$ $f(x_2)=1.5$ We are tasked with finding the distance between $x_1$ and $x_2$ My attempt so far: I ...
hoff's user avatar
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3 answers
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Need to simplify formula $ \pi \sim \frac{8}{n^2} \cdot \sum\limits_{i=0}^{n} \sqrt{i \cdot (n - i)} $

This formula can be simplified? $$ \pi \sim \frac{8}{n^2} \cdot \sum\limits_{i=0}^{n} \sqrt{i \cdot (n - i)} $$ I am trying to find an alternative formula for circle area. I stacked to analyze this ...
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0 answers
31 views

A graph made of half circles

Taking inspiration from the question What kind of curve is made of half circles? I'm trying to build up a function $f$ whose graph is similar to a sine function made of semicircles. While finding an ...
Mathland's user avatar
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3 votes
1 answer
101 views

how do you know when there is no antiderivative written in elementary functions? [duplicate]

I know that there are integrals that cannot be written in this way such as $e^{-x^2}$ but there is a general rule that can be used to recognize these types of integrals?
marcoturbo's user avatar
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0 answers
94 views

How to solve the equation ${\log_{x} 9} + {\log_{2} x} = {\log_{2} (9+x)}$

Please advise on how to solve the following equation. $$ {\log_{x} 9} + {\log_{2} x} = {\log_{2} (9+x)} $$ I tried to rearrange and got this. $$ {\log_{x} 9} = {\log_{2} (1+\frac{9}{x})} $$ $$ [{\log_{...
iamrellik's user avatar
2 votes
1 answer
81 views

Can $\sin(\frac{n\pi}{m})$ with $n,m \in \mathbb{Z}$ always be represented using only algebraic functions?

Can $\sin(\frac{n\pi}{m})$ with $n,m \in \mathbb{Z}$ always be represented using only algebraic functions? In other words can $sin$ of a rational multiple of $\pi$ always be represented using only ...
MorganS42's user avatar
  • 135
3 votes
2 answers
165 views

Are there basic techniques for proving that no elementary antiderivative exists? [duplicate]

Background: It seems there is an algorithm called the full Risch algorithm that can decide whether a given function has an antiderivative that can be expressed in terms of elementary functions. There ...
WillG's user avatar
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5 votes
2 answers
830 views

How to approach this tricky log problem

Consider the following equation: $$(7 + 4\sqrt{3})^{t^2 - 5t + 5} + (7 - 4\sqrt{3})^{t^2 - 5t + 5} = 14$$ I have tried and exhausted all the methods I know of and resorted to brute force to solve this ...
yerdeth's user avatar
  • 69
4 votes
0 answers
143 views

Coefficients of Chebyshev polynomials

Not long ago, I derived the formula for Chebyshev polynomials $$T_{n}\left( x\right)= \sum_{k=0}^{\lfloor \frac{n}{2} \rfloor}{n \choose 2k}x^{n-2k}\left( x^2-1\right)^{k}$$ How to extract the ...
Fty56's user avatar
  • 139
4 votes
1 answer
215 views

Prove that $e^{-ax} \leq (1 - x)^a + ax^2/2$ for $a > 1$, $0 \le x \le 1$

Here we assume that all variables are reals. I would like to show that $$ e^{-ax} \leq (1 - x)^{a} + \frac{1}{2} ax^2 \tag{1} $$ for all $a > 1$, $0 \leq x \leq 1$. By Taylor’s theorem, I know ...
Kazune Takahashi's user avatar
1 vote
1 answer
78 views

if $f(x) = |3x - 1|$ so what is the sum of all $x$ such that $f(f(x)) = x$?

Question: if $f(x) = |3x - 1|$ so the sum of all values of x that satisfies $f(f(x)) = x$ is? alternatives: a) $11/10$ b) $21/20$ c) $27/20$ d) $23/20$ e) $5/4$ My try: I've thought that I had ...
user avatar
2 votes
2 answers
118 views

Inverse function of $x \mapsto \coth x - 1/x$

Consider the function $f$ defined over $\mathbb{R}$ as $$f(x) = \coth x - \frac{1}{x}$$ if $x \neq 0$ and $f(0)=0$. Since the function $\coth$ can be developed in series as $\coth x = \frac{1}{x} + \...
Goulifet's user avatar
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1 vote
0 answers
109 views

can anyone find me or disprove existence of an elementary function such that $f(f(x))=\sin(x)$

I don't have a lot of ideas or a bold direction, I tried to generate such function in Desmos and it kind of seems to exist but I don't know the formula or even if it is elementary or not. I do know ...
עמית חי לרמן's user avatar
2 votes
1 answer
77 views

What is $\,\lim\limits_{x\to0}\frac{\ln\left(\frac{e^x}{x+1}\right)}{x^2}\;?$

I've been trying to solve this limit, but can't really seem to be able to without using l'Hospital's rule (which the textbook specifically forbids). Here goes: $$\lim_{x\to0}\frac{\ln\left(\frac{e^x}{...
Heribert Greinix's user avatar
1 vote
0 answers
88 views

Determine all roots of $\sinh(z^2) = \dfrac{15}{8}$

$\sinh \left(z^2\right)=\frac{15}{8}$ $$ \begin{aligned} & \text{Let } z=x+jy, \quad j^2 = -1 \\ & \frac{e^{z^2}-e^{-z^2}}{2}=\frac{15}{8} \\ & e^{z^2}-\frac{1}{e^{z^2}}=\frac{15}{4} \\ &...
MamaConan 's user avatar
1 vote
0 answers
36 views

Find $k$ such that $F(x;k)$ is elementary.

I’m trying to understand core concepts about the closure of the set of elementary functions. Specifically, i would like to adress the following problem: Find the all the numbers $k\in\mathbb{Z}$ such ...
Simón Flavio Ibañez's user avatar
0 votes
2 answers
52 views

Why does squaring an equation before differentiation give a different result?

Consider $t = \sqrt{x -1}$. Suppose I want to find the velocity $\frac{dx}{dt}$ at $x =1$. I proceed with two different approaches: $1st$ APPROACH: I differentiate both sides with respect to $t$: \...
Atom's user avatar
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0 answers
45 views

What makes logarithmic algebra more special?

among the elementary function sets: Polynomial Trigonometric function Exponential Function Hyperbolic Function I don't see the development of algebraic identities, and then the detailed construction ...
Hisham's user avatar
  • 282
3 votes
2 answers
177 views

Innocent-looking family of inequalities

I recognize that one should give "context" to questions, but sometimes it is hard because you could end up writing much and bring in needless flooding of the post. Just to say: the question ...
T. Amdeberhan's user avatar
4 votes
1 answer
42 views

Amount of compositions until $f(x)=x-\lfloor\frac{x}{n}\rfloor$ becomes constant

Suppose I have positive integers $a$ and $n$. I want to find the number of times $f(x)=x-\lfloor\frac{x}{n}\rfloor$ could be composed on itself with initial argument $a$ until a number less than $n$ ...
Hanson Bai's user avatar
0 votes
1 answer
89 views

Is there a set of functions that span all "nice" functions?

I have been reading as to why there are elementary functions that have non-elementary antiderivatives and I have come to the conclusion that our notion of "elementary" functions is somewhat ...
Brothersquid's user avatar
2 votes
0 answers
260 views

Median of three numbers... but not the typical way

Let's say I have three numbers $a$, $b$, and $c$. To find the median of these three numbers, you order them from least to greatest, then take the second (middle) number. For example, if the numbers ...
Tyrcnex's user avatar
  • 572
4 votes
1 answer
71 views

A problem on compositions of functions

$$f(x) = \begin{cases} x-1 & x \geq 0 \\ 1-x & x <0 \end{cases}$$ $$g(x) = \begin{cases} x& x\geq 0 \\ x^2 & x < 0 \end{cases}$$ It asks me for the compositions $f(g(x))$. What I ...
Martin and Friends's user avatar
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0 answers
63 views

How to calculate the inverse cotangent of a real number in the range of $[0,\pi)$?

Since Fortran does not have an inverse cotangent function, I need to implement one. However, when I tried to use the following formula: ${\rm acot}(x)={\rm sgn}(x) \frac{\pi}{2} - {\rm atan}(x)$ I ...
TobiR's user avatar
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0 votes
0 answers
20 views

Constructing Valuation As a Function of Condition

Suppose there is an item that deteriorates with use, its condition expressed as a number between $0$ (completely broken condition) and $1$ (perfect condition), for which I want to assign a monetary ...
user10478's user avatar
  • 1,872
3 votes
4 answers
117 views

Solving inequality $\frac{7x+12}{x} \geq 3$

I was sitting with my friend in a park, when he thought of and asked for solving this inequality: $$ \frac{7x+12}{x} \geq 3 $$ I was confident enough that I could solve this example as continued: $$ ...
moonbag1212's user avatar
2 votes
2 answers
228 views

What does "$f$ is a function of $x$" mean?

Rigorously speaking, a function $f: X \to Y$ is a mapping that maps each element from set $X$ to an element in set $Y$. (Or more rigorously it can be defined using cartesian product). For $x \in X$, ...
lbwnb123's user avatar

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