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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Find the minimum of $f(x)=x^2-x+1+\sqrt{2x^4-18x^2+12x+68}$.

WA gives the result $9$. But how to solve it by applying inequalites?
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1 answer
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Finding functions $f, g, h : \mathbb R_{> 0} \to \mathbb R_{> 0} $

Undergraduates at my university showed me this problem, which I found intriguing and now want to see the solution of: Find all functions $f, g, h : \mathbb R_{> 0} \to \mathbb R_{> 0} $ such ...
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1 vote
1 answer
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Elementary function that $f(0)=1$ and $f(x)=0$ for all $x>0$.

Here is a short question: finding a function that 1) $f(0)=1$ and 2) $f(x)=0$ for all $x>0$. I have two functions in mind: Something like Dirac delta: $\delta_0(x)$. Something like $f(x)=1-\text{...
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$\frac{f(cx)}{f(x)}=\frac{f(cy)}{f(y)}$, solve for $f$ [duplicate]

For all constant $c>0$ and all $x,y\in\mathbb R_+$, $$\frac{f(cx)}{f(x)}=\frac{f(cy)}{f(y)}.$$ $f$ is continuous. Is it correct that $f(x)=ax^b$ ? I cannot find any other elementary functions as ...
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3 votes
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What is the right exponent $x=x(p, q)$ such that both $a^p\le (a+b)^x$ and $b^q\le (a+b)^x$?

Let $a, b\in\mathbb{R}^*_+$ and $p, q\in\mathbb{R}, p, q>1$. I would like to find the right exponent $x$ (clearly $x$ depending on $p$ and $q$), such that $$ a^p\le (a+b)^x \quad\mbox{ and }\quad b^...
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Is Sqrt(x) not well defined? [duplicate]

I recently came across the notion of a function being "well defined" in some exercises in an abstract algebra book I'm reading. What I could gather about the definition (of which there seems ...
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Are the functions here well defined?

Suppose that $X$, $Y$ and $Z$ are finite vector spaces of dimension $k$, $l$ and $m$ such that $x\in X$ , $y\in Y$ and $g(x,y)\in Z$. I define a function $\pi:X\times Y\to X\times Y \times Z$ such ...
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60 views

Are there any bounded elementary functions which are not Riemann integrable?

Let an elementary function be a function which is made up of sums, products, and compositions, of finitely many polynomial, rational, trigonometric, and exponential functions, and of their inverses. I'...
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7 votes
4 answers
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Examples of non-elementary integrals, but whose definite integral IS solvable with power series.

In a high-school level calculus course you learn about Taylor series and some basic integration techniques. In my experience, most definite-integration exercises boiled down to finding an ...
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6 votes
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When is it necessary to define a new function?

For example: Lambert $W$ is a non-elementary function that can be defined as a solution for $x$ to $x\cdot e^x$, but $\int{\frac{1}{\ln(t)}dt}$ is also supposed to be nonelementary. How do we know ...
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4 votes
3 answers
417 views

Can you solve any mathematical function?

For any finite mathemathical function (consisting of addition, subtraction, division, multiplication, exponentiation, trigonometry) can you find $x$ in $f(x) = y$ where $y$ is a number you want? Is it ...
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1 vote
0 answers
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inverse of AM-GM inequality for converting sums to products [closed]

I have an inequality in the form $\alpha + \sqrt{\beta}$ where both $\alpha,\beta \in [0,1]$ I want to find an upper bound for this quantity in terms of the product. Is there any inequality in the ...
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1 vote
1 answer
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$-\ln{x}<\frac{1}{x}$ when $0<x<1$

I am trying to prove this inequality as described in the title: $0<-\ln{x}<\frac{1}{x}$ when $0<x<1$ Graphing it will tell you this inequality is true. Here are my attempts and I could not ...
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2 answers
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Is there a way to find the antiderivative of $x^k \cos(mx)$? [closed]

Is there a way to find the antiderivative of $x^k \cos(mx)$? $k$ and $m$ are just different constants, of course, which may or may not be integers. Integration by parts doesn't lead me anywhere as it ...
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0 votes
1 answer
57 views

Would this be an even or odd function?

Given that $f(x)$ is odd, I need to find if $f(\sec x)$ is odd. We (or I, at least) have always defined an odd function in the following way: $f(x)$ is odd if $f(x)=-f(-x)$, for all $x$ in the domain ...
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1 vote
0 answers
54 views

Why is riemann zeta function non-elementary?

Why are functions like ln(x), sin(x), cos(x) etc. considered elementary functions but something like the Riemann Zeta function isn't? All of them can be defined as an infinite series. What makes a ...
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1 vote
1 answer
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What does this theorem of continuity of elementary function means?

I saw in my lecture not about this theorem of elementary function continuity:” All elementary functions are C1 in the interior of their maximal domain except the norm when the argument is zero and the ...
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2 votes
1 answer
183 views

Proof Check: Non-existence of the inverse function in a given class of functions

Are my conjecture and proof below mathematically and linguistically correct? Are they well formulated? How can they be improved? How can they be shortened? Is the conjecture obviously? As far as I ...
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4 votes
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Explicit elementary primitive (integral) of $x / \sqrt{P(x)}$ and Galois group of $P$

I am reading some stuff on Risch's algorithm here (Wikipedia in French), about finding explicitly some primitive of functions in terms of "elementary functions" (composition of polynomials, ...
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5 votes
2 answers
114 views

Most simple way to prove $2^n \geq A n^k$?

What is the most simple and less demanding way to prove that exponentials grow faster than powers? So far I have found a proof that relies on Berloulli's inequality(*) which is elegant but doesn't ...
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0 votes
0 answers
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Is there a systematic way to find commonalities between two sets of whole numbers as long as the sets can be defined as functions?

Today I found myself pondering two series of numbers: One series is the result of $2^x$ (2, 4, 8, 16, etc.) and the other being the result of $3^x$ (3, 9, 27, 81 etc.). What I initially wondered was: ...
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1 answer
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How to find such a complex function?

Let $D$ be the first open quadrant ($x,y>0$). Find an elementary function that maps $D$ onto the interior of the complement $int(\mathbb{C}-D)$. Edit: I thought of using De-Moivre's formula, $(\...
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0 votes
1 answer
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Inverse of a log and linear function

How would you find the inverse of a function that is both linear and logarithmic? Take this for example: $f\left(x\right)=ln\left(x\right)+x+1$ Writing it as $y=ln\left(x\right)+x+1$ won't work, at ...
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1 vote
0 answers
56 views

Do the maps $g$ and $g’$ define the same mathematical object?

Let $A$ and $B$ be arbitrary sets with arbitrary elements $a\in A$ and $b\in B$, and powersets $\mathcal{P}(A)$ and $\mathcal{P}(B)$. Further, let $f:A\to\mathcal{P}(B)$ be a map; and for any set $f(a)...
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0 votes
2 answers
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How to write the Cartesian product of all elements of an uncountable set?

Let $A$ be an uncountable set with arbitrary element $a\in A$. I want to define the Cartesian product of all its elements. In other words, all the vectors belonging to \begin{gather} \underbrace{A\...
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2 votes
1 answer
23 views

How to write the realisation of a function whose domain is a function?

Let $A$ and $B$ be arbitrary sets with arbitrary elements $a\in A$ and $b\in B$ and power sets $\mathcal{P}(A)$ and $\mathcal{P}(B)$. Further, let $f:A\to\mathcal{P}(B)$ be a function. I want to ...
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4 votes
2 answers
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Domain of composite function $\left( f \circ g \right)\left( x \right).$

Question: Given that $f\left( x \right) = \sqrt{x - 3}$ and $g\left( x \right) = x + 1$, find the domain of $\left( f \circ g \right)\left( x \right)$. My attempt: $\left( f \circ g \right)\left( x \...
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0 votes
2 answers
80 views

Can we solve $\;-(\beta+1) + \lambda \beta x^{-\beta} (1 + e^{-\lambda x^{-\beta}} \ln \alpha) = 0\;$ analytically?

I want to solve this equation for $x$, but I'm stuck. Is it possible to solve this equation analytically? $$-(\beta+1) + \lambda \beta x^{-\beta} (1 + e^{-\lambda x^{-\beta}} \ln \alpha) = 0$$ where $\...
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0 answers
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Is the set of functions required to define any integral finite?

Let $F$ denote the set of elementary functions, as well as as functions defined as the integral of a non-elementary integral, for example $Li(x)$. Is it possible to have a finite $F$ such that any ...
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2 votes
1 answer
109 views

About the inverse function of $\sin(x)$

The function $f(x)=\sin(x)$ is monotone in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ thus it is invertible and its inverse function ${\rm arcsin}(x)$ is defined from $[-1, 1]$ to $\left[-\frac{\pi}{...
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0 votes
2 answers
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Directly proof $S$ is countable, where $S$ is set of function from $\{0, 1\}$ to $\mathbb{N}$

Suppose $S=\{f_1,f_2,f_3,f_4,f_5,........\}$ where $f_i$ is a function $f:\{0, 1\}\to\mathbb{N}.$ I have to prove $S$ is countable.Then need to prove direct one-to-one correspondence between $S$ and $\...
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3 votes
1 answer
71 views

When is $\int\frac{dx}{\sqrt{a-bx^n-x^2}}$ solvable in terms of elementary functions and why?

The integral in the question appears in the solution to the orbit of a particle subjected to a central force. It is written in Goldstein's Classical mechanics that the solution is possible in terms of ...
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0 votes
0 answers
32 views

What is special about a function being elementary? [duplicate]

Functions such as $ \sin(x) $ are considered to be elementary, however functions like $ \text{erf}(x) $ are considered to be non-elementary. What makes elementary functions different from non-...
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1 vote
4 answers
139 views

Solve $e^x = 2x$ with algebra

I’m trying to solve $$e^x = 2x.$$ I have already understood that there isn’t any solution when I plot both functions, but I would know how to get this conclusion by trying to solve it algebrically. I ...
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4 votes
1 answer
93 views

Evaluating $\int e^{-\sqrt{a^2 - b^2} \cosh(x)} \, dx$

I've been trying to solve the definite integral \begin{align*} I = \int_{0}^{u} e^{-\sqrt{a^2 - b^2} \cosh(x)} \, dx \, , \end{align*} with $u = \mathrm{arctanh}(\frac{b}{a})$ and $a > b, \, a &...
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9 votes
2 answers
180 views

If $f(x)=\frac{-8(1-\sqrt{1-x})^3}{x^2}$, then $f(f(x))=x$. Slick proof?

In the paper A tribute to Dennis Stanton Richard Askey gives the following problem If $f(x)=\frac{-8(1-\sqrt{1-x})^3}{x^2}$, then find $f(f(x))$. I managed to solve it, but spend several hours and ...
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4 votes
3 answers
106 views

A 'non-numerical\analytic' proof that $\binom{n}{k}$ > $\binom{n}{k-1}$ for large $n \in \mathbb{N}$

The number of $k$-subsets of $[n]$ is given by the formula $\binom{n}{k}$ or $^nC_k$. They famously occur in the expansion of $(1+x)^n$ and they are given by the formula $$\binom{n}{k}=\frac{n!}{(n-k)!...
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3 votes
0 answers
63 views

Can you define and elementary $f(x)$ such that $2^{x}<f(f(f(x)))<2^{2^x}$?

Can you define a elementary real-valued function $f$ such that $2^{x} < f(f(f(x))) < 2^{2^x}$ for sufficiently large $x\in \mathbb{R}$? I know that there is no elementary function $f$ such that. ...
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0 votes
1 answer
44 views

Does the function $G(u)=1+2u^2$ satisfy this property involving the supremum?

Let $G:(x, u)\in\mathbb{R}^N\times\mathbb{R}\mapsto G(x, u)\in\mathbb{R}$ be a function such that for any $r>0$, it is $$(S)\qquad\qquad\sup_{|u|\le r} |G(\cdot, u)|\in L^{\infty}(\mathbb{R}^N).$$ ...
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3 votes
0 answers
171 views

When does $\sqrt{f(x)}\exp{g(x)}$ have an elementary antiderivative?

Liouville's original criterion for elementary anti-derivatives states: If $f,g$ are rational, nonconstant functions, then the antiderivative of $f(x)\exp{g(x)}$ can be expressed in terms of ...
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6 votes
1 answer
562 views

How do we know if a function has an elementary inverse?

There are certain elementary functions where the inverse (or the branches of the inverse in a non-injective function, or the inverse over its range for a non-surjective function) is non-elementary. ...
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2 votes
2 answers
153 views

How to find a point on the perimeter of a square?

Given the four corners $(x, y)$ of a square, the center $(x, y)$, and a starting direction (ex: $45^\circ$) around a $360^\circ$ rotation ($0^\circ/360^\circ$ at the top), how do you find out the ...
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0 votes
1 answer
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Inequality for convex function / Jensen's inequality

We know that for a concave function $f\left(\frac{x_1+x_2}{2}\right) \geq \frac{f(x_1)+f(x_2)}2$. Is it true that $ f\left(\frac{\sum x_i}n\right) \geq \frac{\sum f(x_i)}n$ for finite $x_1, ... x_n$? ...
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  • 370
0 votes
1 answer
55 views

Proving this polynomial is irreducible

Let $a_1, a_2,..., a_{2n} \geq 1$ be $2n$ distinct positive integers such that at least two of them are even. Show that the polynomial $$(X^2-a_1)(X^2-a_2)...(X^2-a_{2n})-1$$ is irreducible over $\...
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1 vote
2 answers
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Why can't $a=\theta-\sin(\theta)$ be solved for $\theta$ in terms of $a$ in closed form?

Context: I had taken an interest in alchemical symbols. Many of the ancient drawings are understandably crude, given the tools available at the time. In spite of their rough appearance, I imagined ...
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0 votes
2 answers
152 views

Is there a method for solving equations like $\sin x=x\cos(x)$ in closed form?

Is there a method for solving equations like $\sin x=x\cos(x)$ in closed form? I was looking into involute curves and ran into two equations that I'd like to find closed form solutions to: $\sin(t)=t\...
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0 votes
0 answers
19 views

For a=b^c, what is the name and properties of the function in which b is depdendent and c varies continuously while a stays constant.

For a=b^c, the named functions are: Exponential in which the dependent variable is "a" while c varies and b is constant (and its inverse the log). Power in which the dependent variable is &...
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1 vote
1 answer
69 views

Solutions of $ 5^x+5^{x^2}=4^x+6^{x^2} \quad \left(x\in \mathbb{R}\right)$

For equation $$ 5^x+5^{x^2}=4^x+6^{x^2} \quad \left(x\in \mathbb{R}\right)$$ is there any nontrivial solution? Easy to find $x=0,1$ are the trivial solutions, and also, easy to figure out that $ x>...
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0 votes
1 answer
37 views

Besides circles, are there other curves for which $\vec{r}'(t)$ and $\vec{r}''(t)$ are always perpendicular, described by only elementary functions?

Provided that $\vec{r}_1$ and $\vec{r}_2$ are a quarter turn away from each other, a circle can be described using the equation $\vec{r}(t)=\left(\vec{r}_1-\vec{r}_c\right)\cos(t)+\left(\vec{r}_2-\vec{...
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2 votes
1 answer
97 views

Is it decidable whether the value of a given definite integral has a closed-form expression?

There are many elementary functions such as $e^{-x^2}$ which don't have an elementary antiderivative, but a definite integral of the same integrand has a closed-form value, e.g. $$\int_{-\infty}^\...
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