# 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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### Elementary function producing primes

I know there is no known elementary function $f:\mathbb{R}\to\mathbb{R}$ such that $f(n)$ is the $n$-th prime number, but is there a proof for the non-existance of one? How about one which only takes ...
1answer
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### Which kinds of rational functions of one variable have an inverse relation that contains a branch that is a rational function?

Let's consider the rational functions whose numerator and denominator of the function term are coprime. Which kinds of rational functions of one variable have an inverse relation that contains a ...
2answers
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### Proof of the derivative of an elementary function is also elementary

I heard of the result that the derivative of an elementary function is also elementary long ago. Now I want to prove it rigorously. I found this answer(I didn't comment because it was an old post):...
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### How can I prove that $\frac{a}{2} + \frac{x}{a} \geq \sqrt{2x}$ [closed]

How can I prove the inequality $$\frac{a}{2} + \frac{x}{a} \geq \sqrt{2x}$$ for positive $a$ and $x$. It seems simple, but I don't see a way to prove it.
2answers
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### Finding the Degree of a Polynomial, as a function of the Polynomial

Recently, I was wondering about the following question: Given a polynomial $P(x)$ with real coefficients, express its degree $d$ as a function of only $P(x)$, i.e. $d(P(x))$. Only elementary ...
2answers
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### Can a sum of an elementary and non-elementary definite integral be elementary or evaluated without special functions?

Hi I'm asking this in context of this: Suppose we have non-elementary integral $\chi(x) = \int K(x)dx$ and we want to find $$\int^b_aK(x)dx$$ Now lets say there is some function $F(x)$ which has ...
2answers
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### Does a function from the empty set to the empty set exist?

I know that every function from the empty set to any other set is the empty function. I also know that there is no function to the empty set from any other set. Now, what if both the domain and ...
3answers
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### Polynomial Long Division:

My question essentially boils down to this: (it was part of a question about polynomial ring ideals) Find an integer $b$ such that the rational function \begin{equation} \frac{x^5-bx}{x^2-2x}\in \...
1answer
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### Elementary function $f:\mathbb{R}\to \mathbb{R}$ so that $f(0)=0, f(n)=2^n$ for $n\in \mathbb{N}$

Is there any elementary function $f:\mathbb{R}\to \mathbb{R}$ such that $f(0)=0$ and $f(n)=2^n$ for every positive interger $n$? By elementary functions, I mean functions that are sum, product, ...
1answer
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### Solve for $x$: $a_1\ln(1+b_1x)+a_2\ln(1+b_2x)=0$, maybe with non-elementary function?

Given: $a_i>0$, $b_1>0$, $b_2<0$ Solve for: $a_1\ln(1+b_1x)+a_2\ln(1+b_2x)=0$ My try: The equation is equivalent to $(1+b_1x)^{a_1}=(1+b_2x)^{-a_2}$, Which seems to be solved with only non-...
1answer
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### Prove that there's a unique integer n where : $2n^2 − 3n − 2 = 0$.

I have to prove that there's a unique integer n where : $2n^2 − 3n − 2 = 0$. I factorised and got $(2n+1)(n-2) = 0$ I can then find both $n$ solutions : $-1/2$ and $2$. We can therefore see there's ...
1answer
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### Solving exponential equation with negative base

I have to solve the equation: $(x-2)^{x^2-6x+1}\leq 1$. If $x-2>1$ i.e. $x> 3$ we have $3< x \leq 3+2\sqrt{2}$. If $0< x-2<1$ i.e. $2< x<3$ we get no solutions. If $x -2=1$ ...
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### Is the relation $fRg$ iff $\exists k > 0 \mid f(x) + k < g(x + k)$ on real functions transitive?

Is the relation on the set of real functions $fRg$ iff exists $k > 0$ such that for all $x \in \mathbb{R}$, $f(x) + k < g(x + k)$ transitive? I have proved that it's not reflexive, not symmetric,...
0answers
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### When is the continuous extension of an elementary function elementary?

Some continuous extensions of elementary functions are elementary. For example $f(x) = \frac{x}{x}$ is elementary in $\mathbb{R}\setminus\{0\}$ and its continuous extension to $\mathbb{R}$ is $g(x)=1$ ...
1answer
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### System of modular equation

Given the following graph I gotta find the values of $r$ and $s$, with $r,s\in\mathbb{Z}$ such that they satisfy $$f(x)=|x-r|+|x-s|$$ I know the solutions are $2$ and $4$ but I don't know how to ...
1answer
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### What is the currently accepted “correct” definition of a “transcendental function”?

Caveat: this question has already been asked on this site more than once, but the question has not been addressed completely. The question I want to ask is: there are two common definitions of a "...
1answer
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### Show that $2^{ab}+1=(2^{a}+1)(2^{ab-b}-2^{ab-2b}+2^{ab-3b}+…+1)$ where b is an odd number and( a, b) are natural numbers

Show that : $$2^{ab}+1=(2^{a}+1)(2^{ab-b}-2^{ab-2b}+2^{ab-3b}+...+1)$$ where b is an odd number and( a, b) are natrual numbers. So far I’ve tried to use a similar identity as that used to prove the ...
0answers
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### Why are the elementary functions the ones that they are?

Many antiderivatives exist but cannot be expressed in terms of elementary functions. This got me thinking, well, what if we expanded the set of elementary functions? Would it work then? Let's think of ...
1answer
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### Conditions for the inverse of a bijection to be “almost explicit”

The following map $$f:\left[-1,+\infty\right[\to\left[-\frac1e,+\infty\right[,\,x\mapsto x\,e^x$$is a bijection and it can be shown that its inverse (known as the Lambert function W) is representable ...
1answer
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### Cartesian product — understanding the definition

I just started with Schaum's Outline of General Topology by Seymour Lipschutz a few days ago, and now I am working with chapter 2 about functions. Most of the material I have already seen in other ...
1answer
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### How does one go about solving a linear maximization problem using elementary maths?

I was checking a list of elementary math problems, and one of them was of the kind "a factory produces products $A$ and $B$, the profit of each product is $P_A$ and $P_B$ (The values were given ...
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### How do I understand if an indefinite integral is not solvable in terms of elementary functions? [duplicate]

Many indefinite integrals cant be solved in terms of elementary functions.e.g.-$$\int \frac{\sin(x)}{x}dx$$ But many hard looking ones are still solvable by weird substitutions and other tricks. Is ...
2answers
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### Is it possible to express values such as $\sin^{-1}(\pi/12)$ without inverse trig functions?

Pretty self explanatory. If I had for example something like $\sin^{-1}(\pi/12)$ in an expression, is it ever possible to express that expression without inverse trig functions?
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### primitive roots of unity to solve equations over $\mathbb Z_p$

So, I have been thinking of the equation $x^n-1 \equiv 0 \in \mathbb{Z}_p$, $p$ prime. So, I noticed something weird and I wonder if there is a theory for that. Let $\omega_n$ be the nth primitive ...
4answers
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### Are there any elementary functions $\beta(x)$ that follows this integral $\int_{y-1}^{y} \beta(x) dx =\cos(y)$

Are there any simple functions $\beta(x)$ that follows this integral $$\int_{y-1}^{y} \beta(x) dx =\cos(y)$$ I think there is an infinite amount of solutions that are continuous everywhere but how can ...
1answer
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### is there a function $\gamma(x)$ where when $a$ & $b$ and $a+1$ & $b+1$ are co-prime, $\gamma(\frac{a}{b})>\gamma(\frac{a+1}{b+1})$

is there a function $\gamma(x)$ where when $a$ & $b$ and $a+1$ & $b+1$ are co-prime, $\gamma(\frac{a}{b})>\gamma(\frac{a+1}{b+1})$ when you start with $\gamma(\frac{1}{2})$ you get an ...
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### Can you write $1-x-\frac{x^2}{2!}+\frac{x^3}{3!}-\dots$ with elementary functions

Can you write $$1-x-\frac{x^2}{2!}+\frac{x^3}{3!}-\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}-\frac{x^7}{7!}-\dots$$ with elementary functions, where the function is related to the Thue-Morse ...
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### A conjecture about the solvability of rational equations of transcendental functions by elementary numbers

Is my conjecture below mathematically and linguistically correctly formulated and well formulated? How can the conjecture be improved and shortened/simplified and made more intelligible? The ...
1answer
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### Why are $\sin,\cos,\tan$ continuous

I'm done with two courses in Analysis, but just can't seem to work out how I'll show the base trigonometric functions to be continuous. Any references or indications for a simple, preferably ...
1answer
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### On an equation that involves the number-of-divisors function and the formula for a sequence of figurate numbers

In this post we try to relate a sequence from The On-Line Encyclopedia of Integer Sequences and a sequence that solves an equation that involves the number-of-divisors function \$\sigma_0(n)=\sum_{1\...
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### How to shift and compress a parameterized log curve

Suppose I have some nice way to create a log curve that I need for a certain task (here in Python). ...