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

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3
votes
2answers
33 views

Proving that a function is absolutely monotonic on a given interval

According to the following Wolfram Alpha article: http://mathworld.wolfram.com/AbsolutelyMonotonicFunction.html, the function $f(x) = -\ln(-x)$ is absolutely monotonic on the interval $[-1,0)$. My ...
3
votes
1answer
46 views

Asymptotic elementary expression for the antiderivative of $x^x$

It is well known that there exists no elementary function $f$ with $$\int x^x\,dx \quad = \quad f$$ Is there an elementary function $g$ such that $$\int x^x\,dx \quad \tilde{} \quad g$$ in the ...
0
votes
4answers
61 views

How do I figure out if a 'function is odd or even'?

I am currently doing an highschool math problem and I do not know what the question is asking for when it asks 'state which functions are odd and which are even for the below'. $f(x)= x^2+1 $ ...
0
votes
1answer
24 views

Is there an intersection point for $f_1 = 100n²$ and $f_2 = 2ⁿ$?

As far as i know, there is, and it can be answered by resolving the equation $100n^2 = 2^n$. But how can it be resolved? I already tried to transform the equation using $\log 2$, but i just don't get ...
0
votes
4answers
66 views

How to produce 100 with only four 3s and mathematical symbols

An approach like this doesn't count!        $\underset{k \in \mathbb N, k \leq (3+3/3)} {\sum k^3}$ Thanks for your answers but I need something more strong... You are ...
0
votes
2answers
36 views

Help with parametric quadratic equation

We have the equation $x^2+ax+2a+1=0$ which has real roots $x_1$ and $x_2$ and a is a parameter. I need to answer to the following questions: Find all values of a for which $x_1=(a-1)x_2$. For which ...
1
vote
4answers
37 views

Simplfiying $x(5xy+2x-1)=y(5xy+2y-1)$

I want to simplify: $x(5xy+2x-1)=y(5xy+2y-1)$ to $(x-y)(5xy+something-1)=0$ but I can't figure out what to do with the $2x$ and $2y$ on both sides.
0
votes
2answers
37 views

Inverses / Bijections

Let $f:A\to B$, and $g:B\to A$ such that $$ g(f(a))=a \ ,\ \forall\ a \in A, $$ and $$ f(g(b))=b\ ,\forall\ b \in B. $$ Does this mean that $f,g$ are inverses and bijections? Bests
0
votes
0answers
25 views

Solve this Simple quadratic equation $cU^2-2(a+ b)U+2(a-b)V- cV^2=0$

I need help solving this symmetrical quadratic equation where $\gcd(a,b,c)=1$$cU^2-2(a+ b)U+2(a-b)V- cV^2=0$ Is there an easier method than the quadratic formula? Any hint?
2
votes
2answers
27 views

If $\sin \theta = \frac{3}{5}$, what are the possible values of $\cos\theta$?

If $\sin \theta = \frac{3}{5}$, what are the possible values of $\cos\theta$ ? I do know that .. $\sin^2\theta + \cos^2\theta = 1$ .. and you can solve that equation, resulting in $\cos \theta = ...
2
votes
1answer
26 views

Finding an upper bound for this expression

Given two non-negative numbers $a$ and $b$, I'm trying to found an upper bound for $-\sqrt{a} + \sqrt{a+b}$ But I'd like the bound not to be dependable on any square roots and have only one term. ...
4
votes
5answers
45 views

Tool for expressing $x=f^{-1}(y)$ if $y=f(x)$ is given

I have many equations of nature - $y=ax^{12}+bx^5+5x^4+1$ For all these equations, I need to express x in terms of y. What tool or software would you recommend for this? I would much prefer to ...
3
votes
0answers
33 views

Prove that an equation has no elementary solution

There are methods proving that a polynomial isn't solvable in radical extensions (see Abel–Ruffini theorem) or proving that an integral or a differential equation has no solutions expressible through ...
0
votes
2answers
44 views

Asymptotic approximation of the arctangent?

That is, I am looking for an algebraic function $f(x)$ that approximates $\arctan x$ for large values of $x$. The approximation could be reasonably modest -- perhaps something like $$\tan (f(x)) = ...
1
vote
0answers
38 views

Are there any functions which were proposed as elementary by mathematicians but not considered elementary now?

Are there any functions which were proposed by various mathematicians to be included in the set of elementary functions because of their properties but not considered elementary as of now?
0
votes
0answers
44 views

Monotonically decreasing function for multiplication product?

I have a set of numbers $S = [100,999]$ for which I want the maximum product $p$ such that $p = a \times b$ for all $a,b \in S$ also fulfilling some condition $C$. I would like $p$ to be monotonically ...
2
votes
1answer
496 views

Need to prove that the equation has only 1 solution.

I have been trying to solve the following equation: $5^x+7^x=12^x.$ Obviously, x=1 is a solution but how do I prove that there are no other solutions.
3
votes
2answers
76 views

prove a sequence without knowing its convergence

Let $(X_n)$ be the sequence with $X_1=2$ and $X_n=\sqrt{5X_{n-1} + 6}$ for all $n\ge 2$. How can you prove that it is convergent? IF given convergence, I know that its limit is $6$, but the question ...
1
vote
2answers
34 views

Inequality Expression

what happens when an inequality expression is divided by the same number. Did the sign stay the same each time or change directions? If the sign changed, explain what made the direction of the sign ...
0
votes
0answers
44 views

Autocorrelation of Raised Cosine Function

Let us define the raised cosine function as follows: $f \left( x \right) = \dfrac{\left( 1 + \cos \left( x \right) \right)}{2}$, for $- \pi < x < \pi$. $f \left( x \right) = 0$, elsewhere. I ...
0
votes
2answers
35 views

Evaluate $\lim_{x\to 0}\frac{1}{x^4}\int_{-x}^{0}\sin(t^3)\,dt$

Evaluate $$\lim_{x\to 0}\frac{1}{x^4}\int_{-x}^{0}\sin(t^3)\,dt.$$ I use L'Hopital's Rule and get -1/4. The solution says 1/4. Any ideas?
5
votes
1answer
127 views

Is there a theory of integration in elementary terms for definite integrals?

Let's call a real number explicit if it can be expressed starting from integers by using arithmetic operations, radicals, exponents, logarithms, trigonometric and inverse trigonometric functions. For ...
3
votes
2answers
42 views

Solution of $(1-x)^p= x$ in $(0,1/n)$

Let $n,p$ be positive integers. The equation $$ (1-x)^p = x $$ has a unique solution $x_p$ in the interval $(0,1)$. This follows by the monotonicity properties of $(1-x)^p$ and $x$. My question is: ...
2
votes
1answer
67 views

Injective map from real projective plane to $\Bbb{R}^4$

Consider the mapping $\Bbb R^3\rightarrow\Bbb R^4$ given by $$f(x,y,z)=(x^2-y^2,xy,xz,yz)$$ which passes to the quotient and can therefore be viewed as a map from the projective plane $\Bbb ...
9
votes
11answers
426 views

How to prove $(1-\frac1{36})^{25}\lt\frac12$?

How to prove the inequality? $(1-\frac1{36})^{25}\lt\frac12$ I'm in trouble. Thank you very much for your help
3
votes
7answers
115 views

Find the derivative of $y=x\sqrt{9-x}$

"Find the derivative of $y=x\sqrt{9-x}$." So this is what I have and now I'm stuck. \begin{align} y' &= x \frac{d}{dx}\left[(9-x)^{1/2}\right] + (9-x)^{1/2} \frac{d}{dx}(x)\\ &= x ...
0
votes
0answers
74 views

On a differential equation problem of international mathematical competition for university students

I am trying to solve problem 2 of this competition: http://www.imc-math.org.uk/imc2009/imc2009-day2-solutions.pdf I have other thought but i couldn't fill in the detail. Consider the initial value ...
1
vote
0answers
49 views

Power series solution for a DE with Frobenius method

The given DE is $(x²-3)y"+2xy'=0$ Since there is a singular point ($x=\pm\sqrt{3}$) I used the Frobenius method. I found two indicial relationships: $-3r(r+1)=0$ and $-3(r+1)(r+2)=0$ because I have ...
1
vote
1answer
21 views

Composition of mappings on finite sets

If I'm working in the realm of finite sets on the form $\underline{n} = \{1, \ldots, n\}, n \in \mathbb{N} $. Consider any two transformations $f :\underline{n} \to \underline{m}$ and $g ...
0
votes
1answer
37 views

Find a polynomial with certain conditions.

Suppose that: $$f(x) = 3\frac{x^4+x^3+x^2+1}{x^2+x-2}.$$ Find a polynomial $h(x)$ such that $f(x) + h(x)$ has horizontal asymptote of 0 as $x$ approaches positive infinity.
1
vote
0answers
35 views

Is it possible to find set of functions so that their antiderivatives are also in this set?

Antiderivatives of elementary functions sometimes are not elementary functions. Is it possible to add finite number of non-elementary functions to elementary function set so that their ...
1
vote
4answers
55 views

Equation , powers of two

I want to find the sum of the roots of the equation $$4(4^x + 4^{-x}) - 23(2^x + 2^{-x}) + 40 = 0 $$ in real numbers. I tried the substitution $ 2^x = t $ but then it turns into a quartic equation ...
28
votes
0answers
578 views

Generalization of Liouville's theorem

As proposed in this answer, I wonder if the answer to following question is known. Let $E = E_0$ be the set of elementary functions. For each $i > 0$, inductively define $E_i$ to be the closure ...
37
votes
4answers
3k views

Why can't we define more elementary functions?

$\newcommand{\lax}{\operatorname{lax}}$ Liouville's theorem is well known and it asserts that: The antiderivatives of certain elementary functions cannot themselves be expressed as elementary ...
1
vote
0answers
27 views

Definition of standard functions [duplicate]

In many texts and books about calculus we see There are functions $f$ for which the anti-derivative cannot be expressed in terms of standard functions or there are many integrals that cannot ...
3
votes
3answers
128 views

$4 \sin 72^\circ \sin 36^\circ = \sqrt 5$

How do I establish this and similar values of trigonometric functions? $$ 4 \sin 72^\circ \sin 36^\circ = \sqrt 5 $$
1
vote
0answers
62 views

Logarithm and “basic” functions.

To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm. But how can we prove that ...
1
vote
0answers
49 views

Repository of functions which do not have elementary integrals [duplicate]

If there is some function and I suspect that the primitive function cannot be expressed using elementary functions, I would like to have some argument that there indeed is no such expression. One ...
4
votes
2answers
77 views

value of $a+b$ of the following function

$f(x)=x^3-3x^2+5x\;$ and $\;f(a)=1,f(b)=5.\;$ Find $a+b$. I know only one real root exist for each equation as derivative of the function is always positive .I do not intend to use the formula of ...
1
vote
0answers
13 views

efficiently solve for values of a coefficient in a function, so for those values, the function intersects another function a specific number of times.

This is my summer assignment for my freshman "Intro to Numerical Methods with Matlab: Unit 2" course. The task: "Write an efficient Matlab code, which will take any closed $f(x)$ and $g(x)$ and ...
3
votes
2answers
91 views

Properties of $f(a)-f(b)=a-b$

Let: $f(a)-f(b)=a-b$ and $f(a)>f(b)$ and $f(a),f(b)>0$ and $a>b$ Based on the above facts is it sufficient enough to say that $f(a)=a$ and $f(b)=b$?
2
votes
3answers
72 views

Solving equation involving logarithm

How do I find $x$ in the equation $x - \ln x = 1.9$? Next I have $x - 1.9 = \ln x$ we are learning about logarithm but as I tried to take $\ln$ of both side it leads me to nothing. Could you help ...
0
votes
1answer
42 views

How to define a continuous function $f:I\times I\longrightarrow I$ such that $f(0, t)=t$ and $f(1, t)=1$ for all $t\in I$?

Let $I=[0, 1]$. I need some help to define a continuous function $f:I\times I\longrightarrow I$ such that $$f(0, t)=t\quad \textrm{and}\quad f(1, t)=1$$ for all $t\in I$. The nearest I got was: $$f(s, ...
1
vote
1answer
184 views

What does “versus” mean in the context of a graph?

Say you have a graph of say $y=mx+b$, with $x$ on the horizontal axis and $y$ on the vertical axis. You need to give the graph a title, would you say: This is a graph of "$y$ versus $x$?" or This ...
1
vote
1answer
39 views

Inverse of Higher logarithms

Th polylogarithm function is defined by $$Li_s(z)=\sum_{k=1}^\infty\frac{z^k}{k^s}.$$ At $s=1$, we have the natural logarithm function. We have the inverse of natural logarithm function as the ...
0
votes
1answer
23 views

How to work out the probability of 2 people having a different birthday

I'm trying to reproduce the 'birthday problem' where you work out the probabilities of n people having the same / different birthday. Theres a good example here: ...
0
votes
1answer
62 views

How to graph a sin & cos wave

I've got a time column of numbers(x) from 0 - 20 in 0.1 intervals and i've got a sin(x) and cos(x) column. I've got to produce some kind of wave like graph from it. Although I'm not sure which numbers ...
0
votes
1answer
56 views

Inverse of $a f(x)$ and inverse of $a f(x) + b$

Is there a general rule for the inverse of the function $ g(x) = a f(x) $, where $a$ is a constant, assuming $f^{-1}(x)$ is defined? Follow up: $g(x) = a f(x) + b$. Is the following correct, given ...
0
votes
2answers
32 views

Distributive property

I have the following question on distributive property. If I multiply $3$ to the given expression $\frac{1}{3}\pi r^2h$. Question: $3\cdot\frac{1}{3}\pi r^2h$ Based on what I understand for ...
8
votes
0answers
57 views

Is there a name for the class of functions which are infinitely integrable in elementary functions?

Is there a name for the class of functions which are infinitely integrable in elementary functions, that is whose consecutive integrals also elementary not depending on how much times we took the ...