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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3answers
49 views
Mathematical Systems Question Help
Alright, this is another question for my math for teachers course. This question is not actually in the homework, but there are problems similar to it. I'd really like to learn how to do problems like ...
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2answers
124 views
Given $L(y) = \int_ {1}^{y}\frac{1}{t}dy$ , $y\in(0, \infty)$
I am proving Theorem 37.4 , Elementary Analysis, Kenneth Ross.
Stating $$L(y) = \int_{1}^{y}\frac{1}{t}\operatorname{d}t,\quad y\in(0,\infty)$$
I have to prove that
(i) the function L is strictly ...
1
vote
1answer
49 views
A problem with an elementary function
Prove or disprove:
For every $n\in\mathbb N$ there exist $x_n\in (10^n,+\infty)$ such that $f(x_n)<10^{-n}$ where $f(x)=(x \sin x - \cos^2 x)^2 x^6 + x^2 \cos^8 x.$
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votes
1answer
38 views
When is (a restriction of) the map $y = f(x) = x + \frac{n}{x}$ bijective, if $x, y \in \mathbb{Q}$ and $n \in {\mathbb{Z}}^{+} \cup \{0\}$?
A good day to everyone.
If $0 < x \in \mathbb{Q}$, $0 < y \in \mathbb{Q}$, $n \in {\mathbb{Z}}^{+} \cup \{0\}$, and $n$ is squarefree, then the function
$$y = f(x) = x + \frac{n}{x}$$
is not ...
2
votes
2answers
93 views
Approximating the error function erf by analytical functions
The Error function
$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt$
shows up in many contexts, but can't be represented using elementary functions.
I compared it with another function ...
62
votes
6answers
2k views
Find a real function, $f$, s.t. $f(f(x)) = -x$.
I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success:
Find a function $f: ...
0
votes
0answers
131 views
Linear Fractional Transformation
We are given $f(x)=(ax+b)/(cx+d)$.
We let $f$ be a linear fractional transformation. I need to show that $f$ is one-to-one, and find $f^{-1}(x)$ and also the domains of both $f$ and $f^{-1}$.
Any ...
1
vote
0answers
25 views
Primitive of an irrational function
Does anybody know how to determine a primitive of the function
$f(u)=\frac{u^2}{(au^4+bu^2+c)^{3/2}}$?
Here $a,b,c$ are positive real numbers and one assume that $u>0$.
Thank you in advance.
4
votes
0answers
39 views
Is the inverse of any elementary function asymptotic to some elementary function?
Is the functional inverse of any elementary function asymptotic to some elementary function ?
For instance Lambert's $W(z)$ is asymptotic to $ln(z)$. See ...
0
votes
0answers
185 views
Supremum or infimum of a function
Does there exist an elementary function on some subset $I$ of $R$ such that you can prove that $A = \{\sup(f(x)): x \in I\}$ exists, but you cannot find the value of $A$ ?
If the answer is "no", then ...
0
votes
0answers
37 views
Question about elementary and nonelementary functions.
Let $E(z)$ be an entire elementary function of (complex) $z$ and $N(z)$ be an entire nonelementary function of (complex) $z$.
$e^{N(z)}$$N'(z) = E(z)$
The ' means derivative with respect to $z$.
...
3
votes
0answers
132 views
Does the function $\frac{e^x-1}{x}$ have a conventional name?
I've seen the function $f(x)=\displaystyle \frac{e^x-1}x$ many places, most notably the definition of the Todd class. Is there a consensus on what it's name is? I couldn't find one on Wikipedia or ...
33
votes
8answers
1k views
What makes elementary functions elementary?
Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, ...
