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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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4
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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:
$$
...
2
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3
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What does "$f$ is a function of $x$" mean?
Rigorously speaking, a function $f: X \to Y$ is a mapping that maps each element from set $X$ to an element in set $Y$. (Or more rigorously it can be defined using cartesian product).
For $x \in X$, ...
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Constructing a function $f$ with $f (0) = 0$, $\lim_{x \to \infty}f(x) = 0$, $f^{'} (0) = 0$, and $\lim_{x \to \infty}f^{'}(x) = 0$.
I am trying to define a function $f:\mathbb{R} \to \mathbb{R}$ that has the appearance of a bell curve, but has the following properties:
\begin{align*}
f (0) = 0, \\
\lim_{x \to \infty}f(x) = 0, \\
f^...
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How many intersections of two functions with the same monotonicity?
For example, consider two functions monotonically increasing (when they have at least one intersection) in a particular interval，how many intersections do they have? Which factor may influence the ...
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Finding constants a and b, given a function and its inverse
Given a function and its inverse, where a and b are constants, find the constants a and b.
$$h(x) = x + a $$
$$h^{-1}(x) = b(2x + 3)$$
I tried simultaneous equation(not quite sure is it done like ...
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Given an arbitrary composite elementary function, does there exist some method to give us an intuition of its graph?
Consider this example from the wikipedia page on elementary functions:
$$ \ \ \frac{e^{\tan x}}{1+x^2}\sin(\sqrt{1+(\ln x)^2})$$
Although one might know how any single simple elementary function that ...
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Calculating $\frac{1.01^5}{1.01^5-1}$ without calculator with good accuracy?
Question (may SKIP reading this):
A computer is sold either for $19200$ cash or for $4800$ cash down payment together with five equal monthly installments. If the rate of interest charged is $12\%$ ...
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What are the closed-form inverses of $x+\sinh(x)$, $x+\cosh(x)$?
What are the closed-form inverses of the injective pieces of $x+\sinh(x)$, $x+\cosh(x)$?
I assume these functions don't have inverses that are elementary functions.
Can the inverses be represented ...
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Very trivial doubt on fractional exponents.
Consider the following:
$$x^{2/3}$$
I am having some really trivial doubts, but I would like to clarify them all, once for all. My questions are:
Is $x^{2/3}$ the same as $\sqrt[3]{x^2}$? And if &...
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What are the closed-form inverses of $x \sinh(x), x \cosh(x), x \tanh(x), x\ \text{sech}(x), x \coth(x), x\ \text{csch}(x)$?
What are the closed-form inverses of the injective pieces of $x\sinh(x)$, $x\cosh(x)$, $x\tanh(x)$, $x\ \text{sech}(x)$, $x\coth(x)$, $x\ \text{csch}(x)$?
I assume these functions don't have inverses ...
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Equivalence between $f(x)e^g(x)$ being integrable and $r'(x) +g'(x)r(x)=f(x)$ having a function $r(x)$ that solves it.
I was wondering why $\int{e^{x^2}dx}$ was not integrable using elementary functions and looking for a proof. I found this video by Michael Penn which explains that these two statements are equivalent:
...
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How can we show that $A(z,e^z)$ and $A(\ln (z),z)$ have no elementary inverse?
I'm generally interested in equation solving in closed form.
The algebraic functions are meant with complex coefficients.
Let
$A$ an algebraic function of two complex variables,
$F$ with $F(z)=A(z,e^z)...
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Center of mass of a circular arc of radius R subtending an angle at the center
This is not a question, but just a theory post.
Here we would like to calculate the position of center of mass of the circular arc subtending an angle φ at the center of the circle which it is a ...
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But is it really true that $\log(xy) = \log(x)+\log(y)$
We probably all know that
$$\log (x y)=\log x + \log y$$
However the expression on the left needs only $x y >0$ to be defined, whereas the expression on the right requires $x>0$ and $y>0$.
...
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Normal forms for elementary functions
Are there any normal forms for elementary functions?
(Any square matrix can be written as a conjugate of Jordan one. Any polynomial can be expressed as a linear combination of monomials...)
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When exactly must two elementary functions be identical if they are sufficiently close?
$$\sin(x)\cos(x) = \frac{\sin(2x)}{2}$$
Plotting the L.H.S. and the R.H.S. of this identity shows that these functions “seem” to be identical, but it is not a proof.
Or is it?
Is there a mathematical ...
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Why are "Most" integrals of elementary functions themselves not elementary functions? [duplicate]
I've been reading through Stewart's calculus, and it mentions how most elementary functions don't have elementary antiderivatives. I don't understand why the derivative of an elementary function is an ...
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If an integral on a Riemann surface is an elementary function is the genus zero?
I was hoping to find a simple proof that an integral on a Riemann surface is elementary if and only if the genus, $g = 0$.
For example, if I take a genus $1$ algebraic curve (Riemann surface) such as
\...
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Abel's theorem on integration in terms of elementary functions
I read here (page 15) that Abel proved the following two theorems:
Theorem 1: (Abel (1829), Liouville (1833)) If the abelian integral $\int y dx$ is an elementary function then it must have the form
\...
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Classification of integrals not expressible in elementary functions?
Let's consider the smallest set of functions (with real coefficients) closed under addition, multiplication, division, root extraction, and finite compositions containing
constants
exponential and ...
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Domain of the $f(x) = \sqrt{x}$
Domain of the $f(x) = \sqrt{x}$
So typically, when introducing functions to students, teachers will say that the domain of this function is $[0,\infty)$.
However, in the curriculum I'm following, one ...
user637978
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Elementary closed form of $\exp\left(b\int_0^x \frac{t^{a-1}}{1-t^b}dt\right);a,b\in\Bbb N$
Our goal is an elementary expression of $$\exp\left(b\int_0^x \frac{t^{a-1}}{1-t^b}dt\right)=e^{\text B_{x^b}\left(\frac ab,0\right)}=e^{x^a\Phi\left(x^b,1,\frac ab\right)};a,b\in\Bbb N,0\le x\le1
\...
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Prove that $x^3 - 3x + c$ has at most one root in $[0,1]$, no matter what $c$ may be
Prove that $x^3 - 3x + c$ has at most one root in $[0,1]$, no matter what $c$ may be.
$f(x)$ is a decreasing function in the interval $[0,1]$ is evident by substituting the values of $0$ and $1$.
$f(...
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$T_n=4T_{n-1}\Rightarrow T_n=4^{n-1}T_1$ but how?
It's been a lot time but I have got no response, so asking separately though I know it seems a noob thing to ask but what to do, I'm unable to wrap my head around it :/ .
In this question both the ...
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Exact solution for an exponential equation involving powers
A student asked me this exercise of a past exam:
Find the smallest solution to the equation: $$\exp(\sqrt{2x})=16x^3-1.$$
Give the anwser up to $4$ decimal places.
Is there a way to find an exact ...
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Why we cannot solve $y^{2}+\sin y =2 x^{3}+C$ for y?
This is from Stewart - Calculus - Early Transcedentals
Writing the equation in differential form and integrating
both sides, we have $$ \begin{aligned} (2 y+\cos y) d y &=6 x^{2} d x
\\ \int(2 y+\...
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Does there exist a finite set of solutions to integrals such that any function composed of elementary functions is integrable?
For indefinite integrals whose solutions cannot express with elementary functions, special functions are often defined, such as those shown below.
$$ \mathrm{Si}(x) = \int_0^x\!\frac{\sin t}{t}\,\...
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Operation on functions, which of the following is true?
If $f(x) = 2x + 1$ and $g(x) = x + 3$, then which of the following is true?
(A) $(f + g)x = f(x) + g(x)$
(B) $(f \circ g)(x) = (g \circ f)(x)$
(C) $(fg)(x) = g[f(x)]$
(D) $(fg)(x) = f[g(x)]$
I came ...
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Is there any example of a real-analytic approach to evaluate a definite integral (with an elementary integrand) whose value involves Lambert W?
I have never seen a real-analytic approach before to evaluate integrals of the form below
$$\int_a^b\text{elementary function}(x)\,dx=\text{constant involving}\,W(\cdot)\,\text{in its simplest form}\...
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Representation of Functions with Multiple Inputs and Outputs
Can any function $f :\mathbb{F}^m \rightarrow \mathbb{F}^n$ over some field $\mathbb{F}$ be written as
$f(x_1,...,x_m) = \begin{align}
\begin{bmatrix}
f_{1}(x_1,...,x_m) \\
...
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Solving $\frac{2x^2}{x-1}\cdot 2^x+8=0$
So recently, a friend of mine in grade $12$ got this question on her homework.
$$\left(\frac{2x^2}{x-1}\right)\left(2^x\right)+8=0$$
I tried rearranging this expression into $$\frac{-4x+4}{x^2}=2^x$$
...
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Is the following a surjective function on a boxed codomain?
Naive question, but my mind cannot reason today. Say I have the following function
$$f(x): \mathbb{R} \to [0, 1] ~~~~~~~~~~~ f(x) = 1 - e^{-x}$$
The function is
$\Box\ $ Injective and Surjective
$\Box\...
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Set of functions that diverges but its integral is finite, with various slopes at axis intercept
To design some type of mathematical model, I need a set of functions that diverge within a finite range but its integral is a real number. More specifically, what I need is a design of the form:
My ...
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How to solve the following equations on $\mathbb{R}$?(the result must be expressed by elementary function) [closed]
Solve the following equations on $\mathbb{R}$(the result must be expressed by elementary function)
\begin{eqnarray}
\begin{cases}
x^5-5x^3y+5x^2+5xy^2-5y+1=0 \\
y^5-5xy^3+5y^2+5x^2y-5x+2=0 \
...
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Find four smooth $C^{\infty}$ real functions satisfying a certain property around the origin
In the book "A Singular Mathematical Promenade", there's a Theorem of Kontsevich:
It's impossible to find polynomials $P_1(x)$, $P_2(x)$, $P_3(x)$, $P_4(x)$ satisfying the following two ...
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How to effectively rewrite $\lim_{x \to 0} \frac{e^{x}-1}{x}$ to properly assess its value
I am tasked with evaluating the following limit:
$$\lim_{x \to 0} \frac{e^{x}-1}{x}$$
There are two things I have tried. The first, substituting $x$ for $\ln t$ gave me
$$\lim_{x \to o} \frac{t-1}{\ln ...
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The relationship between $f_{i}$
Let $\lambda_{i}$ be a real number for any $1 \le i \le n$, defining $f_{i}$ as following $$f_{i} = \sum^{n}_{j=1} (\lambda_{j})^{i}$$
Suppose we know $f_{1},f_{2},f_{3}$ and $f_{4}$, then can we ...
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Class of differentiated Gamma functions: are there any algebras where they are elementary?
There is a class of differentiated gamma functions (DGFs) that basically can be expressed via derivatives of Gamma function.
They include the Gamma function, Polygamma function, and Hurwitz Zeta ...
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Evaluate the integral $\int_2^n \frac{e^{\frac{1}{x^2}}}{x} \;\text{d}x$ for real values $n$
I would like to compute the following (semi-definite?) integral
$$\int_2^n \frac{1}{x}\cdot e^{\frac{1}{x^2}} \;\text{d}x \;\;\; \text{ for } n\in \mathbb{R}\,.$$
Wolfram gives the following for the ...
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1
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Proof that the inverse of $f(x)={10}^x$ is $f^{-1}\left(x\right)={\mathrm{log} x\ }$.
I know that the inverse of $f(x)={10}^x$ is $f^{-1}\left(x\right)={\mathrm{log} x\ }$, geometrically this implies reflecting the original function across the line y=x. I am also aware that this can be ...
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$\sqrt[3]{10-x}+\sqrt[3]{30-x}=\sqrt[3]{15-x}+\sqrt[3]{25-x}$
I just happened to find a problem and an elegant solution.
The question asks us to solve the following equation
$$\sqrt[3]{10-x}+\sqrt[3]{30-x}=\sqrt[3]{15-x}+\sqrt[3]{25-x}$$
I am answering this ...
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0
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26
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Property for a continuous function with the same value
I'm doing Real Analysis and I'm at a point where I'm exploring continuous functions.
I stumbled on this question
Let A ⊆ R and f : R → R a continuous function such that f(x) = 6 for all x ∈ A. What ...
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1
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Elementary real function that is increasing and positive?
What are some increasing and always positive functions that is not an exponential function and very simple?
I think it might be very difficult to find a simple function satisfying these two criteria. ...
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0
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Proof that if p: $\mathcal{P} \rightarrow \mathcal{P}$ is a monotonic function, than there exist a set $Z$ such that $P(Z) = Z$
I am attempting to learn set theory, but a specific lemma in my book is confusing me. It states
Let $p: \mathcal{P} X \rightarrow \mathcal{P} X$ be a monotonic function on a set $X$ such that, $A \...
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Differentiable function that has minimum when all the variables are equal
I want to find a function $f:R^n\rightarrow R$, such that $f$ has minimum when $x_1=x_2=...=x_n$.
I came up with this one, but this is not differentiable, do you have any ideas?
$$f(x)=\sum_{i=1}^n \...
