# 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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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 $1$. $f(... 0 votes 4 answers 91 views ###$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 ... • 3,046 0 votes 0 answers 23 views ### 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 ... • 1,505 0 votes 0 answers 44 views ### Why we cannot solve$y^{2}+\sin y =2 x^{3}+Cfor y? This is from Stewart - Calculus - Early Transcedentals Writing the equation in differential form and integrating both sides, we have \begin{aligned} (2 y+\cos y) d y &=6 x^{2} d x \\ \int(2 y+\... 6 votes 0 answers 64 views ### Does there exist a finite set of solutions to integrals such that any function composed of elementary functions is integrable? For indefinite integrals whose solutions cannot express with elementary functions, special functions are often defined, such as those shown below. \mathrm{Si}(x) = \int_0^x\!\frac{\sin t}{t}\,\... 1 vote 2 answers 95 views ### Operation on functions, which of the following is true? Iff(x) = 2x + 1$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 ... • 3,046 9 votes 0 answers 271 views ### 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}\... • 26.5k 0 votes 1 answer 29 views ### 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) \\ ... • 57 1 vote 2 answers 162 views ### 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$$... 1 vote 1 answer 31 views ### Is the following a surjective function on a boxed codomain? Naive question, but my mind cannot reason today. Say I have the following function$$f(x): \mathbb{R} \to [0, 1] ~~~~~~~~~~~ f(x) = 1 - e^{-x}$$The function is \Box\ Injective and Surjective \Box\... • 1,985 0 votes 0 answers 93 views ### Set of functions that diverges but its integral is finite, with various slopes at axis intercept To design some type of mathematical model, I need a set of functions that diverge within a finite range but its integral is a real number. More specifically, what I need is a design of the form: My ... -1 votes 2 answers 83 views ### How to solve the following equations on \mathbb{R}?(the result must be expressed by elementary function) [closed] Solve the following equations on \mathbb{R}(the result must be expressed by elementary function) \begin{eqnarray} \begin{cases} x^5-5x^3y+5x^2+5xy^2-5y+1=0 \\ y^5-5xy^3+5y^2+5x^2y-5x+2=0 \ ... • 311 1 vote 0 answers 21 views ### Find four smooth C^{\infty} real functions satisfying a certain property around the origin In the book "A Singular Mathematical Promenade", there's a Theorem of Kontsevich: It's impossible to find polynomials P_1(x), P_2(x), P_3(x), P_4(x) satisfying the following two ... 0 votes 3 answers 73 views ### How to effectively rewrite \lim_{x \to 0} \frac{e^{x}-1}{x} to properly assess its value I am tasked with evaluating the following limit:$$\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 ... 1 vote 1 answer 31 views ### The relationship betweenf_{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 ... • 113 0 votes 0 answers 26 views ### 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 ... • 8,417 0 votes 1 answer 72 views ### 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 ... 0 votes 1 answer 65 views ### 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 ... • 15 5 votes 2 answers 110 views ###$\sqrt{10-x}+\sqrt{30-x}=\sqrt{15-x}+\sqrt{25-x}$I just happened to find a problem and an elegant solution. The question asks us to solve the following equation $$\sqrt{10-x}+\sqrt{30-x}=\sqrt{15-x}+\sqrt{25-x}$$ I am answering this ... • 1,259 0 votes 0 answers 26 views ### 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 ... 0 votes 1 answer 47 views ### 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. ... • 3,286 1 vote 0 answers 43 views ### 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 \...
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 \...