# Tagged Questions

Elementary questions about functions, notation, properties, and operations such as function composition.

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### If $f$ function then $f^{-1}$ function iff $f$ function injective (one-to-one).

During the lecture we learned this phrase: "If $f$ is a function then $f^{-1}$ is a function iff $f$ is injective (one-to-one)." But why? What with onto? $f$ doesn't need to be Surjective ...
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### Is there a simple distribution/function that behaves like fermi distribution but with a $\exp(-x^2)$ tail?

I have a data file with the following data (see picture). I try to find a simple function/distribution that follows the same trend : A behaviour like a Fermi-Dirac distribution A behaviour like a ...
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### A bound for error function

I am looking for a bound (or dominated function) of $erf(x)$ where $erf$ is defined here https://en.wikipedia.org/wiki/Error_function Thank you very much.
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### Don't know what this expanding periodic-ish function is

I plotted a function $c(x)$, which returns $3x + 1$ if $x$ is odd, and $x/2$ if $x$ is even. It's the Collatz conjecture. I get this interesting function. I don't know what it's called, so I can't ...
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### Finding number of functions from a set to itself such that $f(f(x)) = x$

The questions states that $f: A\rightarrow A$ is a function which satisfies $f(f(x)) = x.$ We have to find the number of such functions with $A = \left\{1,2,3,4\right\}$. The given condition clearly ...
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### Dude with taylor polynomial

Good night, i'm working with an problem of polynomial taylor, but i have a problem with the residue. Get a quadratic approximation $f\left(x,y\right)=\sin\left(x\right)\sin\left(y\right)$ near the ...
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### $e^z=3z^5$ - Rouche's theorem

Question : Show that the equation $e^z=3z^5$ possesses five distinct real roots. In using the Rouche's theorem with the function $f(z)=-e^z+3z^5$ and $g(z)=-3z^5$, I succeeded to prove the ...
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### Substituting into a function

$\hat{T}$ is a constant Are the parts I have underlined in green a mistake? Should they not be $C(\frac{p}{a}\hat{t}-\hat{T})$ as opposed to $C(\frac{p}{a}(\hat{t}-\hat{T}))$? If I have ...
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### Lower semi continuity for the norm of the speed

Let $\Omega$ be a smooth bounded open domain in $\mathbb R^d$. Let $\gamma_n: [0,1]\to \overline{\Omega}$ be a sequence of Lipschitz functions which converges uniformly on $[0,1]$ to a Lipshitz ...
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### Uses of step functions

My highschool teacher has informally told us about what continuity is and used step functions as an example of a discontinuous function. The Wikipedia page for it links to a lot of other kind of step ...
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### What exactly does f'(x)=0 imply from the definition of differentiability?

Let f be a real valued function satisfying $|f (x) −f (a)| ≤ C|x−a|^γ$, for some γ > 0 and C >0. (a) If γ = 1, show that f is continuous at a; (b) If γ > 1, show that f is ...
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### Would such a function be of any importance (primality test)?

While experimenting with some Maths, I came up with a really cool function. Let's call this function $\space \beta \space$. Which is a function of a real variable $\space r \space$. Here is the ...
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### Very difficult functions to prove with O notation

I am trying to prove some O notations as is it one of the tasks for my assignment in my course in algorithms and data structures. First of all I'd like to be sure that I got the "recipe" right. I use ...
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### Function of metric with a fixed point

I'm trying to prove that given a metric space $(X, d)$, for a fixed $x\in X$, define the function $g(y)=d(x,y)$, then $g(y)$ is continuous, using triangle inequality. My first question is that can I ...
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### Isometries in $\mathbb{E}^2$

We define $\mathbb{E}^2 = (\mathbb{R}^2, d_E)$, where $d_E$ is the usual Euclidean metric on $\mathbb{R}^2$. We say that a function $f:X \rightarrow Y$, where $X, Y$ are metric spaces, is an isometry ...
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### Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$

Let $f : [0,1] \to \mathbb{R}$ be a continuous function such that $$\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$$ Determine all such ...
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### Composition of functions Discrete Math question

How do I do this? All help is appreciated! Would prefer a step by step tutorial but any help is ok :) Let $h= g\circ f\circ g$ where $f \colon \mathbb R \to \mathbb Z$ is the floor function and ...
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### Changed codomain of inverse trigonometric functions

If codomain of $\arcsin(x)$ is $(\pi/2 , 3\pi/2)$ and codomain of $\arccos(x)$ is $(\pi , 2\pi)$ then what should be $\arcsin + \arccos$ equal to ? I thought of putting $x = \sin \theta$ But then ...
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### Why can we calculate the Fourier series of $x^2$ in any interval $[-l,+l]$?

We know that a function must satisfy Dirichlet's Conditions before it can be expanded in Fourier series. And Dirichlet's Conditions strictly require a function to be periodic in the interval in which ...
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### Cardinality of strict extrema of a real function

I recently encountered a problem seeking to prove that a real function can only have a maximum number of #$\mathbb{N}$ strict maximums. It may be that I have copied the problem improperly since there ...
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### If $\lim_{x\rightarrow\infty}f(x)=0$, does $||f||_{L^2}=0$? [closed]
If $\lim_{x\rightarrow\infty}f(x)=0$, does $||f||_{L^2}=0$? Thank you very much.