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

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3
votes
3answers
70 views

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 ...
1
vote
0answers
29 views

Finding the inverse of a function involving logarithms

Let $A \asymp B$ mean that there exists universal constants $m,M >0$ such that $mA \leq B \leq MA$. Let $k,n \in \mathbb{N}$ be such that $\log n \leq k \leq n$. I want to prove that $$ k ...
7
votes
6answers
618 views

Is there a simpler function with this shape?

I need a function that has the shape shown below. I don't care what the function does for $x < 0$ or $x > 1$. I've experimented with a lot of different functions, configured first and second ...
0
votes
0answers
14 views

Given 2 real numbers $a < b$ , let $d(x,[a,b]) = min\{|x-y| : a \leq y \leq b \}$ for $-\infty\leq x \leq \infty$

Then the function $f(x) = \frac{d(x,[0,1])}{d(x,[0,1])+d(x,[2,3])}$ satisfies (A) $0 \leq x < \frac{1}{2} $ for every $x$ (B) $0 < x < 1$ for every $x$ (C) $f(x) = 0$ if $2\leq x \leq 3$ ...
1
vote
2answers
35 views

Domain of log function having fractional base.

Find the domain of the function $${\sqrt{\log_{0.4} (x-x^2)}}$$ Where $0.4$ is the base $0.5$ is the power on the whole bracket.
0
votes
2answers
25 views

Infinite Sets Proof - Integer Sets

Let $Z^- $ be the set of negative numbers. Prove $Z^-$ ≈ $Z^+$ by finding a bijective function $f : Z^+-> Z^+$. Prove that the function is bijective. Could someone tell me how to get started on ...
0
votes
0answers
21 views

show function has a uniformly convergent subsequence

alright, so I have this question from my analysis class and I believe I have answered it correctly. I would be grateful if you could read it and give me your thoughts. A sequence $f_n$ of real valued ...
2
votes
2answers
59 views

Functions : $ f(x) = {2x-1\over x^2} $

We have : $$ f(x) = {2x-1\over x^2} $$ 1- Determine $ D_f $ and solve the equation $ f(x) = 1 $ 2- Show that for every $ x $ from $\mathbb{R}^*_+ $ ; $f(x) \le 1 $ The first exercise is ...
2
votes
3answers
62 views

How to show that $f(x) = x \arctan \left(x \sin^{2} \left(\frac{1}{x}\right)\right)$ is strictly increasing for $x \geq 1$?

I am trying to prove that $f(x) = x \arctan \left(x \sin^{2} \left(\frac{1}{x}\right)\right)$ is a strictly increasing function for $x \geq 1$. I try to do this by showing that $f'(x)>0$ for all ...
1
vote
3answers
27 views

Does the graph $y=\sin(x)\times\sin(x^{-2 })$ cross the $x$ axis an infinite amount of times in a finite interval?

Vsauce made a video recently on counting past infinity, and he represented the set of natural numbers to infinity with a set of lines, where each successive line is a smaller distance away from the ...
0
votes
1answer
28 views

Maximal Sets and Bijections

I'm struggling with this question (The function $f(x) = x^2 -3$): Let $A = \{x \in R : x \geq 0\}$. Determine a maximum set $B$ such that $f : A \rightarrow B$ is a bijection. Let $g : B \rightarrow ...
0
votes
1answer
12 views

How would it looks the Fourier series of this signal?

This is a kind of digital signal I'd like to re-create. i.e. I'd like to get N samples that will describe this signal: even better if it satisfy the Nyquist theorem (thus, sample-rate is 2x ...
7
votes
1answer
26 views

Functions with rational image of algebraic elements

Does there exist a non constant continuous fonction $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for any real algebraic number $x$, $f(x)$ is rational? Thank you for your answers!
0
votes
0answers
6 views

Two-point functions and spatial homogeneity

Consider a two-point function $f(\mathbf{r}_{1},\mathbf{r}_{2})$. If one requires homogeneity, then this implies that for a constant vector $\mathbf{a}$ we must have ...
0
votes
3answers
28 views

Can p(1,2)= 1 and p(2,4)=3 be linear opperators?

This is a linear algebra question... So I know that the two conditions for linearity are additivity and homogeneity. Typically Ives seen examples where the ...
1
vote
1answer
22 views

Proving that the tangent to a convex function is always below the function

Consider a real-valued convex function f defined on an open interval $(a,b) \subset \mathbb{R}$. $x,y \in (a,b)$. I want to prove that \begin{equation} f((1-\lambda)x + \lambda y) \leq ...
1
vote
1answer
27 views

Value of $a$ such that range contains the interval $[0,1]$

Find the number of integral values of $a$ in the interval $[0,100]$ so that the range of the function $y= \frac{x+a}{x^2-1}$, $x\in R$ contains the interval $[0,1]$? After rearranging $y= ...
-2
votes
1answer
23 views

Determine some values of a given piecewise-linear function

I am unsure how to begin. The question is $f : \Bbb R \to \Bbb R$ is defined as follows: $f(x) = \begin{cases} 2x - 4, & x > 0 \\ -3x + 1, & \text{otherwise} \end{cases}$. Determine ...
0
votes
1answer
29 views

Proving that a function is a bijection

Let $m_1,\ldots,m_n$ be pairwise coprime and let $m=m_1m_2\cdots m_n$. Show that the map \begin{align} \theta\,\colon \mathbb{Z}_m &\to \prod_{i=1}^n \mathbb{Z}_{m_i}\\ a+m\mathbb{Z} &\mapsto ...
1
vote
1answer
16 views

If $f(x) = f(y) \Longrightarrow g(x) = g(y)$, then determine $\phi$ so $g(x) = \phi(f(x))$

Let $x$,$y$ $\in I=\{1,...,n\}$, let $F$ and $G$ be some sets, and let $f:I\to F$ and $g:I\to G$ be two maps. I want to show that $$f(x) = f(y) \Longrightarrow g(x) = g(y) \ \ \text{if and only if}\ ...
0
votes
0answers
15 views

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 ...
0
votes
1answer
13 views

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.
1
vote
1answer
27 views

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 ...
4
votes
3answers
77 views

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 ...
1
vote
1answer
40 views

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 ...
1
vote
2answers
40 views

$e^z=3z^5$ - 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 ...
0
votes
0answers
8 views

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 ...
1
vote
2answers
43 views

Preimage of sets, complement of sets, continuity of functions

I just got some simple questions in real analysis regarding preimage and complement of sets and continuity. Suppose $f:X\to Y$, then does $f^{-1} (Y\setminus F)=f^{-1} (Y)\setminus f^{-1} ...
-1
votes
2answers
48 views

Example of a function that converges to 0 pointwise but integral is 3/2?

Give an example of a sequence of continuous functions $(f_n)$, $f_n : [0, 1] \to \mathbb{R}$ that converges to zero pointwise, and such that the integral of each function within the given domain is ...
0
votes
3answers
23 views

How to determine intervals where $f$ is greater than $g$?

I have two functions, $f(x) = 2x$ and $g(x) = \frac{x^3}{3}$. I solved for $x$ where $f = g$, finding $x = \pm 6^{1/2}$, then solved for $x$ where $f > g$, $x > \pm 6^{1/2}$, and where $f < ...
0
votes
0answers
16 views

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 ...
2
votes
3answers
156 views

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 ...
0
votes
2answers
37 views

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 ...
5
votes
1answer
57 views

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 ...
0
votes
0answers
62 views

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 ...
0
votes
1answer
25 views

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 ...
-1
votes
1answer
22 views

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 ...
13
votes
3answers
690 views

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 ...
0
votes
1answer
18 views

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 ...
1
vote
2answers
23 views

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 ...
0
votes
0answers
23 views

Runtime of Algorithms (Recurrence&Induction)

Two algorithms are given: $$T_A(n) = (\log_4(n) + 1) \cdot n\quad\text{and}\quad T_B(n) = 4 T_B\left(\frac{n}{4}\right) + n^\alpha$$ $$T_B(1) = 1; \alpha \in \mathbb R_+; n = ...
2
votes
1answer
44 views

Functions: Relations

I'm stuck on this question and would appreciate the solution, thanks! Let $R$ and $S$ be the relations on $A = \{1,2,3,4,5\}$ given by$$R = ...
1
vote
0answers
33 views

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 ...
1
vote
1answer
23 views

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 ...
-2
votes
1answer
32 views

If f(x) is continuous on $[a,b],$ differentiable on $(a,b)$, and $f'(x)\neq 0\ \forall x\in(a,b),$ then f'(x) is stable. [closed]

If $f(x)$ is a continuous function on $[a,b]$ and differentiable on $(a,b)$, and $f'(x)\neq 0\ \forall x\in(a,b)$ then $f'(x)$ is stable $\left(\text{i.e.},\ f'(x)<0\ \ \text{or}\ \ ...
2
votes
1answer
25 views

Invertible Uniform “PseudoRandom” Function

Perhaps this is better suited to a cryptography stack exchange, but I thought I'd try in mathematics in case this question is more obvious than I initially thought. I'm looking for a function ...
1
vote
3answers
31 views

How to phrase a proof of a function from a set A to a set B

Here is a problem: Let $f \subseteq A \times B$ be a function. In many situations you may want to restrict the domain of $f$ or expand its range. If $C \subseteq A$ then define the restriction of $f$ ...
0
votes
0answers
18 views

How to represent the function of variables?

I have a function as $$E=\int_\Omega -\log\big( p_i(x)\big) dx$$ where $p_i(x)$ is density distribution which estimated by Parzen window method. $p_i(x)=\frac{1}{\Omega_i} ...
-1
votes
1answer
49 views

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.
0
votes
1answer
31 views

Can someone draw a plot for this function?

Can someone draw a plot for this function? $ f(x) = \begin{cases} \sum_{i=2}^{x}\left(\frac{\prod_{k=1}^{i-1}\left(2k-1\right)\,\cdot\,-\left(-\frac{1}{2}\right)^{i}}{i!}\right) + \frac{3}{2} & x ...