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

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0
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1answer
34 views

How to count even numbers in the first n natural numbers?

So let's say that I count $1,2,3,4$ and ask the question: How many even numbers are there in this sequence. Well, there are $2$ even numbers or we can say $4/2$ even numbers. But if the count is ...
1
vote
2answers
92 views

Why are all non-polynomial functions are basically exponents?

There's paucity of really "original" functions in Math. Aside from power functions/ polynomials, really the only other function widely used is exponential. For example, $\log$ is simply inverse of ...
0
votes
1answer
8 views

Clarification on the idea of absolute maxima

$$f(x)=-|x|\:,\:\:x≠0$$ If f(x) did not have an point of discontinuity at x = 0, then it is obvious it would have an absolute maximum there. However, now that that point no longer exists, does it ...
0
votes
0answers
18 views

Using integration to find volume of a parabolic prism.

Suppose I have a solid with its profile being a parabola given by the equation $y = f(x)$. Its depth at any point is given by $g(x)$ and its width is a constant $k$. This essentially results in a ...
-5
votes
2answers
34 views

Functions Solving Equations (URGENT) (EDITED) [on hold]

Edit: NOTE: All the questions are linked. I.E the function for $a$ may be in the first question etc I'm really having issues with this so your assistance is very much appreciated. I am having ...
0
votes
1answer
29 views

If $f$ and $g$ are invertible and $f\circ g$ is defined, is $g^{-1} \circ f^{-1}$ defined?

In the proof for that any invertible functions $f$ and $g$ with $f \circ g$ defined, $(f\circ g)^{-1} = g^{-1}\circ f^{-1}$, it seems to me that there is an assumption that $g^{-1}\circ f^{-1}$ is ...
3
votes
3answers
67 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 ...
2
votes
5answers
77 views

Solving $\arcsin\left(2x\sqrt{1-x^2}\right) = 2 \arcsin x$

If we have $$\arcsin\left(2x\sqrt{1-x^2}\right) = 2 \arcsin x$$ we have to find the set of $x$ for which this is true. I tried to solve it by putting $x = \sin a$ or $\cos a, but got no ...
1
vote
0answers
31 views

Solving equations for functions [on hold]

Question removed due to new post adding in details.
0
votes
1answer
54 views
-4
votes
0answers
27 views

How are non trivial zeros of the Riemann Zeta Function distributed? [on hold]

I think I have found it https://drive.google.com/file/d/0Bx1D5GKafNvEamtfdGZUaFdMaDg/view?usp=sharing https://www.dropbox.com/s/ufbvb8p2dbsxmhc/images.pdf?dl=0 Regards Subhendra
2
votes
1answer
37 views

Example of a diffeomorphism from all of $\mathbb{R}$ to itself

I can think of diffeomorphisms from an interval to $(a,b)\rightarrow \mathbb{R}$, scaling the tangent function, and from the punctured plane, polar coordinates, or some odd polynomial, but does anyone ...
7
votes
6answers
596 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 ...
2
votes
1answer
40 views

Functional equation.

I'm trying to solve the functional equation $f(x+f(y)) = f(x)-y$ where $f : \mathbb{Z} \to \mathbb{Z}$. What I got so far is: $f$ is injective and $f(0) = 0$. Thanks in advance for your time.
2
votes
2answers
83 views

Why does definite integral define an area?

I'm kind of new to integrals. I know that $$\int_a^b f(x)\,dx=\int f(b)-\int f(a)$$ Using definite integrals, I can calculate area between the function and the $x$ axis between $x=a$ and $x=b$. For ...
3
votes
3answers
41 views

Fundamental Period of $\tan x \cot x$

What is the period of $\tan x \cot x?$ I was given this question today. What I did was simplify the expression , and it reduced to a constant function. So it had no fundamental period. But my teacher ...
0
votes
0answers
30 views

If $f'$ of a continuous $f$ exists except on a countable set $E$ and $f'$ is Lebesgue integrable over $[a, x]$ then $f(x)-f(a)=\int_a^x f'dx $. [duplicate]

If $f'$ (derivative of continuous function $f$) exists except on a countable set $E$ and $f'$ is Lebesgue integrable over $[a, x]$ then $f(x)-f(a)=\int_a^x f'dx $. Of course I'm aware of the proof ...
0
votes
1answer
707 views

Determine whether the graph of the function is the graph of a one-to-one function.

the graphs in #44, 48, 63, 65, 68 For each graph, does it represent a one-to-one function? my solution are: 44_ Yes _ 48____No____ 63____Yes___ 65____NO___ 68___Yes____ I know that the ...
2
votes
3answers
60 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 ...
0
votes
1answer
26 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 ...
3
votes
1answer
819 views

Plotting discrete time signals involving sumations in matlab.

Many of the examples I've encountered while looking for an answer are simple functions that do not involve summations. Suppose I have the following function; ...
2
votes
2answers
32 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
0answers
13 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
55 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 ...
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
20 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 ...
1
vote
1answer
25 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= ...
114
votes
1answer
4k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
1
vote
3answers
25 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 ...
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
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
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 ...
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
25 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
3answers
46 views

SmoothStep: Looking for a continuous family of interpolation functions

Background: SmoothStep is a simple sigmoid-like function defined as S(x) = 3x^2 - 2x^3. It is monotonically increasing from (0, 0) to (1, 1), is rotationally symmetric over that interval, and has ...
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 ...
27
votes
10answers
11k views

Why does “convex function” mean “concave *up*”?

A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$, $$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$ Equivalently, a line ...
2
votes
2answers
178 views

Recursive Function - mod 5

How do the recursive function for $\mod 5(x) = 0$ rest of division of $x$ by $5$. $$\begin{align} \mod&5(5) = 0\\ \mod&5(6) = 1\\ \mod&5(7) = 2\\ \mod&5(8) = 3\\ \mod&5(9) = 4\\ ...
-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 ...
1
vote
2answers
40 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
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
1answer
84 views

Solving a cubic equation

Solve $y=ax^3+bx^2+cx+d$ I need $x$ in terms of $y$ . I do not need the roots of the cubic equation . I need to express $x$ in terms of $y, x>0$
0
votes
1answer
11 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.
0
votes
0answers
60 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 ...
4
votes
3answers
76 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 ...
0
votes
2answers
33 views

assertion for how to use $x$ in function

in the function $f$($x-2\over2x$)=$2x+5$ why we dont use ($x-2\over2x$) directly as a $X$ ? because as a basic rule in functions we have $f(x)$=$2x+5$ so that why ($x-2\over2x$)cant take $x$ place? I ...
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 ...
1
vote
1answer
25 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 ...
12
votes
2answers
652 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 ...
3
votes
4answers
948 views

expansion of $\cos^k(\theta)$

Does any body know a expansion of : $\cos^k(\theta)$ in function of $\cos$ and/or $\sin$ but without power? For example : $\cos^2(\theta)=\frac{1}{2}(\cos(2\theta)+1)$, but i would want a ...