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

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0
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1answer
15 views

Is there exists other statements equivalent to the analytic rank?

Let $f\colon\mathbb C\to\mathbb C$ be an entire non constant function. We consider its values on the real line. The order of vanishing (the analytic rank) at a point $s=a$ is denoted by $m$ (the ...
0
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1answer
64 views

Example of an infinitely differentiable function f : R → R with f(x) = 0 iff x = 0 and f intersects origin with infinite multiplicity

Is there an infinitely differentiable function f : R → R with f(x) = 0 iff x = 0 for which it is reasonable to say the graph of f intersects the x-axis at the origin with infinite multiplicity. So ...
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2answers
33 views

Horizontal Asymptote of Strange Function

What is the horizontal asymptote as x approaches positive infinity of $\sqrt{4x^2 + 5x} - \sqrt{4x^2 + x}$? The horizontal asymptote is in the form $y = k$. Find $k$.
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1answer
12 views

Can anyone explain how to show the finite difference equation $y'_{0}=\frac{y_{1}-y_{-1}}{2h} + O(h^{2}).$?

I was given that $y_{j}=y(x_{j})$ where $x_{j}=x_{0}+jh$ for integer j and positive h. I need to show that $$y'_{0}=\frac{y_{1}-y_{-1}}{2h} + O(h^{2}).$$ I thought I could start by finding the Taylor ...
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2answers
21 views

Let $f:R \to R$ be a function with $\frac{f(x)f(y)-f(xy)}{3} = x+y+2$ for all real numbers $x,y$. List all possible values for $f(36)$.

So far I have just been plugging in possible $x$ and $y$. $$\frac{f(4)f(9)-f(4\cdot9)}{3}=4+9+2$$ So then $f(36)=f(4)f(9)-45$. $$\frac{f(6)f(6)-f(6\cdot6)}{3}=6+6+2$$ ...
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2answers
17 views

function with moving average on matlab

Do a function that maintains moving average, the function gives the average of all the numbers that have been put in the function. does anyone know how to do this ?
2
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4answers
83 views

value of $x^2 \sin (\frac1x)$ at $x=0$ [closed]

What is the value of $y=x^2 \sin(1/x)$ at $x=0$ $x^2 =0$ but $\sin (1/x)$ is undefined in general if $y= a(x)b(x)c(x)d(x)\dots$ i.e a function made up of a product of functions At a specific ...
0
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1answer
36 views

Functions involving codomains

Problem: Consider the possible $f: [7]\to[9]$ a) How many have $f(i) $even , for all i? b) How many have rng(f) = {5,6} As far problem a goes, I've only gotten to the answer = 4^7. However I'm not ...
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2answers
19 views

Help with demonstration of formula for the axis of a parabola

At school we are studying the parabola and our teacher said that the formula for the axis of a parabola is $x=-\frac{b}{2a}$ without giving us the demonstration; so I tried to come up with a nice ...
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3answers
71 views

Prove this limit (formally)

As I was coursing through Spivak's calculus, more like analysis; I found an interesting, questionable example. Let $\frac{p}{q}$ be in its lowest terms; $p$ and $q$ are integers with no common ...
1
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1answer
682 views

Integers and integer functions

Let $\Bbb{Z}^+$be the set of all non-negative integers where $n$ and $k$ are given natural numbers. We consider the following non-decreasing function, $$f:\Bbb{Z}^+ \to \Bbb{Z}^+$$ such that ...
1
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1answer
21 views

How can I write a formula so that an addition of +1 or -1 changes a number's value less and less, never reaching +10 or -10?

I want to write a formula where 1 or -1 may be added to a starting variable of zero, any number of times, but I want to make it so that the rate of increase / decrease decreases infinitely before ...
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0answers
24 views

Checking this function for differentiability

$f(x) = |x|\sin x + |{x-\pi}|\cos x$ for $x \in \mathbb{R}$ Is the above function differentiable at $x=0$ ? At $x=\pi$ ?
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1answer
14 views

Diagonal of a two variables function and its partial derivative

$L(x,y)$ is a nice function (we can assume nice properties of it if needed), now suppose $$\frac{\partial L(x,y)}{\partial y}|_{y=x}\equiv H(x)$$ is a known function, then what can we learn about the ...
1
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1answer
12 views

Find the vertical and horizontal asymptotes of this function

$$F(x)= \frac{\sqrt{9x^2-5x}}{x+3}$$ Ok, so the horizontal lines $y=3$ and $y=-3$ are horizontal asymptotes because $\displaystyle\lim_{x \rightarrow \pm\infty}F(x) = \pm3$. But, what about vertical ...
6
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1answer
61 views

shape created by parabola

What would be the name of the shape that is the set of all points such that they are equidistant from the point $(0,1)$ and to the parabola $y=x^2$. Here is a desmos graph that generates the ...
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0answers
23 views

Is this a valid proof of surjectivity?

Problem: Determine whether the function $f: [3, \infty) \to [5,\infty)$ given by $f(x) = (x-3)^2+5$ is surjective. Answer. It is surjective. Proof: Let $\omega \in [5,\infty)$. I claim that ...
0
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2answers
12 views

Points of intersection for two polar equations question

Why is it that when I try to find the points of intersection for $r=2$ and $r=4*\cos(2\theta)$, I only get the $\theta$ where the reference angle is $\pi/6$? There is clearly another solution between ...
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0answers
28 views

BusyBeaver growth: “simple” proof

I just try to prove that $BB(n)$ (BusyBeaver-Function) grows faster than any other computable function. Maybe someone can check the proof? $f(n)$ is a computable function which grows to infinity: ...
0
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0answers
39 views

When can I solve in closed form this curve fitting problem?

I have $n$ real values $x_1,x_2,\ldots,x_n$ and $n$ real values $y_1,y_2,\ldots,y_n$; then I have a function $f(x,\boldsymbol\theta)$ from $\mathbb{R}$ to $\mathbb{R}$ and depending on $m$ parameters ...
3
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5answers
76 views

Does $f(x)=ax$ intersect $g(x)=\sqrt{x}$

It maybe a stupid question but I want to be sure how to explain it formally. Does $f(x)=ax$ intersect $g(x)=\sqrt{x}$, when $x>0$ and $a>0$ (however small it is) I think it does. The ...
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2answers
36 views

find a such function $f(x)$

How to find a function $f(x)$ that satisfies: $f(x)$ defines only on the positive axis of X; when $x\to 0$, $f(x)\to +\infty$. For a positive real number $k$, when $x\to k$, $f(x)\to 0$. for $x\geq ...
1
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1answer
14 views

Order-preserving function

I have an order on $\mathbb R ^ \mathbb R: f \le g $ iff $\forall x \in \mathbb R: f(x) \le g(x)$. Now I have a function $\mathbb R ^ \mathbb R \to \mathbb R ^ \mathbb R: F (f)(x)= f(x^2)+1$. I have ...
0
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0answers
22 views

Determine whether $f$ is a valid function if $f$ (a bit string) is defined as a sequence of $0$ or more bits.

Determine if $f$ is valid from the set of all bit strings to the set of integers if $ f(S) $ is defined as the number of $0$ bits in S. I don't understand where to start. I thought of saying that ...
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2answers
34 views

Show that there exists a bijection from $(0,1)$ to $(0,1]$ [duplicate]

I'm having trouble envisioning a bijective relationship that maps $(0,1)$ to $(0,1]$. My professor gave the hint that it can be expressed as a piece-wise function $f(x)$ comprising of two cases: _ ...
0
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1answer
35 views

Show if $f(x_n) \to L$, for every $x_n$ in $(a, \infty)$ which converges to $\infty$ as $n \to \infty$, then $f(x_n) \to L$.

I already proved the → direction. I am having difficulty proving the converse. Prove: Suppose that f:[a, ∞) → R, for some a in R. Then f(x_n) → L, for every x_n in the interval (a, ∞) which ...
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0answers
16 views

Function satisfying the requirement

In my (physics) booklet the following is said: $$\alpha T = f[f(T)] $$ Then $$f(T) = \pm \sqrt{\alpha}T$$ But it gives no motivation for this at all, it just says "it can be proven that..". How can ...
0
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1answer
26 views

Find total number of surjective mappings

Let $A$ and $B$ two sets with $|A|=n$ and $|B|=m$. Then find total number of injective mappings from $A$ to $B$ if $n\leq m$. find total number of surjective mappings from $A$ to $B$ if $n\geq ...
0
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1answer
35 views

Can every smooth sine function be given a smooth argument?

Here's a conjecture that I believe to be true, but I couldn't find a proof: Let $\alpha: \mathbb{R}\longrightarrow\mathbb{R}$ be a function such that $\sin\alpha$ is smooth. Then there is a smooth ...
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0answers
20 views

Computing composite functions

This may not be strictly a math question but is related. Whenever there is some function that computes more than two elements, is it possible that all elements are computed at once? Or is computing ...
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4answers
58 views

Why does $x/(x^2+4)$ have a range?

The domain of this graph is all real numbers. There are no vertical asymptotes but there is a horizontal one at $y=0$. But then when I graph there's an $x$ value at $y=0$ and there is a range (where ...
2
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1answer
23 views

Pre-Calc question, please help.

I have to find a polynomial function of degree 4 with real coefficients with a real zero at i, a zero at -3(multiplicity 2), that passes through the point (-1,16). Please help.
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3answers
55 views

Roots of this third degree polynomial

I've got the following polynomial $$ x^3-6x^2-2x+40 $$ and I want to find its roots. The only option I see at the moment is to compute all the divisors of $40$ and their inverse, and manually check if ...
0
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1answer
14 views

Reverse map for an equation .

I don't know this is actually reverse mapping or what but i have following equation. $$x = \tanh(a \cdot b ) + c $$ How do I solve for $a$? Does it has anything to do with inverse hyperbolic ...
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2answers
42 views

Arguing whether a function is one to one

The function is $x^{2}-1$. How exactly would you go about arguing that it is one to one? (Or not one to one if that were to be the case). I know it isn't since I know what the function looks like.
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3answers
33 views

singularity of $\frac{z}{\sinh(z)}$

I was wondering why $0$ is not a singularity of $\frac{z}{\sinh(z)}$ Thank you for your feedbacks
3
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1answer
80 views

Functions for non-negative integers [duplicate]

Let $\Bbb{Z}^+$be the set of all non-negative integers where $n$ and $k$ are given natural numbers. We consider the following non-decreasing function, $$f:\Bbb{Z}^+ \to \Bbb{Z}^+$$ such that ...
1
vote
1answer
47 views

Find tangent line at given points, no function equation

I have never encountered a problem like this and am a bit confused. Function $f$ satisfies: $f(3)=5$, $f(9)=7$ $f'(3)=11$, $f'(9)=13$ Find an equation for the tangent line to the curve $y=f(x^2)$ ...
0
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2answers
24 views

What is the meaning of the PI function.

I am solving for a configuration problem and i have seen a function π This is a function not 3.14 which is the value of pi. While accessing some lectures i found out that they also call this symbol ...
0
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1answer
21 views

Power function and involution $f(x) = x^a$

For power functions we have a variable $x$ and a constant $a$; we get that $f(x) = x^a$. Find all involutions for $f(x)$. I started out with basic functions such as $f_1(x) = f_1^{-1}(x) = x^1$ and ...
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4answers
481 views

A continuous function defined on an interval can have a mean value. What about a median?

A function can have an average value $$\frac{1}{b-a}\int_{a}^{b} f(x)dx$$ Can a continuous function have a median? How would that be computed?
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1answer
45 views

Can a function be the domain of another function?

Is this the correct way to express a function whose domain is another function?: Let $n$ be any given natural number. Let $s$ be the square root of $n$. Using $s$ as the domain of the prime counting ...
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0answers
52 views

Connection between the Cantor set and the Tent map $T_3$

I would like to prove part c) using the ternary expansion as shown in the second half of the image. I understand how to compute up to what is shown in the image. I am struggling to understand the ...
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0answers
30 views

How to plot this function

How to plot this function in WolframAlpha or some other graphing calculator? $f(x) =\left\{\begin{matrix} 1 & -\dfrac{-2\pi}{3} \leq x \leq \dfrac{2\pi}{3}\\ -1 & ...
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2answers
46 views

Difficult algebraic expression for $f(x) = \frac{x-a}{bx-c}$ to find involutory solution

So I read in another thread about involutory functions, he claims for any real numbers $a$ and $b$, the function: $$f(x) = a + \frac{b}{x-a} = \frac{ax + (b-a^2)}{x-a}$$ satisfies $$f(f(x)) = a + ...
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2answers
38 views

Is a Hilbert space a vector space or a space of functions?

I was learning what Hilbert space was and this is the definition that I have: $\mathcal{H}$ Hilbert Space is a vector space with $\langle \cdot , \cdot\rangle$ inner product and is complete with ...
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1answer
24 views

Image of $f$ in $f(x)=\lfloor x\rfloor$ out of bounds for intervals?

Edit 1. This all being worked on with the real numbers $\mathbb{R}$ Given a function $f(x) = \lfloor x\rfloor$ (Floor function). Find the image of B, $f^{-1}(B)$ if $B = [0,1)$ For easier cases such ...
3
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2answers
56 views

Zeros of a function of degree 4

I'm trying to show that the following function has no zeros $$ 60x^4-44x^3-25x^2-44x+60=0. $$ I already tried using Eisenstein's criterium, but since the first and the last coefficient are both $60$, ...
0
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1answer
36 views

Prove that $f_1=f_2=…=f_n$. [duplicate]

Let functions $f_1,f_2,...,f_n: \Bbb R \to \Bbb R^+$ are periodic functions such that $\lim_{x\to\infty} (f_1(x)+f_2(x)+...+f_n(x))=0$ Prove that $f_1=f_2=....=f_n$. The question was posted few ...
1
vote
1answer
21 views

Number of linear functions from $\{0,1\}^n$ to $\{0,1\}$

For $x,y \in \{0,1\}^n$, let $x \oplus y$ be the element of $\{0,1\}^n$ obtained by the component-wise exclusive or of $x$ and $y$. A boolean function $F:\{0,1\}^n \to \{0,1\}$ is said to be linear ...