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

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0
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0answers
7 views

Searching function starting exactly constant and approaching another constant

for the default of an R API parameter i seek a function that has the property of yielding a good guess. I want the function to be defined for $\mathbb Z^+$ (But no reason not to define it for ...
0
votes
2answers
27 views

Calcultaing function limit of a limit sequence

How do I even start? $$\lim_{x \to 1^+} \lim_{n \to \infty}{x^n \over x^n + 7}$$ I see that it should be $1$ but how do i prove it?
1
vote
2answers
19 views

Basic question on the probability function and the probability distribution function

I have a question on the probability function. In my book it says that if A and B are mutually exclusive events $P(A∪B)=P(A) + P(B)$. Then when it starts talking about the probability distribution ...
0
votes
1answer
22 views

Why can open intervals be used to calculate rate of change?

I was watching this video Question : On which interval does $y(x)$ have an average rate of change of $\frac{1}{2}$? The first option is $-2 < x < 2$ The video narrator just puts in $-2$ and ...
0
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0answers
27 views

Holomorphic function in unit disc?

I am stuck on this exercise from Stein and Shakarchi's Real Analysis: suppose $F$ is holomorphic in the unit disc, and $$\sup_{0\leq r < 1} \frac{1}{2\pi} \int_{-\pi}^\pi \log^+ ...
1
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3answers
31 views

Show that $f: \mathbb{Z} \to n\mathbb{Z}$ defined as $f(m) = mn$ is bijective

Let $n \in \mathbb{N}$. Show that $f: \mathbb{Z} \to n\mathbb{Z}$ defined as $f(m) = mn$ is bijective such that $f(m_1 + m_2) = f(m_1) + f(m_2)$, $\forall m_1, m_2 \in \mathbb{Z}$ To be bijective, ...
-2
votes
1answer
42 views

What does it mean for a function to be closed under linear operators? [on hold]

What does it mean for a function to be closed under linear operators? I'm looking for as informal and intuitive of an explanation as possible.
1
vote
1answer
25 views

How do you create the equation for the Cantor Pairing Function?

According to wikipedia, here is the equation: $f(x,y) = \frac{(x+y)(x+y+1)}{2}+y$ How do you go about creating this function? I understand that the X value is found by the corresponding triangle ...
0
votes
4answers
66 views

Does there exist a bijection [on hold]

Does there exist a bijection from (0,1) to $\Bbb{R}$? How to prove there is or not?
0
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0answers
6 views

Defining a function with an arbitrary number of values

In school we learn that a function may have one value at most for a given input. But if we define its range to be the set $\bigcup_{n=1}^{\infty}{{\mathbb{R}}}^n$, then it can have an arbitrary number ...
1
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1answer
38 views

For what values of $x\in \Bbb N$ is $\tan^{-1}\left(\frac{360}{x}\right)$ rational?

For what values of $x\in \Bbb N$ is $$\tan^{-1}\left(\frac{360}{x}\right)$$ rational? Just wondering if there is any method to accomplish this.
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0answers
3 views

On the zero set of a smooth function which is linear on the first variable.

Let $f:C^n\times R^n\rightarrow C$ be a smooth function which is linear in $z\in C^n$. (1) For any $z\in C^n$, $f(z,R^n)$ is compact and there exists $x_z\in R^n$ satisfying $f(z,x_z)=0$. (2) There ...
1
vote
4answers
81 views

Solving the equation $ f^{-1}(x)=f(x)$

I attempted to solve the equation given in the title for the function; $$f: \mathbb R_{++} \to\mathbb R_{++}; \quad f(x)=x^2(x+2)$$ I understand that the problem is equivalent to solving $f(f(x))=x$ ...
2
votes
1answer
29 views

How to determine the generating function?

So I have $$\overset{*}{F} = \overset{*}{F}_{n-1} + \overset{*}{F}_{n-2} + g(n)$$ where $\overset{*}{F}$ is NOT a Fibonacci number for $n \geq 2$. $g(n)$ is any function $g: \mathbb{N} \to ...
1
vote
1answer
12 views

To find, wether '1' lies in the range of f, where $f(x)=[ln(\frac{7x-x^2}{12})]^\frac{3}{2}$?

$f(x)=[ln(\frac{7x-x^2}{12})]^\frac{3}{2}$, For the given function, the question is whether, f(x) can equal 1 for some real value of x?
3
votes
2answers
39 views

How to precisely define $C^\infty$ in $f(x) \in C^\infty$

In single variable calculus, a common way to denote a function that is continuous for all derivatives is to write $f(x) \in C^\infty$ i.e. $f(x) = \exp(x)$ Is there a more rigorous way to define ...
0
votes
1answer
31 views

Exponential type of $\sin z$

An entire function $f$ is of exponential type if $\,\lvert\, f(z)\rvert\le C\mathrm{e}^{\tau\lvert z\rvert},\,$ for all sufficiently large values of $\lvert z\rvert$. The exponential type of $f$ is ...
0
votes
0answers
16 views

Graphical transformation : reflect and shift

I know that x[-n] will be reflection of x[n] along y-axis and x[n+k] will shift x[n] to left by k points. Now if I take x[n] 1. x'[n]=x[-n] should reflect along y axis 2. x'[n+k]=x[k-n] should shilf ...
1
vote
1answer
52 views

$f '(x) = -f(x)$ and $f(1) = 1$, Solve for $f(2)$

I am honestly not even sure how to start this problem... My first thought was that $f(2) = 2$ ... But now I don't even know where to go from there.
0
votes
2answers
44 views

Determine if z is a function of x and y. $6x-4y+2z=10$

"Determine if z is a function of x and y. $6x-4y+2z=10$. Find the formula" All i did was equate for z $$z = 5-3x+2y$$ That is the formula. And It's pretty obvious that the answers are unique but i ...
0
votes
2answers
58 views

Why is $f'(c) = \text{does not exist}$ a critical point?

In my lecture the prof wrote that when the derivative does not exist at a point it is also a critical point I can understand that $f'(c) = 0$ indicates that we have a flat place on our curve, so ...
0
votes
0answers
13 views

Looking for a formula with that goes with a specific family of line graphs

I need a formula that I can use in my Arduino project that gives me the following graph family. The idea is that 'x' would be drawn randomly, and 'a' would be the number of previous attempts at my ...
0
votes
1answer
19 views

Recursive equation, inhomogeneous for the Fibonacci numbers [on hold]

Recursively applies the inhomogeneous: $$F_n = F_{n-1} + F_{n-2} + g(n)$$ for $n \geq 2$ $$F_1 = g(1)$$ $$F_0 = g(0)$$ where $g: \mathbb{N} \to \mathbb{R}$ is any function. How can the recursive ...
3
votes
2answers
40 views

period of $\cos(x) + x - \lfloor x \rfloor$?

What is the period of $\cos(x) + x - \lfloor x \rfloor$? This is what I have done: $x = \lfloor x \rfloor + \{x\}$ $\cos(x)$ has period $2\pi$ $\{x\}$ has period $1$ so $\cos(x) + \{x\}$ should ...
1
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2answers
58 views

Graph of the function $y = 2 + (x + 1)^3$

I know that this function will have the behavior of $Y = X^3$ but as I will translate for this function $(Y = X^3)$? I do this: $$(x + 1)^3 = x^3 + 3x^2 + 3x + 3 \quad y = x^3 + 3x^2 + 3x + 5$$ But ...
1
vote
1answer
45 views

Find a function with the property, or prove it doesn't exist

Today, I encountered the following problem in my research. I'd like to find a function $f(x_1, x_2, \ldots, x_n)$ such that $$ 0 = \frac{d f(x_1, x_2, \ldots, x_n)}{d a}\bigg|_{c_1,c_2,\ldots,c_n} ...
1
vote
0answers
19 views

How would we generate a basis of sigmoidal functions?

I am trying to figure out how to generate a basis of sigmoidal functions. My issue is thus: there are several possible generating functions for a sigmoid (logistic curves, error functions, arctangent, ...
0
votes
1answer
14 views

The domain of logarithmic functions with sinx and cosx

I need to solve this equation: $\log_{\cos x} \sin x + \log_{\sin x} \cos x=2$ But in order to solve it, I first need to find the domain. What I did was this: $\cos x\neq1 \wedge \sin x\neq1 $ ...
8
votes
2answers
130 views

Does equation $f(g(x))=a f(x)$ have a solution?

Suppose $g: [0,1] \to [0,1]$ is a strictly increasing, continuously differentiable function with $g(0)=0$ and $g(1) < 1$, and $a \in (0,1)$. Is there a function $f:[0,1] \to \mathbb{R}$ such that ...
-2
votes
2answers
42 views

Proving the equation has no root. [on hold]

How to show that for $a\in \mathbb R$, the equation $x^2+12a^2+4ax-8a+8=0$ has no root?
2
votes
0answers
72 views

Inverse of $f(x)=x+\sin(x)$

I'm attempting to find $$f^{-1}(x)$$ for the function $$f(x) = x + \sin{x}.$$ So far I've tried some simple algebraic methods as well as rewriting $\sin{x}$ as a power series. I'm not quite sure where ...
1
vote
1answer
29 views

Integral with an unknown function

I am trying to solve this integral $$ \int \frac{f(x)}{g(x)}\frac{\mathrm dg}{\mathrm dx}\mathrm dx $$ where $g$ is an unknown function of $x$, and $f(x)$ is a known function that can be integrated ...
0
votes
1answer
21 views

What would the multifunctional inverse of $F(x)=|x|$ be?

What would the multifunctional inverse of $F(x)=|x|$ be, assuming $x$ is on the complex plane. Also, how would this usually be represented? Note that this won't be a 'true' function. (But assume a ...
0
votes
2answers
56 views

For a function $f:S\to S$, if $f$ is injective, then $f\circ f\circ f$ is injective

For a function $f:S\rightarrow S$, if $f$ is injective, then $f\circ f\circ f$ is injective. Is my attempt correct? True. Let $S=A$ , $S=B$ and $S=C$ and $f=g$ . Suppose $f:A\to B$ and ...
7
votes
4answers
470 views

Doubt about the domain in logarithmic functions.

According to my book, the logarithmic function $$\log_{a}x=y$$ is defined if both $x$ and $a$ are positive and $x\neq 0$ and $a\neq 1$. So are these not correct? $$\log_{-3}9=2$$ $$\log_{-2}-8=3$$ ...
0
votes
1answer
38 views

Good Triple Well Function [on hold]

I am looking for a good triple-well function with good control over the barrier height. Let's say that $y=9x^{2}-6x^{4}+x^{6}$. In this function there are three wells (even though only two wells are ...
1
vote
1answer
35 views

Proof that a function is unbounded [on hold]

I have this function \begin{equation*} f(x):=\left\{\begin{array}{cl} \frac{3}{2}x^{\frac{1}{2}}(\sin\frac{1}{x})+x^{\frac{3}{2}}(\cos\frac{1}{x})(-x^{-2}), & \mbox{for }0<x\leq 1,\\ 0, & ...
3
votes
0answers
28 views

Identification of a function

I recently came across the following function $$\sum_{k=1}^\infty(\log(k))^n\frac{z^k}{k}$$ I found it while dealing with the polylogarithm function, $Li_n (z)$ (Notice that if instead of ...
2
votes
1answer
37 views

Show that if $f$ is a proper,surjective map which is locally injective then $f$ must be a covering map

Suppose $f :X \to Y$ is a continuous proper map between locally compact Hausdorff spaces. Show that if $f$ is a surjective map which is locally injective then $f$ must be a covering map. It is well ...
3
votes
1answer
30 views

Classification of Proper Maps between domains in $\mathbb{R}^n$

Suppose $f:D_1\to D_2$ is a continuous map between domains in $\mathbb{R}^n$. Show that $f$ is proper iff for every sequence $(x_n)$ in $D_1$ which accumulates only on $\partial D_1\cup\{\infty\}$, ...
0
votes
3answers
70 views

For a function Y : X→X , if Y is injective, then Y∘Y∘Y is injective.

For a function Y : X→X , if Y is injective, then Y∘Y∘Y is injective. My attempt: Using contrapositive, if Y is not injective. then Y ∘ Y is not injective, the there exist x, x' ∈ X with x ≠ x' but ...
-3
votes
1answer
27 views

Find the domain and range of $f$ and $f^{-1}$ [on hold]

Find the domain and range of $f$ and $f^{−1}$ $f(x) = x^2 − 9, \ \ \ x \le 0$ $f^{−1}(x) = -\sqrt{x+9}$
1
vote
1answer
124 views

Shoud a function be defined for all elements in the domain in order to be surjective/bijective?

In other words, the surjection says: for any y in the codomain there should exist x in the domain. Now, do I need for every x in the domain to have an y in the codomain for surjectivity?
-2
votes
2answers
66 views

Proving question on sets

I am unable to understand this question. I have to say whether its true of false and then prove it but I can't proceed with the question unless I understand it. Let $S$ be the set of stars in our ...
1
vote
3answers
147 views

Limit of a Sine Function

Calculate via the limit definition: $$\lim\limits_{x \to \frac {\pi}2^-} \frac{\sin^2(\frac {\pi}2-x)}{\sqrt{\pi-2x}},$$ I tried to calculate this limit using the definition of a limit and got ...
6
votes
1answer
48 views

which functions can be obtained as a composition of a continuous function with itself? [duplicate]

let $f(x)=x^2$, then $f(f(x))=x^4$, so $x^4$ is a continuous function from $\Bbb R$ to $\Bbb R$ which can be obtained as $f\circ f$ for a continuous $f\colon \Bbb R\to \Bbb R$. general example: for ...
1
vote
3answers
68 views

Prove that $f : [a,b] \rightarrow \mathbb{R}$ is a bijection from $[a, b]$ to $[f(a), f(b)]$

I'm a 1st year mathematics student, and in my analysis class I'm having trouble with proving the following: Let $a < b \in \mathbb{R}$, and let $f : [a,b] \rightarrow \mathbb{R}$ be a continuous ...
0
votes
1answer
18 views

How should one define the cross product for two vector valued functions?

In many textbooks regarding vector analysis, the cross product is mention in the section about the properties of the derivative of vector functions, but there seems to be no explanation what that ...
1
vote
2answers
26 views

Characteristic Function in the subset E

Let $E \subseteq \mathbb R$. Then the characteristic function $\chi_{E}:\mathbb R \to \mathbb R$ is continuous if and only if a) $E$ is closed. b) $E$ is Open. c) $E$ is both Open and Closed. d) ...
0
votes
1answer
24 views

What is the definition of a binary relation?

According to http://en.wikipedia.org/wiki/Binary_relation it is first defined as "a collection of ordered pairs of elements of A" and then as "an ordered triple (X, Y, G) where X and Y are arbitrary ...