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

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1
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1answer
20 views

Definition of the Domain of a Function when the sets are the elements

In case I have a function that calculate the normalized distance of elements in two sets $A$ and $B$ I can define the function as $\mathrm{elementDistance} : A \times B \rightarrow [0,1]$. But if I ...
1
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2answers
59 views

If a uniquness for all functions exist shouldn't there be uniquness to recursion?

What I'm specifically saying is every function is definitely unique, as they may be nearly equivalent to another function, for example. Let's make a table of values for $^{x}2$ (0,1) (1,2) (2,4) ...
1
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1answer
19 views

Laplace Transform with sin and cos

Hi I am having trouble figuring out the solution of this Laplace transform: $$L_t{(u(t- \pi)(2\cos(t)-3\sin(3t))}$$ Where I am stuck if I am even on the right track is: $$L_t{(u(t- ...
1
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1answer
18 views

How $n^d \times m([0, \frac{1}{n}[^d) = m([0, 1[^d)$ follows from translation invariance and (finite) additivity

In this StackExchange question (which itself seems to reference to an exercise in Terence Tao's lecture notes on introductory measure theory on his blog here), it's said that assuming "finite ...
3
votes
1answer
31 views

Non-monotonic functions on ordered sets

I'm trying to prove that if $~~(A,<_A)~~$ and $~~(B,<_B)~~$ are linearly ordered sets and $~~f: A \rightarrow B~~$ is non-monotonic function than there exist points $~~a,b,c\in A~~$ such that ...
0
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1answer
42 views

How to identify the function of a graph?

I am trying to identify the function of a graph in order to create a dataset for it. Dataset as in several x/y-values that will lead Excel/PowerPoint to create a graph looking just like the drawing: ...
0
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1answer
24 views

Dirac delta function and well behaved function [duplicate]

whether dirac delta function a well behaved function? Can u please explain the properties of a well behaved function..?
0
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1answer
31 views

Find domain of function with quadratic numerator algebraically

I'm stuck on this problem: $$f(x) = \frac{x^2 -4}{x}$$ I need to determine why this function's domain is not: $$\{x|x \neq \pm 2\}$$ All of the examples that I've seen have the quadratic in the ...
0
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1answer
26 views

Need help creating a special function

I'm creating a special function in a game and needed some help with the maths end of it. Essentially, I need a programmable, non-linear function so that $f(100) = 0$, and $f(0) = 100$ (or some other ...
0
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1answer
24 views

Solving function in difference quotien equation

I have the problem Find the difference quotient $\frac{f(2 + h) - f(2)}{h}$ for $f(x) = \frac{1}{x^2}$. The answer they gave is $\frac{-(4 + h)}{4(2 + h)^2}$ So far I've done: $$\frac{[1/(2 + h)^2 ...
3
votes
1answer
75 views

Is $f(t)=(\cos(t),\sin(t))$ a function?

In the Linear Algebra book we're using (Linear Algebra with Applications, Bretscher, p.129), the author defines this as the function of the unit circle. I understand why the equation of a circle ...
1
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3answers
62 views

Show that $x^3 +x-1$ has a zero between $x=0$ and $x=1$

Show that $x^3 +x-1$ has a zero between $x=0$ and $x=1$, does anyone know how to go about starting this problem? I am basically clueless. I thought maybe at first polynomial division since its $x^3$, ...
-1
votes
1answer
52 views

Find $k$ so that $f(x)$ is a continuous function [closed]

Find $k$ so that $f(x)$ is a continuous function. $$f(x)=\left\{\begin{array}{ll}x^2 &x\leq2\\ k-x^2 & x>2 \end{array}\right.$$ Does anyone know how to go about this problem? Thanks in ...
0
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0answers
7 views

Proper name for the problem (finding optimal discrete function)

Given a set $D = \{d_1, d_2, ..., d_N\}$, a set of some subsets of $D$, $D^\ast$ and a set of classes, $C = \{c_1, c_2, ..., c_M\}$, I want to find function, that maps a sequence $({d_i}_1^\ast, ...
1
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2answers
46 views

To prove : If $f^n$ has a unique fixed point $b$ then $f(b)=b$

If $f: \mathbb R \to \mathbb R$ be a function such that for some $n_o \in \mathbb N$ , the $n_o$th iterate of $f$ has a unique fixed point $b$ , then how to prove that $f(b)=b$ ? I cant think of ...
0
votes
1answer
29 views

Existence of function satisfying given conditions?

Let $f:[0,1]\longrightarrow[0,1]$ be continuous, strictly increasing and $f(1)=1$. Suppose further that $f(x)>x$ for all $x\in[0,1)$. Is there any function satisfying the above conditions? My ...
2
votes
1answer
46 views

Let $f$ be continuous and $U \subset \mathbb{R}^n$ open, if $f: U \rightarrow \mathbb{R}^m $ is injective then $n \leq m$?

I had intended to restrict the image then $f:U \rightarrow f(U) \subset \mathbb{R}^m $ is bijective. Therefore $\dim f(U) = n \leq m$. That's right?
1
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0answers
40 views

A continuous differentiable map of R to (0;1)

Is there a single, continuously differentiable function $g(x,k)$ that approximates the following: $f(x)= \begin{cases} 0 & x<0 \\ x & 0 \le x \le 1 \\ 1 & x>1\end{cases}$ $k$ is a ...
1
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3answers
69 views

Why do we define functions to be set theoretic objects?

Why do we define functions to be set theoretic objects? Functions are so intuitive, why do we define it in complicated set theory language?
9
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4answers
246 views

Why is this proof of $\mathbb{N}\times\mathbb{N}$ being countable not formal?

My copy of Introduction to Real Analysis: Bartle and Sherbert gives: Theorem: The set $\mathbb{N}\times\mathbb{N}$ is countable. Informal Proof: Recall that $\mathbb{N}\times\mathbb{N}$ ...
0
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5answers
172 views

How to prepare this function for integration

I want to prepare $$f(x)=\frac{x}{1+x^2}$$ for integration, how do i get the $1+x^2$ to the top? Is $$\frac{x}{1+x^2}$$ the same as $\frac x1 + \frac{x}{x^2}$? If not please explain how I prepare the ...
0
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2answers
20 views

Getting to answer on difference quotient/function problem

Q: Find the difference quotient $\dfrac{f(x) - f(3)}{x - 3}$ for $f(x) = \dfrac{1}{x}$ Ans a: $\dfrac{1}{3x}$ Haven't been able to get to that answer. I got the bottom $3x$ right once but the top ...
1
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1answer
55 views

What distribution is this?

Top: Uniform, Bottom: ?? Distribution. Ignore the random spikes - those are just binning errors. Looking for a distribution that is on $[0,1]$ and is equal to $0$ at $1$ and some positive ...
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2answers
25 views

Set of reals in a function [closed]

Need help finding the correct answer of the function.
3
votes
0answers
55 views

Functions that are defined by the equation [closed]

How many different functions of $x$ are defined by the equation $x^2+y^2=9$ if the domain is $x\in [-2,2]$? (A) None (B) 1 (C) 2 (D) 4 Need help finding out how many functions ...
2
votes
1answer
31 views

Election measurable in uniform continuity

Let $f:[0,1]\times [0,1] \rightarrow \mathbb{R}$ borel measurable such that for all $x \in [0,1]$ $f(x,-):[0,1] \rightarrow \mathbb{R}$ is continuous, in particular uniformly continuous. Then there ...
1
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1answer
38 views

Existence of injective function in a manifold with special atlas

I am trying do the following question: Let $M$ be a $n$-dimensional smooth manifold that admits an atlas with only two charts. Show that there exists an injective smooth map ...
0
votes
0answers
40 views

How to interpret the indicator function?

I am reviewing a paper titled " Bayesian Sampling Approach to Decision Fusion" by Biao Chen and Pramod K Varshney. This paper uses an indicator function that I am not being able understand. The ...
0
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1answer
35 views

Confused about images, reverse images.

I am confused over a seemingly simple practice question which I will post below. I am confused over the concept as well, but this question just helps to show what it is I am not understanding. ...
1
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3answers
27 views

Search for two Real Valued functions.

Can we have two real valued functions $f_1$ and $f_2$ defined on $[a,b]$ such that $f_1(x)=f_2(x)$ for infinitely many points and $f_1(x)\neq f_2(x)$ for infinitely many points. ?
0
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1answer
32 views

What is the difference between a bijection and a reversible transformation?

I was reading http://arxiv.org/abs/quant-ph/0101012v4 and one of the axioms is that there needs to be a continuous reversible transformation between states. What is the difference between that and a ...
0
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2answers
46 views

Converting a set to a tuple?

Okay, so, let's say I have a set: $\{0,1,2,3\}$ And I want to convert it to a tuple: $(0,1,2,3)$ How would I do this? Would it be as simple as: $f(\{0,1,2,3\}) = (0,1,2,3)$ ??
1
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5answers
52 views

Finding the range and domain of $f(x)=\tan (x)$

I am attempting to find the range and domain of $f(x)=\tan(x)$ and show why this is the case. I can seem to find the domain relatively well, however I run into problems with the range. Here's what I ...
1
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3answers
29 views

Find that the given linear transform is a isomorphism

I'm studying Linear Algebra and I'm having trouble demonstrating that a function is a isomorphism, that is: "Given the linear transform $T: V \rightarrow W$, $T$ is a isomorphism if and only if it is ...
0
votes
0answers
28 views

Tensor Product of Hilbert Spaces: incomplete?

Let $\mathcal{H}$ be an infinite dimensional Hilbert space and $\mathcal{H}\otimes_0\mathcal{H}$ its algebraic twofold tensor product. Define a scalar product on it as ...
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votes
0answers
27 views

Question of set theory [duplicate]

Suppose That A is a set that at least have 2 element prove that exist a function form A to A that f is 1-1 and onto that for any x is an element of A,f(x) is not equal with x.
0
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1answer
14 views

Intersection of 2 Indicator Functions

Let $E$ and $F$ be events. Let $I_E(\omega)= \left\{\begin{array}{cc} 1, & \omega\in E, \\ 0, &\omega\in E^C. \end{array}\right.$ Show that $I_{E\cap F}(\omega)=I_EI_F$ I found the answer ...
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votes
1answer
29 views

Understanding a definition for vector-spaces

Let $V$, a finite dimensional vector space, and $L$, a subspace of $V$. Let $T:V^*\rightarrow L^*$ defined as: $T(\varphi)(x)=\varphi(x)$ for all $\varphi \in V^*$. Prove $T$ is onto. Well, I'm ...
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votes
2answers
35 views

What is a preimage of domain's subset? [closed]

Let f: A->B be a function. Now let D be subset of A. What is a preimage of D? Is it empty set? There is no typo. The actual question has D as subset of A and E as subset of B. Then you need to ...
0
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1answer
33 views

Finding the range and domain of $h(x) = \sec (x)$

I am attempting to show how to find the range and domain of $h(x) = \sec (x)$. Here's my working so far. Consider $h(x) = \sec (x)$, which is defined as $h(x) = \sec (x)=\frac{1}{\cos(x)}$. We know ...
1
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1answer
20 views

$\ker S$ is not contained in $\ker T$ implies $\dim \Im T \ge 1$

Let $T,S:V\rightarrow W$.where $V$ is a finite vector space above $F$ and $W$ is one-dimensional vector-space above $F$ ($\dim W = 1$). It is given that $\ker S$ isn't contained in $\ker T$. Why is ...
2
votes
2answers
37 views

Finding the best possible $\delta$ for a continuous function.

I am trying to understand the following problem... I understand half of it, but I get confused with something. First of all, I was wondering if there is a relation between $\delta$ and $\epsilon$ ...
3
votes
5answers
228 views

A simple function equation

I come from a programming background and I can’t find a simple math function. The request might seem strange, but I needed it a graphical context to alter some points locations: I need a function ...
0
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3answers
33 views

Showing one to one correspondence

Show that there is a one to one correspondence between the set of left cosets of $H$ in $G$ and the set of right cosets of $H$ in $G$. What is the basic technique/principle for showing one to one ...
2
votes
3answers
132 views

Are the pre-image and the domain the same, or not?

Throughout school I thought that the pre-image was a subset of the domain, not that they were necessarily the same. When I spoke of a function f:R->R, I didn't think that this meant that f was defined ...
2
votes
1answer
55 views

Figuring out when $f(x) = \sin(x^2)$ is increasing and decreasing

Regarding the function $f(x) = \sin(x^2)$, I'm supposed to figure out when it is increasing/decreasing. So far, I've found the derivative to be $f'(x) = 2x\cos(x^2)$. So long as I can solve the ...
6
votes
1answer
144 views

Looking for different proofs of “Discrete Liouville's Theorem”.

Good day. There is a question I have already encountered twice, in very different contexts, that is relatively simple looking, but both solutions I know involve some pretty advanced theorems from the ...
0
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1answer
25 views

Functions and Relations - Help!

Given that : $$\begin{align} &f: D_1 \rightarrow \mathbb{R} \\ & g: D_2 \rightarrow \mathbb{R} \end{align} $$ Find, $f + g : D_1 \cap D_2 \rightarrow \mathbb{R} $.
2
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8answers
699 views

What are the most important functions every mathematician should know? [closed]

I am an undergrad in math and was wondering, what are for you the most important functions every mathematician should know? At the moment I think ...
7
votes
6answers
1k views

What do I not understand about one-to-one functions?

Firstly, a definition: Definition 1: A function $\phi : X \rightarrow Y$ is one-to-one if $\phi(x_1) = \phi(x_2)$ only when $x_1 = x_2$. Now the question: Students often misunderstand the ...