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

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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
votes
4answers
250 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
votes
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
votes
2answers
21 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
vote
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 ...
-2
votes
2answers
28 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
32 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
vote
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
votes
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
vote
3answers
28 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
votes
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
votes
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
vote
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
vote
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 ...
-2
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
votes
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 ...
-1
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 ...
-1
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
votes
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
38 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
votes
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
146 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
votes
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
votes
8answers
701 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 ...
1
vote
0answers
63 views

Find function such that $\displaystyle f(1)=10 \ , \ f'(0)=-2$ and $f(x) >0 \ \ \forall x \in \mathbb{R}$

I'm trying to find a function under the following conditions: $f(1)=10$ $f'(0)=-2$ $f'(x)$ is monotonically decreasing. I want to find a function such that $\displaystyle f(x)>0 \ \forall x \in ...
1
vote
2answers
63 views

Can all functions be expressed in terms of elementary functions?

After being introduced to the non-elementary function through an attempt to evaluate $\int x \tan (x)$, an interesting question occurred to me: Can the non-elementary functions be decomposed to ...
1
vote
0answers
22 views

Characterizing a function regarding symmetry

Let us suppose a function $f \colon \mathbb{N} \times \mathbb{N} \to \mathbb{R}$, such that $$\neg\left(\forall a,b \,|\, a \in N \land b \in N \implies f(a,b)=f(b,a)\right)$$ That is $$\left(\exists ...
1
vote
1answer
55 views

finding exact value of $\sec^{-1} 5$

Find the exact value of $\sec^{-1} 5$ (decimal answer). I know that $\sec^{-1}5=\cos^{-1}\dfrac{1}{5}$, but I don't know how to proceed from here. I drew a right triangle with sides $1$ and $5$ ...
1
vote
1answer
25 views

Derive property from continuity - is this proof valid?

Prove that if $f:R^+ \rightarrow R^+$ is continuous on the positive reals and is decreasing, then for all $a$ there exists an $\eta > 0$ such that $(a-\eta)f(a-\eta) > \frac{1}{2}a*f(a)$. EDIT ...
1
vote
1answer
50 views

Find $\lim_{x\to-1} f(x)$ for $f(x) = (x^2 - 2x - 3) / (x+ 1)$

I need to find the following limit: $$\lim_{x\to -1}\frac{x^2 - 2x - 3 }{x + 1}$$ The polynomial is simplified to $\dfrac{(x+1)(x-3)}{x+1}$ Hello, I can solve this by plugging in the value $-1$ ...
0
votes
2answers
45 views

Period of $\frac{\sin(Ny)}{sin y}$ with $N$ odd?

The function $$f(y) = \displaystyle \frac{\sin(Ny)}{\sin y}$$ is periodic with period $2 \pi$ in general. But tracing the graphic of that function for $N$ odd it seems that for $0 \leq x < \pi$ ...
0
votes
2answers
30 views

Values of $a$ for which range of $y=\frac{x+1}{a+x^2}$ contains the interval [0,1]?

Question: For what values of $a$ does the range of $y=\frac{x+1}{a+x^2}$ contain the interval [0,1]? This is how I did it: Cross multiplying and making the discriminant of the quadratic in $x$ to ...
12
votes
2answers
83 views

Function such that zeros$=$order of the derivative

Does there exist a function $f\in C^n(\mathbb{R},\mathbb{R})$ for $n\ge2$ such that $f^{(n)}$ has exactly $n$ zeros, $f^{(n-1)}$ has exactly $n-1$ zeros and so on ? Where $f^{(n)}$ is the nth ...
0
votes
4answers
74 views

Can a limit of a function be not an integer?

I'm just taking calc, and all the teacher's examples gave only integer results. Is it possible to have fractions or decimals?
0
votes
2answers
39 views

Prove that $f(m,n) = 2^m(2n +1 ) -1 $ is a bijection

Basically this proves that set of natural numbers is equinumerous to its cartesian product with itself. f I have tried proving injectivity and surjectivity.Here is what I have done so far. To prove ...
2
votes
2answers
111 views

$f\left(\frac{1}n\right)=\frac{n^2}{n^2+1}$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ of class $C^\infty$ $\forall n\in\mathbb{N}^*,f\left(\frac{1}n\right)=\frac{n^2}{n^2+1}$ Let $p\in\mathbb{N}^*$ What is the value of $f^{(p)}(0)$ ? (by ...
0
votes
0answers
9 views

Guessing next K values of a function

Say we have sampled a function in a constant rate and recieved $x_1,...,x_n$ then we are requested to guess the next $k$ values $x_n+1,...,x_n+k$, is it a known problem? is there a known algorithms or ...
3
votes
1answer
42 views

Fourier transform of $te^{-t^2}$?

How can I find the Fourier transform of: $$f(t) = te^{-t^2}$$
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0answers
13 views

mid-point convex but not a.e. equal to a convex function

I recently stumbled upon Wikipedia's page on convexity (http://en.wikipedia.org/wiki/Convex_function#Properties) and there's reference to Sierpniski's theorem from which we can deduce that for ...
0
votes
0answers
27 views

hessian postiv, but no minimum

I have a problem with a little instance: $f(x,y) = \begin{cases} (x^4-3x^2y^2+y^2)/(x^2+y^2) & otherwise \\ 0 & \text{(x,y)=(0,0)} \end{cases}$ This is a example of a function which ...
1
vote
2answers
48 views

Example of a smooth 'step'-function that is constant below 0 and constant above 1

I need an infinitely smooth non-decreasing function $\ f(x)$, that $$f(x)=0\quad\forall x\leq 0,$$ $$f(x)=1\quad\forall x\geq 1,$$ and all its derivatives in $x=0$ and $x=1$ are $0$. I found that I ...