Elementary questions about functions, notation, properties, and operations such as function composition.
0
votes
2answers
14 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
0answers
33 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
22 views
3
votes
0answers
47 views
Functions that are defined by the equation [on hold]
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
27 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
37 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
36 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
30 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.
...
0
votes
3answers
23 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
31 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
43 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
47 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
27 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
26 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
27 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
32 views
What is a preimage of domain's subset? [on hold]
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
29 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
vote
1answer
19 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
36 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
221 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
32 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
131 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
54 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 ...
5
votes
1answer
129 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
24 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
687 views
What are the most important functions every mathematician should know? [on hold]
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
5answers
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
62 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
60 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
20 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 $$\left(\exists ...
1
vote
1answer
53 views
finding exact value of $\sec^{-1} 5$
Find the exact value of $\sec^{-1} 5$ (decimal answer).
I know that $\sec^{-1}5=\cos^{-1}\dfrac{1}{5}$, but I don't know how to proceed from here. I drew a right triangle with sides $1$ and $5$ ...
1
vote
1answer
25 views
Derive property from continuity - is this proof valid?
Prove that if $f:R^+ \rightarrow R^+$ is continuous on the positive reals and is decreasing, then for all $a$ there exists an $\eta > 0$ such that $(a-\eta)f(a-\eta) > \frac{1}{2}a*f(a)$.
EDIT ...
1
vote
1answer
46 views
Find $\lim_{x\to-1} f(x)$ for $f(x) = (x^2 - 2x - 3) / (x+ 1)$
I need to find the following limit:
$$\lim_{x\to -1}\frac{x^2 - 2x - 3 }{x + 1}$$
The polynomial is simplified to $\dfrac{(x+1)(x-3)}{x+1}$
Hello, I can solve this by plugging in the value $-1$ ...
0
votes
2answers
37 views
Period of $\frac{\sin(Ny)}{sin y}$ with $N$ odd?
The function
$$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 ...
11
votes
2answers
78 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
71 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
37 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
109 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
41 views
1
vote
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
25 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
47 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 ...
0
votes
0answers
25 views
Value of Delta Function
Quick Question. Is $\delta _{mn}=1$ when $m\neq n$. and $\delta _{mm}=0$?
I am not very good at Math. So would you give me the answer and explanation please?
0
votes
3answers
61 views
If $f$ is continuous, so is $g=|f|$ [closed]
Prove that if $f$ is continuous, so is $g=|f|$.
I need help on this. Thank you.
Ok, this is my first time here.
The definition of continuity i am using is that $f$ is continuous at $a$ if for any ...
1
vote
3answers
37 views
How to establish these two facts about polynomials?
Let $f(x) := \sum_{k=0}^n c_k x^k $ be a polynomial of degree $n\geq 0$ with real coefficeints such that $f(x) = 0$ for $n+1$ distinct real values of $x$. Then how to prove that each $c_k = 0$ and ...
0
votes
0answers
43 views
Real Analysis - Proving a function is injective
I just need a nudge in the right direction. I know one-to-one (injective) functions but I have never seen it like this:
Let $i\colon \mathbb Z\to \mathbb Q$ be defined by $ i(x)=[(x,1)]$ for all ...
