Elementary questions about functions, notation, properties, and operations such as function composition.
0
votes
3answers
18 views
how to use predicate logic to express the statement about function
Given R is defined on $A\times A$ where $|A|=n$. How to use the predicate logic to express the:
i) The relation R corresponds to a function from A to A.
ii)The relation R corresponds to an ...
1
vote
1answer
40 views
Piecewise continuous differentiable
if I have a piecewise continuously differentiable function. How do I see that on each open interval, where the derivative is continuous, there is a continous extension on the larger closed interval?
4
votes
1answer
38 views
Linear transformation invertible or not?
Let $ T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a linear transformation defined such that the inner product of $\langle T(v), v \rangle = 0$ for all $v$ in $\mathbb{R}^2$. Is $T$ invertible or ...
1
vote
4answers
97 views
What is rigorous notation for functions?
I have seen many ways to denote a function: $f(x)=x^2, y=x^2, f: x\mapsto x^2$ and so on. What is exact notation for functions? Please include lethal doses of rigor, set theory, and of course ...
2
votes
2answers
67 views
1
vote
1answer
62 views
find the number of injections/surjections
I've been banging my head on this problem for some time now, and could really use help. Bear in mind I'm not very good at this sort of thing and am struggling to get by in class.
Problem:
Given $ ...
3
votes
4answers
50 views
What type of input does trigonometric functions take in
I see in my Book that 45 deg is equivalent of π/4 . Ι also do the conversion if I simply convert degrees into radians like this
45* π/180 = π/4 radians
and ...
0
votes
0answers
31 views
Can someone help me to decode this function?
I used an exponential decay model in R.
But I do not know the mathematical background of it.
According to the Help Page of R (?EXD.3) the model is:
f(x) = c + (d-c)(\exp(-x/e))
but I do not know ...
0
votes
2answers
47 views
Number of functions
Let $F$ denote the set of all functions from $\{1,2,3\}$ to $\{1,2,3,4,5\}$
a) Find and simplify the number of functions $f$ in $F$ so that $f(1)=4$.
b) Find and simplify the number of one-to-one ...
1
vote
3answers
58 views
Are absolute extrema only in continuous functions?
The Extreme Value Theorem says that if $f(x)$ is continuous on the interval $[a,b]$ then there are two numbers, $a≤c$ and $d≤b$, so that $f(c)$ is an absolute maximum for the function and $f(d)$ is an ...
0
votes
3answers
54 views
Does Euler totient function gives exactly one value(answer) or LEAST calculated value(answer is NOT below this value)?
I was studying RSA when came across Euler totient function. The definition states that- it gives the number of positive values less than n which are relatively prime to n.
I thought I had it, untill I ...
0
votes
1answer
67 views
I don't know where to begin with this functions question (one-to-one, onto)
a) Suppose that $f:\Bbb Z\to \Bbb Z$ is a one-to-one function. Define a function $g:\Bbb Z\to \Bbb Z$ by: for all integers $x$, $g(x)= -f(x)$. Prove that $g$ is also one-to-one.
b) Suppose $f:\Bbb ...
0
votes
0answers
21 views
Inverse Variation Function as Real Life Example
What is an example of the Inverse Variation in real life? Given the function y = a/x'. I've tried to Google it, and look in all our math books, but I can find no examples.
1
vote
3answers
55 views
Prove that a map is Injective
How can I prove that $f: (0, \infty) \times (0,\pi) \to \mathbb{R}^2$ where $f(x,y) = (\sinh(x)\sin(y),\cosh(x)\cos(y))$ is injective?
1
vote
2answers
28 views
Square bracket $[X]$ with finite fields and polynomial rings
I understand that parentheses are used for functional notation. I do not have any confusion about this one.
However, in some literature, I find square brackets ($[X]$) after some finite field ...
2
votes
4answers
37 views
Function generation by input $y$ and $x$ values
I wonder if there are such tools, that can output function formulas that match input conditions. Lets say I will make input like that:
$y=0, x=0$
$y=1, x=1$
$y=2, x=4$
and tool should ...
-1
votes
0answers
38 views
function property
Let $f_{n}(x)$ is diffentiable for all n $\in \mathbb N$ & for all $x\in \mathbb R$ .
If $g(x)$ is function which is obtained by replacing $n$ with $n+1$ in $f_n'(x)$, then $g(x)$ is equal to ...
-6
votes
1answer
37 views
how do I solve for the image of this rotation? [closed]
The function notation $\operatorname{Rp60 degrees}(G)=G'$ describes the effect of a rotation. Which point is the image under this rotation?:
A. Point R
B. Point P
C. Point G
D. Point G'
2
votes
1answer
69 views
Analytically determine that $\arctan x$ is an odd function
Without producing the maclaurin series for $\arctan x$, how would determine whether it was odd or even?
2
votes
4answers
149 views
Is the set of surjective functions from $\mathbb{N}$ to $\mathbb{N}$ uncountable?
I want to use Cantor's diagonalisation argument to prove that the set S of surjective functions of the form $\Bbb{N} \to \Bbb{N}$ is uncountable. The normal procedure is creating a matrix and filling ...
0
votes
1answer
36 views
About continuous functions and aritmethic progression
I've try solve this question, but I haven't sucess...
The problem is the following:
A continuous functions $f:[a,b]\rightarrow \mathbb{R}$ assume positive and negative values in its domain, show ...
5
votes
2answers
50 views
number of all inconstant maps f from A to A [duplicate]
Let $ A=\{1, 2, 3,..., n\}$. Find the number of all nonconstant maps $f: A \rightarrow A$ for which $f(k) \le f(k + 1)$ and $f(k) = f(f(k + 1))$ for $k = 1, \dots, n-1$..
0
votes
1answer
45 views
How to find the primitive function of this integral
I am trying to find the primitive function of $\displaystyle\int_{}^{}\frac{dx}{5+2\sin x-\cos x}$. I've got $$\int_{}^{}\frac{dx}{5+2\sin x-\cos ...
-1
votes
1answer
30 views
Can someone help me to decode these functions? [closed]
I need to use the following functions, however, I don't manage to read them... Can someone help me?
Cedergreen hormesis model:
f(x) = c + \frac{d-c+f \exp(-1/x^{α})}{1+\exp(b(\log(x)-\log(e)))}
...
1
vote
1answer
51 views
Why is ess sup $f$ not ess max $f$?
Consider a measure space $(X,\Sigma\,\mu)$. Given that one can easily prove that, $\mu$-a.e., $f \leq \text{ess} \sup_X f$, why is the notation not simply "$ \text{ess} \max_X f$"?
(Here $\text{ess} ...
5
votes
1answer
47 views
A question regarding the Power set
In the proofs that I have seen so far for showing that the power set $2^X$ of a set $X$ cannot be in bijection to $X$, the common idea is to assume that there exists a surjection $f \colon X \to 2^X$ ...
0
votes
1answer
62 views
Does convergence in $C(\Bbb R)$ imply convergence in $\mathcal D'(\Bbb R)$
Sequence of continuous functions $(f_k)$ which converge to $f$ in $C(\Bbb R)$. Does it imply that $f_k$ will converge to $f$ in $\mathcal D'(\Bbb R)$ as well? Or we need something more to make this ...
1
vote
0answers
18 views
Six notions of closure associated with every binary operation (more generally, with every ternary relation).
Let $X$ denote a set and consider a ternary relation $\phi \subseteq X^3.$ If you're more comfortable thinking of binary operations, just imagine that $\phi(x,y,z) \leftrightarrow x*y=z$ for some ...
1
vote
1answer
33 views
If $f:U\to \mathbb{R}$ is continuous and $(x^2+y^4)f(x,y) + (f(x,y))^3=1$, then $f$ is $C^\infty$
Let $f:U\to \mathbb{R}$ be continuous in $U \subset\mathbb{R}^2$, such that
$$(x^2+y^4)f(x,y) + (f(x,y))^3=1$$
for all $(x,y) \in U$. Prove that $f\in C^{\infty}$.
I'm learning the implicit ...
1
vote
4answers
37 views
Given certain conditions, prove a function G(x) is always equal to 4.
Theres a question I've been having trouble with:
$G(x)$ is the function $|x+2|+|x-2|$. Show that if $-2<x<2$ then $G(x) = 4$.
Any help would be greatly appreciated. No calculus please :D
4
votes
3answers
134 views
How to find inverse of the function $f(x)=\sin(x)\ln(x)$
My friend asked me to solve it, but I can't.
If $f(x)=\sin(x)\ln(x)$, what is $f^{-1}(x)$?
I have no idea how to find the solution. I try to find
...
10
votes
2answers
199 views
Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Similarly, let $f :\mathbb{C}→ \mathbb{C}$ be a function such that $f^2$ and $f^3$ ...
2
votes
1answer
27 views
For the following monic polynomial,$f$ of even degree how to prove that that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$
For any monic polynomial $f \in \mathbb {Q[x]}$ of even degree,how to prove, there exists polynomial $g \in \mathbb {Q[x]}$ such that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$
2
votes
1answer
35 views
Finding a PDF from a function
I have a function $y = f(x),\ x\in\mathbb{R}$ (assume $f(x)= \sin(x)/x$ if you need an example). How can I find the probability distribution function (PDF) of $y$, assuming $x\sim U(\mathbb{R})$ ...
0
votes
3answers
67 views
How to solve these?
Inverse Trigonometric Functions
They are incomplete and I don't know how to complete them.
Who can help me?
1st
$$
\int\frac 1{ x \sqrt{x^{6} - 4}}dx
$$
I tried with:
$$u = x^3 $$
$$du= 3x^2dx$$
...
3
votes
3answers
121 views
How many functions $f:\{1,2,3,4\}→\{1,2,3,4\}$ satisfy $f(1)=f(4)$?
I just need a hint or a way to think a about this problem: $f(1)$ can be $1, 2, 3, 4$ and $f(4)$ can be $1,2,3,4.$
3
votes
3answers
135 views
Function such that $f(x) = -1$ for $x < 0,$ and $f(x)=1$ for $x > 0$?
What is a function to returns $-1$ if number is negative, $1$ if positive, and zero if number is equal to 0?
for example:
$$
f(-8) = -1
$$
$$
f(8) = 1
$$
$$
f(0) = 0
$$
for $$x < 0$$ maybe?
$$ ...
1
vote
3answers
65 views
How many functions $ f: \{1, 2, 3, \dots, 10\} \to \{0,1\}$ satisfy $f(1) + f(2) + \dots + f(10) = 2$?
How many functions $ f: \{1, 2, 3, \dots, 10\} \to \{0,1\}$ have this property: $$f(1) + f(2) + \dots + f(10) = 2.$$
I understand just $2$ functions can be $1$, the rest have to be $0$, in total ...
7
votes
8answers
139 views
Why is it important to have a discrepancy between image and codomain?
A function $f:\mathbb{R}\rightarrow \mathbb{R}$ given by $f(x)=x^2$ have $\mathbb{R}^+$ as it codomain and $\mathbb{R}$ as it's image.
What's the need of this discrepancy? Why don't we just write ...
0
votes
0answers
43 views
Let $f,g :\mathbb{R}^n \to \mathbb{R}$, such that $g(x) = f(x) + (f(x))^5$. If $g \in C^k$ then $f \in C^k$.
Someone can help me on this question ?
Section on the implicit function theorem.
0
votes
1answer
30 views
What does this dollar sign over arrow in function mapping mean?
In a certain function mapping like this,
$x \xleftarrow{\$} \{0,1\}^k$
(Lecture Notes on Cryptography by
S. Goldwasser and M. Bellare, page 18)
I fail to understand what exactly does this \$ sign ...
0
votes
0answers
17 views
Marginal Pdfs for Continuous Random Variables
http://oi42.tinypic.com/ddyjph.jpg
this problem is confusing me, i know how to start it, we need to find $f_Y(y)$ so we integrate with respect to x and i get $-2e^{-x}e^{-y}|^y_0$ which then should ...
0
votes
2answers
21 views
A question about function
If $x^6 =64$ and $$\left ( \frac{2}{x}-\frac{x}{2} \right )^2 =b,$$ then what function $f$ satisfies $$f(b+1)=0?$$
1
vote
4answers
36 views
Range of a function
Find the range of
$$f(x)=\frac{(x-a)(x-b)}{(c-a)(c-b)}+\frac{(x-b)(x-c)}{(a-b)(a-c)}+\frac{(x-c)(x-a)}{(b-c)(b-a)}$$ where $a, b, c$ are distinct real numbers such $a\neq b\neq c\neq a$.
1
vote
2answers
35 views
Solving the domain and range of a region satisfying two inequalities?
The question I was provided was:
"Find the domain and range of the region satisfied by the following inequalities:
i) $y \ge (x-1)^2$
ii)$y \le2x+1$
Any help would be greatly appreciated. Would you ...
1
vote
1answer
26 views
Question on basic functions?
I was having trouble with this question:
"If $f(x+3) = (x-1)^2 + 4$, find $f(a-1)$
I think this is simple, but I've completely forgotten what to do. :P
Thanks!
5
votes
3answers
54 views
Open, closed and continuous
I have some troubles to understanding something:
We were asked to find a function that is open and continuous but not closed and actually I found such a function, but our tutor gave us this example
...
3
votes
3answers
58 views
Are the graphs of these two functions equal to each other?
The functions are: $y=\frac{x^2-4}{x+2}$ and $(x+2)y=x^2-4$.
I've seen this problem some time ago, and the official answer was that they are not.
My question is: Is that really true?
The functions ...
0
votes
1answer
34 views
How to make a unit step function?
I am trying to make a unit step function.
I have this function (the equation of an ellipse, not centered at the origin):
$$
f(x,y) = \frac{(x-X_c)^2}{a^2}+\frac{(y-Y_c)^2}{b^2}
$$
What I would ...
-1
votes
0answers
24 views
Flux integrals, parameterization
let S be the cylinder x^2 + z^2 = 9 where -2 /ge y /le 2
parameterization: thi(u,v)= <3cosv, u, 3sinv> where -2 /ge y /le 2 and 0 /ge v /le 2pi
(thi is the symbol of I with the circle in the ...
