Elementary questions about functions, notation, properties, and operations such as function composition.
1
vote
1answer
46 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} ...
4
votes
1answer
45 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
59 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
31 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
128 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
188 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
24 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
63 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
107 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
134 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?
$$ ...
1
vote
3answers
63 views
How many functions $ f: \{1, 2, 3, \dots, 10\} \to \{0,1\}$ satisfy $f(1) + f(2) + \dots + f(10) = 2$?
How many functions $ f: \{1, 2, 3, \dots, 10\} \to \{0,1\}$ have this property: $$f(1) + f(2) + \dots + f(10) = 2.$$
I understand just $2$ functions can be $1$, the rest have to be $0$, in total ...
7
votes
8answers
132 views
Why is it important to have a discrepancy between image and codomain?
A function $f:\mathbb{R}\rightarrow \mathbb{R}$ given by $f(x)=x^2$ have $\mathbb{R}^+$ as it codomain and $\mathbb{R}$ as it's image.
What's the need of this discrepancy? Why don't we just write ...
0
votes
0answers
39 views
Let $f,g :\mathbb{R}^n \to \mathbb{R}$, such that $g(x) = f(x) + (f(x))^5$. If $g \in C^k$ then $f \in C^k$.
Someone can help me on this question ?
Section on the implicit function theorem.
0
votes
1answer
27 views
What does this dollar sign over arrow in function mapping mean?
In a certain function mapping like this,
$x \xleftarrow{\$} \{0,1\}^k$
(Lecture Notes on Cryptography by
S. Goldwasser and M. Bellare, page 18)
I fail to understand what exactly does this \$ sign ...
0
votes
0answers
17 views
Marginal Pdfs for Continuous Random Variables
http://oi42.tinypic.com/ddyjph.jpg
this problem is confusing me, i know how to start it, we need to find $f_Y(y)$ so we integrate with respect to x and i get $-2e^{-x}e^{-y}|^y_0$ which then should ...
0
votes
2answers
20 views
A question about function
If $x^6 =64$ and $$\left ( \frac{2}{x}-\frac{x}{2} \right )^2 =b,$$ then what function $f$ satisfies $$f(b+1)=0?$$
1
vote
4answers
35 views
Range of a function
Find the range of
$$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)}$$ where $a, b, c$ are distinct real numbers such $a\neq b\neq c\neq a$.
1
vote
2answers
35 views
Solving the domain and range of a region satisfying two inequalities?
The question I was provided was:
"Find the domain and range of the region satisfied by the following inequalities:
i) $y \ge (x-1)^2$
ii)$y \le2x+1$
Any help would be greatly appreciated. Would you ...
1
vote
1answer
26 views
Question on basic functions?
I was having trouble with this question:
"If $f(x+3) = (x-1)^2 + 4$, find $f(a-1)$
I think this is simple, but I've completely forgotten what to do. :P
Thanks!
5
votes
3answers
51 views
Open, closed and continuous
I have some troubles to understanding something:
We were asked to find a function that is open and continuous but not closed and actually I found such a function, but our tutor gave us this example
...
3
votes
3answers
58 views
Are the graphs of these two functions equal to each other?
The functions are: $y=\frac{x^2-4}{x+2}$ and $(x+2)y=x^2-4$.
I've seen this problem some time ago, and the official answer was that they are not.
My question is: Is that really true?
The functions ...
0
votes
1answer
34 views
How to make a unit step function?
I am trying to make a unit step function.
I have this function (the equation of an ellipse, not centered at the origin):
$$
f(x,y) = \frac{(x-X_c)^2}{a^2}+\frac{(y-Y_c)^2}{b^2}
$$
What I would ...
-1
votes
0answers
23 views
Flux integrals, parameterization
let S be the cylinder x^2 + z^2 = 9 where -2 /ge y /le 2
parameterization: thi(u,v)= <3cosv, u, 3sinv> where -2 /ge y /le 2 and 0 /ge v /le 2pi
(thi is the symbol of I with the circle in the ...
0
votes
1answer
33 views
A particular weak subadditivity
Given a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, consider the following property.
For all $(x^1, ..., x^n) \in \left(\mathbb{R}^n \right)^n$ such that $f(x^i) \geq 0$ $\forall i \in [1,n]$, ...
0
votes
0answers
19 views
How can we define the closedness and boundedness in continuous functions space?
How can we define the closedness and boundedness in continuous functions space?
I'll be very happy if you help me..
1
vote
1answer
43 views
Given $f(x) =x^2+bx+c$, $f(1) = 2$, and $f(-1) = 12$, how do I get $b$ and $c$?
Is there a different way to get $b$ and $c$ values and then the value of $f(2)$?
Here's how my book does it:
Given that
$$f(x) =x^2+bx+c, \qquad f(1) = 2,\qquad f(-1) = 12$$
we see
$$
f(1) = 2 ...
10
votes
5answers
353 views
What's the difference between arccos(x) and sec(x)
My question might sound dumb, but I don't really see why the graphics of arccos(x) and sec(x) are different, because as far as I know arccos is the inverse cosine function (cos(x)^-1) and sec equals ...
4
votes
4answers
96 views
what is the relation of smooth compact supported funtions and real analytic function?
What is the major difference between real analytic and test function (smooth compact supported functions). Can we find a real analytic function $f$ on $R^n$ which is also smooth compact supported? If ...
5
votes
2answers
75 views
Continuous function that is only differentiable on irrationals
Can you help me finding a function $f : \mathbb{R} \rightarrow \mathbb{R}$ that is continuous in $\mathbb{R}$ and differentiable at $x$ iff $x \notin \mathbb{Q}$ ?
Thank you very much !
4
votes
1answer
43 views
Iterated function?
$$f(n) = \frac{n}{\lg n}$$
$$g(n) = \min (i \ge 0: f^i(n)\le 2)$$
In other words, $g(n)$ is the number of times $f(n)$ needs to be iterated to reduce $n$ to 2 or less.
What's a tight bound on ...
1
vote
3answers
59 views
How can you show that $\delta′=f(0)\delta′−f′(0)\delta$ for a function f that is infinitely differentiable?
Assume that $f$ is infinitely differentiable. Let $\delta$ be the (Dirac) delta functional.
I know that $f\delta = f(0)\delta$, but I'm not sure how to derive the equation ...
0
votes
0answers
21 views
Multiplication and Division of functions
Suppose that you have two continuous functions, $f(x)$ and $g(x)$.
Suppose that you have numerical approximations for these functions, stored a vectors, $f^*$ and $g^*$.
If I want to approximate ...
3
votes
1answer
25 views
Preimages of a function: Is the following proposition true or false?
Let $g: ℤ \times ℤ → ℤ \times ℤ$ be defined by $g(m,n) = (2m, m – n)$.
Is the following proposition true or false? Justify your conclusion.
For each $(s, t) ∈ ℤ \times ℤ$, there exists an $(m, n) ∈ ℤ ...
3
votes
2answers
39 views
$f(g(x))=x$ implies $f(x)=g^{-1}(x)$
Is it possible to find a necessary and sufficient condition to conclude when
$$f(g(x))=x \implies f(x)=g^{-1}(x) \wedge f^{-1}(x)=g(x),$$
if both functions are well defined?
1
vote
2answers
44 views
Conditions for f(x,y) to coincide with some g(x) + h(y)
Given a function of two variables, say f(x,y), how to know/check whether it can be simplified into some g(x) + h(y)?
Some property of some condition that condition that f(x,y) satisfies to know ...
5
votes
2answers
54 views
Inverse and derivative of a function [duplicate]
Find an example of an inverse function f(x) such that its derivative is the same as its inverse.
I tried many different functions but non of them worked.
4
votes
2answers
129 views
Is $y=|x^3|$ a smooth function?
Is this a smooth function? $y=|x^3|$
The graph of this function has no sharp cuts or corners, so I think it is a smooth function but someone told me that it's not.
7
votes
2answers
139 views
$\cos(x)+\cos(x\sqrt{2})$ is not periodic
Show that the function $$f(x)=\cos(x)+\cos(x\sqrt{2})$$ is not periodic. I tried $x = a$ and $a\sqrt{2}$. I am guessing that the method of contradiction would be of some help over here. What else ...
-3
votes
0answers
37 views
Give an example of a complete analytic function.
Give an example of a complete analytic function for which over a point a = 0 is the branching order of 2 and 4, as well as a significant feature.Exactly I dont know how to do it.We know just we have: ...
0
votes
0answers
36 views
Help me prove the supremum property.
Let $A$ and $B$ be nonempty sets and $f$ be a function from any nonempty set $S$ to subset of real number. Prove that
$$\sup_{x \in A} \{ \min \{ \sup_{y \in B} \{ \min \{ f(y) \} \}, f(x) \} \}= ...
1
vote
1answer
43 views
is this function injective?
Is this function injective?
$f:\mathbb{N}\to \mathbb{N\times N}$ defined as
$f:n\to (n, n+1)$
$f(n_{1})=f(n_{2})\Rightarrow (n_{1},n_{1}+1)=(n_{2},n_{2}+1)\Rightarrow$
$n_{1}=n_{2} \wedge ...
29
votes
4answers
752 views
When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?
The question is:
When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?
The most obvious solution is a linear function of the form $f(x)=ax+b$. Is this the only solution?
Edit
I should ...
0
votes
1answer
30 views
prove $f^{-1}(B)=A$
I am given $A_1$, $A_2 \subseteq A$ and $B_1$,$B_2 \subseteq B$. and the function $f: A \rightarrow B$
I want to prove that $f^{-1}(B)=A$.
I just assume that here one is talking about ...
2
votes
2answers
30 views
Proving the coefficient of Power series is “0” always on given condition.
Suppose the power series $P(x) = \sum_{n=1}^\infty b_n x^n$ converges for $|x| \leq 1$ and that for some $c>0$ it is given that $$P(x)=0 \quad \forall x \;\text{such as}\;|x| < c$$ Show that ...
0
votes
1answer
41 views
Proof: $f: A \to B$, if $f$ is bijection then $\forall a\in B \exists ! b\in A (f(b) = a) $
I musti proof the following:
" let $f: A \to B$, if $f$ is injection then $\forall a\in B \exists ! b\in A (f(b) = a)$"
proof: by contraddiction, therefore the negation of $\forall a\in B \exists ! ...
1
vote
2answers
28 views
Proving a condition on a cont-differentiable function on positive real numbers.
Let f be a continuously differentiable function on [0,infinity) such that $f '(x) \le f(x)$ for all $x$.Suppose $f(0)=5$. Show that $f(x) \le 5e^{x} ~~\forall x$.
I am not getting how we will ...
2
votes
0answers
27 views
Inferring simplest method to convert bit array 1 to bit array 2.
Consider the set of all bit arrays of length $n$. Now consider the set of all 1-to-1 functions that map from this set to this set.
Now select a single function out of the latter set. Is there any ...
