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

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1
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2answers
50 views

Is the probability of the union of events nondecreasing in the probability of the events?

Can it be shown that the probability $P(A_1 \cup \dots \cup A_n)$ is nondecreasing in the probability of any event $A_i$? This fact seems intuitive to me, independent of the fact whether $A_1, \dots, ...
-1
votes
2answers
44 views

$f(x) \geq g(x) \Leftarrow \lim_{x \rightarrow \infty}\frac{g(x)}{f(x)}=0 $?

I want to know if function $f(x)$ is greater or equal than $g(x)$. If I prove that $\lim_{x \rightarrow \infty}\frac{g(x)}{f(x)}=0$ then is it so?
2
votes
1answer
47 views

Finding the constant for a quadratic. Two methods; which one is correct and why?

The question reads $kx^2 + (k+2)x - 3 = 0$ has roots which are real and positive. Find the possible values k might have. Now, since it has real and positive roots, the discriminant $\Delta{d} > ...
1
vote
1answer
20 views

Inverting the equality which contains the operation of taking integer part

I was recently presented with the following equality $$ n = \left[\frac{w}{2d+a}\right]\cdot \left[\frac{h}{2d+b}\right] $$ where all participating variables are non-negative integers, and $[\ldots]$ ...
0
votes
0answers
14 views

Big-Oh, Big-Omega, Big-Theta determination

I am given a recurrence relation and told to solve it. Once we solve it we are supposed to determine whether it is in $O(f(n)), \Omega(f(n))$, or $\Theta(f(n))$. The relation is $t_n = 2nt_{n-1}$. ...
1
vote
2answers
50 views

Is there any way to differentiate such function?

Let $S$ be a set. If I had a bijection $f$ mapping each element $n\in \mathbb{N}$ to an element $s \in S$ such that: $$s = f(n) = \sum^{n}_{k=1} {1\over k}$$ Is the function differentiable in ...
4
votes
2answers
58 views

Is there a difference between $f(x,y)$ $f(x;y)$ and $f(x\mid y)$?

While reading I have come across all three of these notations seemingly at random, and as far as I can tell they are all positional arguments to a function, but I can't tell if they mean different ...
1
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2answers
42 views

Does the inverse of $f(x)=x^3$ have a non-negative domain to have a real output?

I'm not familiar with complex analysis. While playing with Mathematica (a mathematics software), I found that it keeps spitting out unexpected results, and the reason was that it considers differently ...
0
votes
0answers
8 views

Does it make sense to define continuity and monotony on Monte-Carlo simulations?

Suppose a continuous and monotone function $f:\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ to be given. So, in the general case, if I slightly change parameters $a$ and $b$, the function ...
0
votes
0answers
15 views

Has the following concept for integer valued functions been studied

Let $[a] = \{1,2,..a\}$, $[b] = \{1,2,..b\}$ and $F(a,b)$ be the set of all functions from $[a]$ to $[b]$. I define a subset $S \subset F(a,b)$ to be $k$-distinct if each pair of functions in $S$ ...
1
vote
2answers
27 views

Limit of $h(x)= \frac{f(x)}{g(x)}$

If both $f(x)$ and $g(x)$ make : $$\lim\limits_{x \to 1} \frac{f(x)}{g(x)}= 1 $$ and the serieses $$\lim\limits_{n \to \infty} a_n= 1$$ $$\lim\limits_{n \to \infty} b_n =1 $$ $$\implies ...
2
votes
1answer
13 views

Quasi-homogeneous smooth functions vs. polynomials

I am dealing with the following problem. Assume for simplicity that we are dealing with function of two variables only $f = f(x,y)$. Let $|x|,|y| \in \mathbb{Z}$ be two non-zero integers, called ...
3
votes
3answers
38 views

Is $f(x)=x^{3}+3x^{2}+12x-2\sin x $ one-one and onto?

For linear or simple quadratic equations, it is quite simple to check if the function is onto or not. But I often face questions like the one I posted above, to check whether they are one-one and ...
0
votes
0answers
22 views

Proof for bounding a function in two variables, one real and one integer

I would like to proof that the function $f(x,k)=2xk^{-4x^2}$, where $x$ is a real variable and $k$ is an integer variable, is always smaller than $1$ for all $k>2$ and all $x \ge 0$. This is my ...
1
vote
1answer
12 views

Can you stretch a function with a zero or undefined gradient?

If $y=f(x)$ is either $y=3$ (zero gradient) or $x=2$ (undefined gradient), is it possible to stretch $y=f(x)$ by graphing $y=af(x)$ or $y=f(ax)$? If it is possible to stretch them, can you only ...
1
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0answers
66 views
0
votes
2answers
17 views

Disjunctive Normal Form with Minimum variables

I am trying reduce this DNF function to minimal variables. $f(a,b,c,d)=(ac’+c)(a’bc+d’)+(cd’+b)(cd’+c)+abd’+abc’d$ I have reduced to $ac'd+bc+cd'+abc'$ but I know it can be reduced down to $ab ...
0
votes
1answer
30 views

Monotonicity of the sum/product/max of two monotone functions

Suppose two monotone functions $f$ and $g$ (both weakly increasing or both weakly decreasing) are given. How can it be shown that f+g, f*g, max(f,g) is again monotone (either weakly increasing or ...
1
vote
1answer
36 views

How to find $f(2)+f^{-1}(5)$ if $f(2x^2+3x+4)=6x^2+9x+20$? [closed]

$$f(2x^2+3x+4)=6x^2+9x+20$$ How to solve $f(2)+f^{-1}(5)$ ? Any help or advice on solving is much appreciated. Thanks!
0
votes
0answers
11 views

Matrix notation: How would you apply a function to every column/row of a matrix?

Let's consider a real matrix A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ ...
0
votes
1answer
18 views

The function $y=f(x)$ has the property that the chord joining any two points $A(x_1,f(x_1)),B(x_2,f(x_2))$ always intersect $y-$axis at $(0,2x_1x_2)$.

The function $y=f(x)$ has the property that the chord joining any two points $A(x_1,f(x_1)),B(x_2,f(x_2))$ always intersect $y-$axis at $(0,2x_1x_2)$.Given that $f(1)=-1$.Find ...
1
vote
1answer
94 views

Can this equation be solved without numeric calculation? [closed]

I want to know the function $f(x)$ which is shown below the integral equation. Can this equation be solved without numeric calculation? Then $\alpha \in \mathbf{R}$ is a scalar. $$ f(x) \cdot \int ...
0
votes
4answers
99 views

Does there exist the function $f(x)$?

Does there exist the function $f(x)$, satisfying all of the following conditions: (a) $f(x+y)=f(x)+f(y)$ for all real $x$ and $y$. (b) $f(1)=\sqrt 2, f(\sqrt2) =1$. (c) $f(x)$ is bounded on the ...
1
vote
1answer
23 views

Surjectivity and the non-existence of maps.

This question comes from Jacobson's Basic Algebra. It asks: Show that $S \overset{\alpha}{\to} T$ is surjective iff there exist no maps $\beta_1,\beta_2$ of $T$ into a set $U$ such that $\beta_1 ...
1
vote
3answers
70 views

Is $f(x) = \frac{5x^2}{1+x^2}$ bounded?

Show that $$f(x) = \frac{5x^2}{1+x^2}$$ is a bounded function? I know that if $x=0$ the function is undefined, but how can you prove that it is bounded? Help is much appreciated!
1
vote
1answer
13 views

Log transformations of function domain and inequalities

If I know that for some function $f$, the following is true for $x, y \geq 0$: $f(\log (x^a y^b)) \leq f(\log x)^a f(\log y)^b$ Can I make the claim that $f(x^a y^b) \leq f(x)^a f(y)^b$ If I ...
-2
votes
1answer
29 views

Finding the inverse of a function $f(x_1,x_2) = x_1/x_2$? [closed]

I'm trying to understand the inverse of the function $$y_1 = \frac{x_1}{x_2} \to x_1 = \sqrt{y_1y_2}$$ and $$y_2 = x_1 x_2 \to x_2 = \sqrt{\frac{y_2}{y_1}}$$
1
vote
0answers
14 views

Limit of convergent sequence of contraction maps

Let $f_n$ a sequence of contractions on a metric space $(Y,d)$, with a Lipschitz constant $0<\lambda<1$. Suppose that for all $y\in Y$ the sequence $f_n(y)$ converges to $f(y)$. Then $F$ is also ...
1
vote
2answers
37 views

Absolutely Continuous and Continuously Differentiable

Suppose $f(x)$ is absolutely continuous and hence differentiable almost everywhere. Is it true that there are regions where $f$ is continuously differentiable?
0
votes
0answers
20 views

Convex Analysis - How To Find Non Convex Set

I have a problem regarding the following exercise (I considered to put this question on mathematica.stackexchange, but I changed my mind and though this was the right place for this particular ...
4
votes
1answer
92 views

Proving nonexistence of sequence of continuous functions convergent pointwise to Dirichlet function (definition only)

A fellow member of the community asked: "there isn't a sequence of continuous function on $[0,1]$ that converges pointwise to the function $f$ on $[0,1]$ defined by $f(x)=0$ if $x$ is rational and ...
5
votes
3answers
123 views

What is the period of $f(x) = \sin^4(x)+ \cos^4(x)$?

This is an elementary problem but I'm just not getting the right answer. My reasoning is as follows: The period of $g(x) = \sin^4(x)$ is $\pi$ and that of $h(x) = \cos^4(x)$ is $π$ as well, so the ...
-1
votes
0answers
22 views

Can these summations ever be equal? [closed]

Let $n, t\neq 0$ be real numbers and $f(n)\neq 0$ be a function of $n$. Is it possible to have $\sum_{n > 1/t} ...
-3
votes
0answers
34 views

Why is it a function? [closed]

Ok. So there is this question asking why are these domains and ranges function and even though I know, I can't explain that in words. Can somebody explain why are these functions?
-1
votes
1answer
65 views

Can such an equation exist? [closed]

$$y(x)=\lim_{h\to 0}\tan\left(\frac{2\pi}{h}x\right)$$ $$L(z)=y(x)z+c$$ I found such an example in a strange maths book in the dusty section of the library. It said this equation produces rotating ...
0
votes
1answer
25 views

Part of a sigmoid function?

I revised a sigmoid function to use in my research. The function looks like this. $$ f(x) = 0.4 \cdot \frac{1}{1 + e^{-5x}}+ 0.3 $$ where $ x \in [-1,1] $. Is there a specific name to refer to this ...
0
votes
0answers
14 views

Functions to represent set operations?!

Assume you have set of real positive numbers $a_1,...,a_n$. And a strictly decreasing convex function $f$. Assume the intervals $A_i = [0,f(a_i)]$ to represent $i^{th}$ set, $i = 1,...,n$. Can we ...
1
vote
0answers
22 views

N-th root of functions

Given $n \in \Bbb Z$ and a function $g(x)$, is it possible to find an $f(x)$ such that $f^n(x) = g(x)$ where $(a \circ b)(x) = a(b(x))$ given such a function exists? Examples: Given $$g(x) = ...
0
votes
1answer
78 views

Function such that $f(a, b) = c$, but even if I knew $c$ and $b$ I cannot (practically) find $a$? [on hold]

I need a function where $f(a, b) = c$. a,b,c are all positive integers. But even if you knew $b$ and $c$ you cannot practically discover $a$ or narrow $a$ down to fewer than ~1 billion ...
0
votes
3answers
82 views

Tricky Integration And Functions Question

If there is a functions $f(x)$ such that $$ f(x) = x+\int_0^{\frac{\pi}{2}} \sin(x+y)\cdot f(y) \, dy $$ I tried doing it but it seems to get more and more complex as I proceed. Find $f(x)$ Thanks
0
votes
2answers
27 views

Continuous Density Function and Cumulative Distribution Function Question

Suppose you choose, at random, a real number $X$ from the interval $[2, 10]$. Find the density function $f(x)$ and the probability of an event $E$ for this experiment, where $E$ is a subinterval $[a, ...
17
votes
6answers
2k views

Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$.

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$. I do not understand how to go about completing this problem or even where to start.
0
votes
3answers
54 views

What does y=y(x) mean?

In many diciplines that utlizes mathematics, we often see the equation $$y=y(x)$$ where $y$ might be other replaced by whichever letter that makes the most sense in context. My question is what does ...
-1
votes
0answers
15 views

How to find $\text{min}_{b \in R} \ f(b)$ for $f(b) = \text{sup}_{x \in R} \left| \ \sin(x) + \frac{2}{3+ \sin(x)} +b \right|$? [closed]

Let $f(b) = \text{sup}_{x \in R} \left| \ \sin(x) + \frac{2}{3+ \sin(x)} +b \right|$. How to find $\text{min}_{b \in R} \ f(b)$?
0
votes
1answer
46 views

Show that f is continuous at 0, but discontinuous at c for all c/=0.

I showed that f is continuous at 0 by simply using the definition of continuity. However I have no idea how to prove the second party that f is discontinuous at c for all c/=0. Thanks
1
vote
1answer
19 views

Roots of polynomials combined with Trigonometric Functions

If $$ f(x) = x^2 + ax + d \cos x $$, where $a$ is an integer and $d$ is a real number, what are all possible values of the tuple $(a,d)$ such that $f(x)$ and $f(f(x))$ have the same set of real roots? ...
2
votes
1answer
37 views

Show that for any subset $C\subseteq Y$, one has $f^{-1}(Y\setminus C) = X \setminus f^{-1}(C)$ [duplicate]

Let $f: X\rightarrow Y$ be a map Show that for any subset $C\subseteq Y$, one has $f^{-1}(Y\setminus C) = X \setminus f^{-1}(C)$ In this case $f^{-1}$ refers to preimage I started off with trying ...
1
vote
2answers
24 views

How to find all relations of a set and determine which of them aren't functions?

Given the following question: "How many relations are there on {2, 3}, that aren't functions from {2, 3} to {2, 3}?" The answer gives 16 relations, of which 12 aren't functions. How did they ...
19
votes
4answers
7k views

Overview of basic results about images and preimages

Are there some good overviews of basic facts about images and inverse images of sets under functions?
2
votes
1answer
33 views

Proof of $f^{-1}(B_{1}\setminus B_{2}) = f^{-1}(B_{1})\setminus f^{-1}(B_{2})$

I want to prove the following equation: $$ f^{-1}(B_{1}\setminus B_{2}) = f^{-1}(B_{1})\setminus f^{-1}(B_{2}) $$ Is this a valid proof? I am not sure, because at one point I am looking at $f(x) \in ...