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

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0
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3answers
54 views

Show that $\frac{1}{1+k}=\frac{\frac{1}{k}}{1+\frac{1}{k}}\leq \ln(1+\frac{1}{k})\leq\frac{1}{k}$

Prove the following: $$\frac{1}{1+k}=\frac{\frac{1}{k}}{1+\frac{1}{k}}\leq \ln(1+\frac{1}{k})\leq\frac{1}{k}$$ I know I can prove it with induction if the values were naturals. However, the "problem" ...
0
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0answers
16 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
51 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 ...
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 $S$ ...
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
vote
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
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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 ...
1
vote
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
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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 ...
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 ...
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 ...
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 ...
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
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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 \\ ...
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
1answer
19 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
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
1answer
95 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 ...
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 ...
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 ...
-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 $$y_2 = x_1 x_2 \to x_2 = \sqrt{\frac{y_2}{y_1}}$$
0
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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 ...
1
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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
1answer
31 views

Find the equation of a non-linear relation given 2 points

So I ask my question, let me just begin by stating that I'm in grade 9, and have decided to start learning calculus to aid me in the development of an undisclosed project that I am working on. Now, ...
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 ...
-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
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} ...
-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
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 ...
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 ...
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
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, ...
0
votes
3answers
55 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
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 $[\ldots]$ ...
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? ...
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 ...
0
votes
2answers
23 views

Show $f^{-1}(A^c)=(f^{-1}(A))^c$ [duplicate]

Let $f: X \to Y$, and $A\subseteq Y$. Show that $f^{-1}(A^c)=(f^{-1}(A))^c$ I know how to prove that $f^{-1}(A^c)\subseteq(f^{-1}(A))^c$, but stuck on proving $(f^{-1}(A))^c\subseteq f^{-1}(A^c)$. ...
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
0answers
46 views

Why this is non negative

Assume you have a strictly decreasing non negative and convex function $f$ , and let $a_i$ for $i \in \{1,2,3\}$, be some positive real numbers, then $$g(t,a_1,a_2,a_3) = \sum_i^3 f(a_i t) ...
0
votes
1answer
24 views

Are the following families of sets closed under intersection?

Problem Statement Let $X$ be any set whatsoever, and let $f:X\to X$ be any function. Note that in general, no structure is imposed on $f$ whatsoever (i.e. continuity, linearity, etc). The problem is ...
1
vote
4answers
44 views

What is the polar formula for $y=x$?

$y=x$ is a basic cartesian equation, but I'm at a loss as to what it is in polar form. It seems the only way I've found to express it is with $r$ on both side of the equation, but is there a way of ...
1
vote
1answer
27 views

Proving P(x) > 0 given a condition.

$P(x)$ is a polynomial function such that, $P(1) = 0, P′(x) > P(x), ∀ x > 1. $ Prove that $P(x) > 0, ∀ x > 1.$ I was trying to do by taking the P(x) in the denominator and then ...
1
vote
2answers
38 views

What is the correct method of finding the leading order behavior of a function in a given limit?

I am kind of confused about finding the correct leading order behavior of a function. Example: If I want to understand the behavior of the following function $$f(x)=\coth(x)-\frac{1}{x}$$ I can ...
2
votes
1answer
26 views

Assume that $f : X \rightarrow Y $ is surjective. Show that $f(A^c) = (f(A))^c \ \forall A\subset X$ iff f is also injective.

Assume that $f : X \rightarrow Y $ is surjective. Show that $f(A^c) = (f(A))^c \ \forall A\subset X$ iff $f$ is also injective. So I tried starting with the right implication $\Rightarrow$ Since ...
0
votes
0answers
46 views

Continuous non-differentable functions

I'm looking for some examples of everywhere continuous functions which are nowhere differentable. I found already Takagi curve and Weierstrass function. Can you point out some online courses or pdf ...
1
vote
0answers
26 views

Meanvalue theorem for quadratic arguments

I have trouble proving the following result and I would be happy about any kind of suggestion how the precise argument looks like. Let $f : [0,\infty) \rightarrow \mathbb{R}$ denote a continuously ...
0
votes
2answers
55 views

Is $x\sin x$ Surjective?

I have to determine whether $x\sin x$ Is a surjective function, in $\Bbb R$. My solution: Let $f(x)=x\sin x$. $f(\frac{\pi}{2})=\frac{\pi}{2}$ $f(-\frac{\pi}{2})=-\frac{\pi}{2}$ Therefore in ...
2
votes
1answer
37 views

$C ([1,2] \times [0,1] \to \mathbb R)$ dense in $C ( [1,2] \rightarrow L^{2} ([0,1] \to \mathbb R))$?

Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense ...