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

learn more… | top users | synonyms (1)

6
votes
6answers
141 views

Removable discontinuity question

A quick and easy question on terminology: If we have a function $$f(x) = \frac{x(x-2)}{(x-2)}$$ then clearly the function is not defined at $x = 2$. If we cancel out "$(x-2)$" then the function is ...
6
votes
2answers
69 views

shape created by parabola

What would be the name of the shape that is the set of all points such that they are equidistant from the point $(0,1)$ and to the parabola $y=x^2$. Here is a desmos graph that generates the ...
0
votes
1answer
22 views

representation for Banach algebra [on hold]

How we can represent any Banach algebra as a subspace or Subalgebra of Cb(X)?( in isometrically isomorphic concept)
-1
votes
1answer
38 views

Suppose all we know about y=f(x) is that it is continuous for all x and f(4)=5. Which must be true?

Suppose all we know about $y=f(x)$ is that it is continuous for all $x$ and $f(4)=5$. Which must be true? a. $f'(4)=5$ b. Every number x is in the domain of f c. The function is increasing near x=4 ...
0
votes
1answer
15 views

Superfunctions with complex iteration indices: Interpretation

Superfunctions are a fascinating concept, allowing us to generalize functional iteration to arbitrary real and complex iteration indices. We have $$ \begin{equation} \begin{split} S_f(0) & ...
0
votes
1answer
32 views

Limit on a continuous differential equation

Let $f$ be a continuously differentiable function and let $L=\lim_{x\to\infty}(f(x)+f'(x))$ be finite. Does this imply that if $$\lim_{x\to\infty} f'(x)$$ exists, then it is equal to $0$?
1
vote
0answers
32 views

Converse of Euler's homogenous function theorem proof follow up question

So basically I read someone else's answer to a question regarding Euler Homogeneous function theorem source: http://quant.stackexchange.com/questions/8911/what-is-exactly-eulers-decomposition Also ...
0
votes
3answers
135 views

Function with zero description

Is there a nice expression (possibly differentiable outside $0$) for a function $f(x)$ that satisfies the following property other than the delta? $$f(x)=1\iff x=0$$ $$f(x)=0\iff x\neq0$$ Is it ...
1
vote
1answer
50 views

Proving this equivalence relation

If $X,Y$ are reflexive, symmetric, and transitive, then $X \times Y$ is an equivalence relation where ${(a,b):a\in X, b\in Y}$. I am trying to self learn these topics. I do know what an ...
0
votes
0answers
53 views

Prove $X\times Y$ is an equivalence relation

(Relation between two sets) If $X$ and $Y$ are sets, a relation between $X$ and $Y$ is a subset $R \subset X \times Y.$ For a relation $R \subset X\times Y$ and $a \in X$ and $b \in Y$ if $(x,y) \in ...
-1
votes
2answers
52 views

How do I take the 100th derivative of a polynomial [on hold]

How could I find $$f^{100}(x)$$ for $$f(x)=2x^{100}-7x^{80}+15x^{60}-27x^{40}-18x^{20}+300$$
0
votes
2answers
35 views

Is there any standard terminology for this property?

Let $f$ be a map whose domain is $X$. If $f$ satisfies the property that for all $x\in X$, $$f(f(x))=f(x)\text{,}$$ is there any standard name for such a function? Not sure if "projection" is the ...
0
votes
3answers
42 views

Proving by Cauchy's definition $\lim_{x\to -1} x^2+3x-5=-7$

Prove by Cauchy's definition $\displaystyle\lim_{x\to -1} x^2+3x-5=-7$ From definition: $|x+1|<\delta\Rightarrow |x^2+3x+2|<\epsilon \iff |x+1||x+2|<\epsilon$. Now I'm not really sure ...
4
votes
1answer
73 views

Find the inverse function 3

Find the inverse function for the following function: $$f(x) = \log_{\sqrt{4-x^2} } \left(x^3+5x^2-x \right)$$ Thanks.
2
votes
1answer
28 views

Proving by Cauchy's definition $\lim_{x\to 0} x^2\cos x=0$

Prove by definition that $$\displaystyle\lim_{x\to 0} x^2\cos x=0$$ So take $\delta=\sqrt\epsilon$, and from definition we have: $|x|<\delta\Rightarrow|x^2|<\delta^2\Rightarrow|x^2\cos ...
2
votes
2answers
51 views

Proving $\lim_{x\to9}\sqrt x=3$ using Cauchy's definition

Prove: $\displaystyle\lim_{x\to9}\sqrt x=3$ using Cauchy's definition for a limit. After doing the scratch work I get that: $\delta=\epsilon^2+6\epsilon$, so going back, I have to show that ...
0
votes
1answer
27 views

In the three angles, A, B, C of a triangle, angle B exceeds twice angle A by 15 degrees. Express the measure of angle C in terms of angle A.

In the three angles, A, B, C of a triangle, angle B exceeds twice angle A by 15 degrees. Express the measure of angle C in terms of angle A. I know it looks simple, but my reasoning does not agree ...
0
votes
0answers
11 views

Multitangent to a polinomial function

I'm trying to build some exercises on tangents of functions for beginner students in mathematical analysis. In particular I would like to suggest the study of polynomial functions $ y = p (x) $ of ...
2
votes
1answer
47 views

$f$ is surjective iff it has a right inverse: using the axiom of choice and errors in ProofWiki

Paraphrased from Munkres' Topology: Lemma 9.2. Given a collection $\mathcal{A}$ of nonempty sets, there exists a choice function \begin{equation*} f: \mathcal{A} \to \bigcup\limits_{A \in ...
1
vote
4answers
33 views

Cancelling common factors and equality of functions

Suppose we have two expressions: $\frac{x-1}{x-1}$ and $1$. In the first expression we cancel the nominator and the denominator and are left with $\frac{1}{1} = 1$ and the first two expressions are ...
2
votes
1answer
34 views

Finishing a proof: $f$ is injective if and only if it has a left inverse

I've already done a lot of searching (in particular: https://www.proofwiki.org/wiki/Injection_iff_Left_Inverse) to try to prove this statement: $f: A \to B$ is injective if and only if it has a ...
0
votes
4answers
59 views

$|(0,1)| = |\mathbb R|$

For this problem in proving that the cardinality of (0,1) is equal to that of the set of real numbers, would I just prove that (0,1) is uncountable, and then use the theorem that the subset of an ...
0
votes
1answer
641 views

How to combine an amount of money with the compound interest function?

Tommy has some money at home from his graduation modeled by the function $h(x)=350$. He read about a bank that has savings accounts that accrue interest according to the function $s(x)= 1.04 ...
1
vote
0answers
42 views

Is the inverse function continuous at a fixed point?

Show that $f:I=(-1,1) \rightarrow \mathbb{R},$ it follows that $$ f(x)=\begin{cases} \quad1-x & \text{ as } -1<x\leq 0, \\ \frac{{x}^{-1}+ \lfloor {x}^{-1}\rfloor}{1+{x}^{-1}+\lfloor ...
2
votes
7answers
220 views

How the cardinality of $\mathbb{R^+}$ and $\mathbb{R}$ same?

Let me first confirm you that this question is not a duplicate of either this, this or this or any other similar looking problem. Here in the current problem I'm asking to disprove me(most probably ...
1
vote
1answer
36 views

Looking for a special kind of injective function

Does there exist an injective function $f:\mathbb R \to \mathbb R$ such that for every $c \in \mathbb R$ , there is a real sequence $(x_n)$ such that $\lim\big(f(x_n)\big)=c$ but $f$ is neither ...
18
votes
2answers
480 views

Maximizing curious symmetric function from simple combinatorics

A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture). (This question has been posted at ...
2
votes
1answer
1k views

into function vs injective function

In many mathematical books that I have read and from lectures from professors, the words 'into' and 'injective' were used interchangeably, but in Patrick Suppes book Axiomatic Set Theory he gives a ...
1
vote
2answers
649 views

Trying to figure out a formula with given input and outputs.

I'm playing this video game where people can get kills, deaths, and assists , and all this is recorded on a stats website. The stats website gives you a rating by directly manipulating these numbers. ...
1
vote
4answers
64 views

Show the inverse of a bijective function is bijective

We have a function $\varphi:G\rightarrow H$ is an isomorphism, show its inverse $\varphi^{-1}:H\rightarrow G$ is also an isomorphism I am fine with showing it to be a homomorphism and surjective, ...
0
votes
2answers
983 views

Graphing: Given two points on a graph, find the logarithmic function that passes through both.

Is there such a method to do this? I would like to come up with a logarithmic function (a graph that looks like a square root graph) that passes through two given points. Haven't had any luck in ...
1
vote
1answer
50 views

How to compute $\int^{1}_{-1}f(x)dx$?

I need to compute $\displaystyle\int^{1}_{-1}\,{\rm f}\left(\, x\,\right)\,{\rm d}x$, where $$ \,{\rm f}\left(\, x\,\right) =\left\{\begin{array}{lcrcl} x & \mbox{if} & x & \leq & 0 ...
3
votes
1answer
87 views

What is the purpose of removable discontinuity?

I've just learned about removable discontinuities. So, when we have such a function we redefine it, making a new function that is defined at the point the first isn't. What is the point of this? What ...
-3
votes
1answer
34 views

Injective function from rational numbers to rational numbers [on hold]

Suppose we have $f\colon\mathbb{Q}\to\mathbb{Q}$, $f\circ g=f$ and $g\circ f=f$. Question: is $g$ the identity function $g\colon\mathbb{Q}\to\mathbb{Q}$? Is $g$ and injective function? (meaning ...
1
vote
3answers
52 views

How to answer the question “what is the domain of this function”?

Could you please help me understand and solve this problem about domain of function? All that is written for the question is: What is  the  domain of this function? $$ 2\sin\sqrt{2x-1}+1 $$ ...
1
vote
0answers
97 views

Is there a formula telling if number is prime? [on hold]

Like the topic.. . I mean.. let's say i'm wondering if 15 is prime or not. Could i calculate it, like function roots? EDITED: I mean something like columbus8myhw said: How about: Define ...
27
votes
4answers
3k views

What is the “fastest” increasing function that's useful in some area of math?

Context: I just completed the first quarter of an Intro to Real Analysis class, and while I was thinking about how some functions (like $x^2$) aren't uniformly continuous because they, roughly ...
3
votes
2answers
27 views

Optimization - find the dimensions of a box as functions of volume - minimal surface area

Had a basic calculus course exam today. This was one of the problems: We have a rectangular box of a given volume V. Present the width, height, and length of the box as functions of V so that the box ...
0
votes
0answers
29 views

What is the smallest positive period of the fractional part function $\mathbb R \to \mathbb R:x \to\{x \}$ ?

Is $1$ the smallest positive period of the fractional part function $\mathbb R \to \mathbb R:x \to\{x \}$ ?
0
votes
1answer
18 views

How to find find $f(x)$ such that $f'(x)=\sin^2(x)$ & $f\left(\frac{\pi}{2}\right)=\frac{\pi}{4}$?

I need to find $f(x)$ such that $f'(x)=\sin^2(x)$ & $f\left(\frac{\pi}{2}\right)=\frac{\pi}{4}$. How to do it?
3
votes
2answers
60 views

Show that $\mathbb{Q}\times \mathbb{Q}$ is denumerable [duplicate]

I am new to functions and relations, and with some concepts I am not so familiar. I have a question in an homework: Show that $\mathbb{Q} \times\mathbb{Q}$ is denumerable. From what I ...
0
votes
1answer
41 views

What is at the difference bijection and equinumerous?

I have to explain what a bijection function is, but it seems that equinumerous is a synonym for bijection. Is that correct?
2
votes
5answers
407 views

Find value of f(2013)?

Given a function $f(x)$ such that: $f(1) + f(2) + f(3)+\cdots+f(n) = n^2f(n)$ Find the value of $f(2013)$. It is given that $f(1) = 2014$. I tried attempting the question as a bottom-up DP, but ...
3
votes
1answer
29 views

Project Motorola: setting up and solving an equation

Stuck on a homework project in a highschool college algebra question. I'm given the following information: Tact time is the average time to pick and place one part. Throughput is the number of ...
1
vote
1answer
47 views

Is it true that the relation |A| < |B| is a sufficient condition for claiming that $f$ is a bijection?

This is an exercise of an assignment I have: Suppose $A$ and $B$ are finite sets and $f\colon A\to B$ is surjective. Is it true that the relation “$|A| < |B|$” is a sufficient condition for ...
1
vote
1answer
147 views

Inverse of $f(x)= x+\sin(x)$? [duplicate]

How to find the inverse of $f(x) = x+\sin(x)$, analytically? Well how should I proceed to find the inverse of $f(x)$? Basically I have applied graphical approach to solve the equation, but I want to ...
1
vote
0answers
29 views

Derivative of a composition of function - nice proof

Let's consider the well known "fake" proof below for the derivative of the composition of functions: Let $E,G$ be intervals of $\mathbb{R}$, let $F$ a subset of a normed vector space, let ...
3
votes
1answer
66 views

Derivability of a function with an infinity of zeroes

Let $F$ be a normed vector space and $a\in F$. Is there a non zero function $f:\mathbb{R}\rightarrow F$, such that $f'(a)=0$ and $f$ is $0$ an infinity of times in any neighborhood of $a$ ? If not, ...
0
votes
1answer
34 views

Why a trigonometric function doesn't satisfy a polynomial equation?

Why can't I have a trigonometric function as an input to a polynomial equation?
2
votes
2answers
52 views

Jacobi Elliptic Functions Special Case

I have spent some time analysing the pendulum problem, and hence the Jacobi elliptic functions recently, and have come across what seems to me to be a slight inconsitency. I define my $am(t|k)$ as the ...