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

learn more… | top users | synonyms (1)

1
vote
1answer
41 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 ...
2
votes
1answer
33 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 ...
1
vote
0answers
35 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 ...
1
vote
1answer
32 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 ...
3
votes
1answer
84 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 ...
1
vote
4answers
63 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, ...
-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
50 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? $$ 2\sin\sqrt{2x-1}+1 $$ ...
0
votes
0answers
95 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 ...
3
votes
2answers
26 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 ...
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 ...
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?
1
vote
1answer
49 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
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 ...
2
votes
5answers
406 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 ...
1
vote
0answers
24 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
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 ...
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?
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 ...
0
votes
1answer
40 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?
0
votes
0answers
8 views

How can I make this tangent function only appear once (or be spaced very widely)?

I only want the function to go from $x=5$ to whenever the function is 4.5 (in other words, when $y=4.5$). Is there any way to do this without specifying the domain? It has to have the shape of the ...
2
votes
2answers
32 views

Function with only one real root

I'm trying to show that the function $f(x)=2x+3sinx+xcosx$ has only one real root (which is $0$) I've noticed that this is an odd function and therefore if it has a second real root $x_0>0$ then ...
3
votes
1answer
65 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, ...
1
vote
0answers
22 views

an example of functions which is essentialy bounded but not continuous in circle

Can you give me an example of a function which is essentially bounded but not continuous in the unit circle and bounded in the open unit ball?
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 ...
0
votes
1answer
20 views

uniformly continuous functions have a uniformly continuous composition

If you have a function $f: A \rightarrow B$ and $g: B \rightarrow \mathbb{R}$. And I want to show that $g(f(x))$ is uniformaly continous, where both functions are uniformally continous. Do I just ...
0
votes
0answers
7 views

Can functions with a non-analytic point always be approximated with power laws around the special point?

I'm interested in continuous functions from $\mathbb{R}^n$ to $\mathbb{R}$ that fail to be analytic at a given point (let's say the origin), while still being analytic in a region surrounding it. ...
3
votes
2answers
52 views

Opposite of a function being bijective?

A function is bijective if it is both surjective and injective. Is there a term for when a function is both not surjective and not injective?
0
votes
2answers
23 views

Can a surjective function have an element in the domain not mapped to the codomain?

I have seen a lot of definitions for surjectivity stating that every element in the codomain must be mapped to something in the domain. But does the opposite also have to hold true for a function to ...
1
vote
1answer
20 views

Which function will fit this curve best?

I am trying to do a test of normality on this data set here. My QQ Plot looks like this . It looked like an arctan function to me. So my idea was to do a reverse "tan" function transformation on it. ...
0
votes
3answers
51 views

let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true?

let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true? (a). If $g$ is continuous, then $f\circ g$ is continuous. ...
4
votes
1answer
41 views

How I can simplify this inequality or how I can solve it?

How I can simplify this inequality or how I can solve it: $$\left\lceil\dfrac{\ln(t+2)}{\ln 2}\right\rceil-\left\lfloor\dfrac{\ln(t+1)}{\ln2}\right\rfloor>1$$ where $t$ is a positive integer. ...
0
votes
0answers
18 views

Existence and uniqueness of a maximum

Consider $\alpha \in [0,1]$, $\beta>0$, $\delta \geq 0$. Let $1\{...\}$ be the indicator function taking value 1 if the condition inside is satisfied and zero otherwise. Let $$ f(x,y;\alpha, \beta, ...
0
votes
1answer
15 views

Given $f(x)$ and $g(x)$ find the following and state if the composite function exists

This is a two part question. I want to find the following and determine whether or not the composite functions exist. I'm fairly sure of my functions, but I would like to confirm that they are correct ...
2
votes
2answers
46 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 ...
1
vote
1answer
17 views

Given the graphs of $y=f(x)$ and $y=g(x)$, sketch the graph of $y=k(x)$

I think I have solved this problem correctly, but I am a little unsure of whether or not the asymptotes I found are correct. My apologies for the picture, I realize it is a little small for clear ...
6
votes
1answer
63 views

Given $f(x)$ and $g(x)$, find $(fg)(x)$

I've attempted to solve the problem below, and here is what I got for a solution: Given $f(x)=x^2-9$ and $g(x)=x^2+3x-1$, find $(fg)(x).$ $$ \begin{align} (fg)(x)&=(x^2-9)(x^2+3x-1)\\ ...
1
vote
1answer
23 views

Derivation: How do I derivate this

How do I deveriate the following expression? The problem I have is the n in d^n. This expression is part of a bigger task of mine : Show via complete induktion that is true for all n from ...
1
vote
1answer
18 views

Given $f(x)$ and $g(x)$, find the following and state any restrictions

Given $f(x) = 3/(5-x)$ and $g(x) = 2x-1$ find the following and state any restrictions i) (f(g(x)) ii) (g(f(sqrt2)) Here what I got for part i: $g(x) = 2x-1$ ...
3
votes
2answers
102 views

Confusion about the definition of function

Yesterday I was talking to one of my friends about the definition of function. The formal definition of function is given by Cartesian Products but my friend's question was whether it is possible to ...
3
votes
0answers
39 views

Number of functions from domain to codomain

Let A and B be finite sets. Let a be the size of A. Let b be the size of B. Assume 0 < a < b. (a) How many functions are there with domain A and co-domain B? (b) How many one-to-one functions ...
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 ...
0
votes
4answers
20 views

Find the domain, co-domain and range of a function

The function is $$g:\Bbb R\setminus\{0\}\to\Bbb R\setminus\{1\}\;,$$ where $$g(x) = x-\frac1x\;.$$ Please pardon my formatting as I am new to this. I know what a function is of course and their ...
1
vote
2answers
32 views

Existence of continuous functions $f,g:(0,1) \to (0,1)$ such that $f\big((0,1)\big)=(0,1)$ ; and what if $(0,1)$ replaced by $[0,1)$ ?

Does there exist continuous functios $f,g:(0,1) \to (0,1)$ such that $f\big((0,1)\big)=(0,1)$ \ $g\big((0,1)\big)$ ? The problem I am having is that since $(0,1)$ is not compact I am not able to tell ...
0
votes
0answers
23 views

Sketch the graph of $y=(g-f)(x)$ given the graphs of $g(x)$ and $f(x)$

Sketch the graph of the combined function of $y=(g-f)(x)$ for the following functions of $f(x)$ and $g(x)$. I've attempted the problem already (see attached work), but I am unsure of whether I ...
0
votes
0answers
19 views

Does $t^2y''-ty'+y=2t$ with $y(t)=tz(t)$ only if $z'$ as $t^2u'+tu=2$?

I have this question: We have $y$ and $z$ functions with reals as $y(t) = t\times{z}(t)$ for $t \in I = ]0,+\infty[$. Then $y$ satisfies $t^2\times{y}''-t\times{y}'+y=2t$ on $I$ if and only ...
1
vote
2answers
28 views

Use the graph of f(x) and g(x) to evaluate (f+g)(1)

Use the graphs of f(x) and g(x) to evaluate the following. I've done what I think is all the work for this problem, but I'd like to make sure I'm on the right track with this. Feedback on whether ...
0
votes
3answers
53 views

How to come up with bijections?

Is there a good technique for finding bijections in general? Like between the integers and the natural numbers or between [0,1) and (0,1).
2
votes
7answers
205 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 ...