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

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3
votes
2answers
44 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 ...
3
votes
5answers
307 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
22 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
28 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
42 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
34 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
6 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
28 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
64 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
19 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
142 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
17 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
48 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
22 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
17 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. ...
3
votes
1answer
39 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
17 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
36 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
60 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
19 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
101 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 ...
26
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
19 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
31 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
22 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
51 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
5answers
143 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 ...
0
votes
1answer
30 views

Are $\Re(z)$ and $\Im(z)$ solutions of $z' = az$?

I'm having trouble with a question (I have to answer "true" or "false" and explain it): We have $a:I \to C,$ a continuous function on $I$, interval of $R$. If the function $z:I \to C$ is a ...
2
votes
1answer
37 views

The function f is defined as follows: $f:A \to A$

The function f is defined as follows:$f:A$ to $A$ where$$ f(x)=\frac{3(x +1)}{x^2-1}$$ Along my proof in showing that show that there exists an x ∈ A with $f(x) = y$ (showing f is onto) ,I ran into ...
2
votes
2answers
37 views

Equality of functions

How a function which is not defined for some value can be equal to a function which is defined for the same value? How is $f(x) = \frac{(x-2)(x-3)}{(x-2)(x-4)}$ equal to $g(x) = \frac{(x-3)}{(x-4)}$ ...
1
vote
3answers
44 views

Is $g$ the unique function with this property?

Prove/Disprove: Let $A$ and $B$ be sets and let $f : A \to B$ be a function. If there is a function $g : B \to A$ such that $g\circ f = \operatorname{id}_A$, then $g$ is the unique function with this ...
0
votes
0answers
12 views

CONFIDENCE LEVEL for Median Interval

A firm wants to estimate the unknown median, m , of the height of their employees. Random Simple Size = 90 $X_{i}$ is the order statistics of the Sample Size X where height of each employee was ...
1
vote
3answers
36 views

Does the method for finding the intersection of 2 single variable functions work for multivariable functions?

I have $2$ multivariable functions $Q(x,y)$ and $P(x,y)$, I was wondering if finding the point of intersection between these 2 functions is as easy as making $Q(x,y) = P(x,y)$ as you would do for most ...
0
votes
1answer
18 views

Linear Algebra - Inner Products, Functions, and Closet Polynomial

This is the question: Formulate the linear algebra problem of finding the closet poly $p \in span \{1, t^2\}$ to the function $f(t)=e^tcos(t)$ with respect to the L$^2$ inner product: $\lt f,g\gt ...
0
votes
1answer
31 views

Can you raise trigonometric functions to a non-integer power?

I don't inmediately see any reason why you could not yet I have never come across it. For any answer given reasoning would also be appreciated! Thank you
3
votes
1answer
42 views

How do I prove this function doesn't exist?

Let $g: S\rightarrow S$ be a function such that $g$ has exactly two fixed points, and $g\circ g$ has exactly four fixed points. Prove that there is no function $f:S\rightarrow S$ such that $g=f\circ ...
0
votes
1answer
30 views

Solving ODE for x instead of y

Find the general solution of the ODE. Give the largest interval over which the general solution is defined. Determine any transient terms in the general solution. $y dx - 4(x+y^6)dy = 0$ This is ...
2
votes
1answer
53 views

The “sin-cos-maximum” function

Is there some specific notation for the function $f(x):=\max\{\cos(x),\sin(x)\}$, or maybe some equivalent compact expression? Improvement: Actually, maybe a compact equivalent expression for its ...
1
vote
0answers
25 views

Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ ...
2
votes
2answers
45 views

Prove that the function: f: $\mathbb{N} \mapsto \mathbb{Z}$ defined as f(n)= $\frac{(-1)^n(2n-1)+1}{4}$ is bijective.

Prove that the function: f: $\mathbb{N} \mapsto \mathbb{Z}$ defined as f(n)= $\frac{(-1)^n(2n-1)+1}{4}$ is bijective. This is rough. I've been staring at this one for a while now. I get stuck on the ...
-1
votes
0answers
42 views

List of simple, common functions with an incomplete domain or range on $\mathbb{C}$

This may seem like a strange question, but it's an interest of mine and I would appreciate the help of the community in addition to brainstorming on my own. As the question states, I'm looking for ...