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

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2
votes
4answers
32 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 ...
-1
votes
1answer
37 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
29 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 $0$?
0
votes
3answers
133 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 ...
-2
votes
2answers
51 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 ...
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
26 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 ...
4
votes
1answer
71 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
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 ...
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 ...
1
vote
0answers
41 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
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 ...
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 ...
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, ...
-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 ...
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 ...
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
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
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
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 ...
1
vote
0answers
28 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
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?
0
votes
0answers
9 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
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, ...
1
vote
0answers
23 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
21 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. ...
2
votes
2answers
59 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
22 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
19 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 ...