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

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52
votes
6answers
5k views

Is there a function whose antiderivative can be found but whose derivative cannot?

Does a function, $f(x)$, exist such that $\int f(x) dx $ can be found but $f' (x)$ cannot be found in terms of elementary functions. For example, if $f(x)=e^{x^2}$, then the derivative is easily ...
3
votes
1answer
23 views

Let $S=\{0,2,4,6,8\}$, $T=\{1,3,5,7\}$. Determine whether each of the following sets of ordered pairs is a function with domain $S$ and co-domain $T$.

Let $S=\{0,2,4,6,8\}$ and $T=\{1,3,5,7\}$. Determine whether each of the following sets of ordered pairs is a function with domain $S$ and co-domain $T$. $\{(6,3),(2,1),(0,3),(8,7),(4,5)\}$ TRUE ...
0
votes
3answers
71 views

Maximizing $f(0)$ given that $f(3)=5$ and $f'(x)\ge1$ [on hold]

Let there be $$f:(-1,4)→ R$$ $$\text{differentiable on} (-1,4) , f(3)=5 , f'(x)≥-1$$ $$\text{which is the maximum value of}$$$$f(0)$$
0
votes
1answer
33 views

Simple formula difficult solution

I've thinking a lot about it, but is there a simple way to get $\frac{A}{C}$ from $X = \frac{A + B}{C + D}$ where it does not depend on A and C anymore? This seems so easy but it's quite hard for ...
1
vote
0answers
20 views

Proof of equivalence between limit of a vector field and limit of a scalar field

I have a doubt with a proof regarding the following implication. Consider $F=(f_1,..,f_m): A \subset \mathbb{R}^n \rightarrow > \mathbb{R}^m$ and $\bar{x}$ a limit point for $A$, then $$\...
1
vote
1answer
165 views

Find the max volume using polynomials with the sum of the height and perimeter less than 100cm

I have to find out which shape of packaging for a fragile object has the most volume to fit the object and styrofoam packing. The sum of the height and the perimeter must be less than $100cm$. There ...
-1
votes
0answers
35 views
0
votes
2answers
25 views

Finding the domain and range of a difficult piecewise composite function

I recently inquired about finding a formula for a composition of two piecewise functions, but I have been thoroughly confused by a slightly different example. In this case, I have a question about ...
6
votes
3answers
306 views

Limit of the sequence $\lim_{n\rightarrow \infty}n\left ( 1-\sqrt{1-\frac{5}{n}} \right )$, strange result

$\lim_{n\rightarrow \infty}n\left ( 1-\sqrt{1-\frac{5}{n}} \right )$ $\lim_{n\rightarrow \infty} n *\lim_{n\rightarrow \infty}\left ( 1-\sqrt{1-\frac{5}{n}} \right ) = \infty * \left ( 1-\sqrt{1-0} \...
0
votes
0answers
17 views

Different ways of decomposing an exponential map

There are many decompositions of an exponential map which has two (or more) operators in the exponent (i.e. $e^{A+B}$, where $A$ and $B$ are operators). For example, the Baker-Campbell-Hausdorff (and ...
27
votes
3answers
1k views

No continuous function switches $\mathbb{Q}$ and the irrationals

Is there a way to prove the following result using connectedness? Result: Let $J=\mathbb{R} \setminus \mathbb{Q}$ denote the set of irrational numbers. There is no continuous map $f: \mathbb{R} \...
5
votes
5answers
8k views

Does there exist a function that is differentiable but not integrable? or integrable but not differentiable?

It has become very complicated to me to find out a function which is differentiable but not integrable or integrable but not differentiable.
1
vote
3answers
79 views

Find $f'(x)$in terms of $f(x)=|\cos(x)|\sqrt{1-\cos(x)}$

I am trying to solve the following exercise : Let $f$ be the function defined by : $$\forall x\in]0,\pi[\;\;\;\;\; f(x)=|\cos(x)|\sqrt{1-\cos(x)}$$ calculate $f '(x)$ in terms of $f(x),$ for all $x\...
2
votes
1answer
38 views

Open sets and annihilator of functions

A topological space $X$ is said to be completely regular provided that it is a Hausdorff space such that, whenever $F$ is a closed set and $x$ is a point in its complement, there exists a function $f\...
1
vote
0answers
37 views

Why use the letter “k” in the function transformation formula $f(x - h) + k$?

This is strictly a historical "why is it the letter k rather than say v for vertical" question -- is it the initial letter of something from a specific language? Is it arbitrary? While we're at it, ...
5
votes
4answers
401 views

Finding a tricky composition of two piecewise functions

I have a question about finding the formula for a composition of two piecewise functions. The functions are defined as follows: $$f(x) = \begin{cases} 2x+1, & \text{if $x \le 0$} \\ x^2, & \...
1
vote
2answers
48 views

I find correct limit of the sin cos function?

This is no homeworks I only do for learn. $$\lim_{x\rightarrow \pi}\frac{\sin^{2}x}{1+\cos x}$$ I use l'Hôpital rule because no idea where limit go for both. Top is called $g(x)$ and bottom is $...
0
votes
2answers
21 views

How would I use the difference quotient on this logarithmic function?

This is no homework, it's for exam practice. Show that $\lim_{x\rightarrow 0}\frac{1}{x}ln(1+ax) = a$ where $a \in \mathbb{R}\setminus \left \{ 0 \right \}$ is chosen definitely / fixed (...
0
votes
2answers
39 views

Additive functions and measure theory

Key reference is the following: Hamel basis and additive functions Let's investigate real-valued functions $f(x)$ with the following (additive) property for all $\,a,b$ : $$ f(a+b)=f(a)+f(b) $$ It ...
1
vote
2answers
52 views

Is $2\prod_{i=1}^{n}{p_i} - \prod_{i=1}^{n}{(p_i + 1)} - \prod_{i=1}^{n}{(p_i - 1)}$ even and negative for $n > 1$?

Is $$2\prod_{i=1}^{n}{p_i} - \prod_{i=1}^{n}{\left(p_i + 1\right)} - \prod_{i=1}^{n}{\left(p_i - 1\right)}$$ even and negative for $n > 1$, where $p_i > 1 \hspace{0.07in} \forall i \in \left[1,n\...
1
vote
3answers
34 views

Greatest Integer Function and Limits - Is GIF of $\sin x/x$ equals to $0$?

Okay, so I read this somewhere that, $$ \lim_{x \to 0^+} \left[ \frac{\sin x}{x} \right] = 0 $$ Where, [] denotes the greatest integer function. But, on the other hand, this is also true, ...
1
vote
1answer
23 views

Functions - Find number of positive integral values of 'x' which satisfy an inequality

Let $f(x) = 30 - 2x - x^3$, then find the number of positive integral values of 'x' which satisfies $f(f(f(x))) > f(f(-x))$. The first thing that I saw in the above question was that the ...
1
vote
3answers
54 views

Find all the angles $v$ between $-\pi$ and $\pi$

Find all the angles $v$ between $-\pi$ and $\pi$ such that $$-\sin(v)+ \sqrt3 \cos(v) = \sqrt2$$ The answer has to be in the form of: $\pi/2$ (it must include $\pi$) I have tried squaring but I get ...
1
vote
1answer
21 views

Need know all ways to show function is continuous, convergent and differentiable [on hold]

Please tell me all ways to show / proof that a function is continuous, convergent and differentiable. continuous: show that function is differentiable if yes then it is continuous also convergent: ...
2
votes
2answers
32 views

Limit of functions - always for both sides (+-) necessary?

I'm very confused when I read some pages on the internet about limits (for functions). Let's say I got any function f(x) given and someone tells me to find the limit (towards 3 or $\infty$ or ...
0
votes
1answer
28 views

Correct Order of Applying Graphical Transformation with Absolute Value

I was going through this website, reading about transformations of graph when $| |$ is applied to various parts of a given function, $y=f(x)$. Going through the fourth example of the page, I came ...
0
votes
1answer
31 views

Finding a delta for the greatest integer function given an epsilon = 1/2

I'm having trouble with the following problem. Given the standard greatest integer function $\lfloor x \rfloor = int(x)$ where $ \lfloor x \rfloor $ returns the greatest integer less than or equal to ...
1
vote
1answer
395 views

Determine the range of f(x)=(sinx)/x

I am having trouble understanding the solution to this question. ''Determine the range of the following function: $f(x)$ = $(1$ $if$ $x=0)$ or (${\sin x\over x}$ if $x$$\neq$$0$) where the domain ...
1
vote
0answers
24 views

Improving inequality $(\int X(x) Y(x) \,dx) \leq (\int |X(y)| \,dy) Y_{\max}$

Want to improve the following inequality: $(\int X(x) Y(x) \,dx) \leq (\int |X(y)| \,dy) Y_{\max}$ Looking to replace $Y_{\max}$ with something that will give a tighter bound. Everything else needs ...
1
vote
3answers
72 views

Prove $ \left[1,2 \right) \bigcup \left(5,6 \right)$ $=A$ has cardinality?

Prove that the following set has cardinality $c$.(which stands for continuum.) $ \left[1,2 \right) \bigcup \left(5,6 \right)$ $=A$ So I know that A would be a cardinal number if it is equivalent to$...
0
votes
0answers
11 views

Bessel function J of fractional order for large complex argument

I am trying to evaluate the bessel function of first kind of fractional order for a large complex argument as input, but I get nans and infs as the result. If for example I have that z=30000-30000i, ...
-1
votes
0answers
20 views

Absolute Value Graph Problem in Gelfand's Functions and Graphs

I am working through Gelfand's Functions and Graphs, where I am currently on the absolute value section. At the end of the chapter practice problems, Gelfand poses a set of problems regarding ...
0
votes
0answers
12 views

proof of preimage of union and intersection of sets

I was learning to proof the following proposition "the inverse image of an intersection or union equals the intersection or union of the inverse image" following these two really good youtube videos: ...
1
vote
1answer
28 views

Proof about Finite set (Surjectivity and Injectivity)

Let $B$ be a non-empty set. Then the following are equivalent: (1) $B$ is finite. (2) There is a surjective funtion $f:\{1,...,n\}\rightarrow B$ for some $n\in \mathbb{N}$ (3) There is an injective ...
0
votes
0answers
19 views

Is the inverse of the following function correct

Function is Q(x) = 1/2 + 1/2*[erf(x/sqrt(2)] Inverse calculate is Q_inverse(x) = sqrt(2)erfinv(2x-1)
4
votes
3answers
62 views

On notation: is it better to say $A^B = \{f| f:B \to A\}$ or $A^B = \{f :B \to A| f \text{ is a function}\}$

The title says it all, let $A^B$ denote the set of all functions from $B$ to $A$, then it is better to write in set notation $A^B = \{f\mid f:B \to A\}$ or $A^B = \{f :B \to A\mid f \text{ is a ...
6
votes
1answer
235 views

the continuity of argmin on convex funtion

Define $$x'=\text{argmin}_{x_1}f(x_1,\lambda),$$ where $f$ is a strictly convex function on $x_1$ and $\lambda$. I would like to ask if there is any theorem about the continuity of $x'$ w.r.t $\...
1
vote
1answer
17 views

tangents conditions

what are the condition for a tangent to be exist . Is it necessary for the function to be continous. but it is necessary to be continous for a function to be differentiable at that point . can ...
22
votes
2answers
2k views

If $f(x)$ has a vertical asymptote, does $f'(x)$ have one too?

So here is what I understand: If $f(x)$ is increasing/decreasing, then its derivative $f'(x)$ is positive/negative and... If $f(x)$ is increasing/decreasing, then the derivative of $f'(x)$ (...
0
votes
3answers
63 views

How to solve $x_{100}$ with $x_{n+1}=\frac{2x_n}{2+x_n}+1$ and $x_{1}=3$

How to solve $x_{100}$ with $x_{n+1}=\frac{2x_n}{2+x_n}+1$ and $x_{1}=3$? Can anybody shed light on this? regards.
1
vote
0answers
20 views

Find a hypergeometric formula embracing three specific cases

For a parameter value $a=\frac{1}{4}$, I have the result \begin{equation} Q(k,\frac{1}{4})=\frac{2^{-2 k-\frac{19}{4}} \Gamma \left(2 k+\frac{13}{4}\right) \, _3F_2\left(1,k+\frac{13}{8},k+\frac{17}...
0
votes
0answers
24 views

I want to show that the function space $C_0(X)$ is Banach [duplicate]

I'm reading some papers but I encountered a problem that "$C_0(X)$ is Banach space". Here $$ C_0(X):= \{ f: X\to \mathbb{C}: f \text{ is continuous and } \forall \epsilon>0, \exists K(\text{compact}...
0
votes
0answers
10 views

Determine the operation based on the conditions given below

\begin{align} f(c, d)&= a;\\ g(c, d)&= b;\\ h(a, b, c)&= d. \end{align} The functions $f$, $g$, $h$ are defined for all $a,b,c,d\in\mathbb R$. For instance: $h$ can be Division; $a$, $b$, ...
-4
votes
0answers
30 views

Solve for the conditions given below [closed]

\begin{align} f(c, d)&= a;\\ g(c, d)&= b;\\ h(a, b, c)&= d. \end{align} The functions $f$, $g$, $h$ are defined for all $a,b,c,d\in\mathbb R$. For instance: $h$ can be Division; $a$, $b$, ...
7
votes
4answers
148 views

inequality $\sqrt{\cos x}>\cos(\sin x)$ for $x\in(0,\frac{\pi}{4})$

How can I prove the inequality $\sqrt{\cos x}>\cos(\sin x)$ for $x\in(0,\frac{\pi}{4})$ ? The derivative of $f(x):=\sqrt{\cos x}-\cos(\sin x)$ is very unpleasant, so the standard method is ...
1
vote
1answer
61 views

Define an $\mathbb{N}$ to $\mathbb{N}$ function that is

Hi I'm preparing for an exam and was going through exercises on functions. I stumbled upon this question and didn't know how to answer it. Give an $\mathbb{N}$ to $\mathbb{N}$ function that is one-...
0
votes
1answer
22 views

How to know describe the set of levels for functions f(x,y)=c when c varies

Hey im having quite troubles trying to understand how to describes the set of levels in functions. In this problem any ideas? $$f(x,y)=x^2+y^2+1$$
1
vote
0answers
25 views

A problem about a continuous iterated function [duplicate]

Let $f:\mathbb {R} \rightarrow \mathbb { R } $ be a continuous function such that $f\circ f \circ f=\text{id}_\mathbb{R} $. Show that $f=\text{id}_\mathbb{R}$. Is there any hint to prove this? ...
2
votes
1answer
19 views

increasing and one one function

If we are given a function f(x)=$x^3$$+$$3x$ for all x belong to real number . Now as the derivative of function is always positive so the function should be increasing function and if it is always ...
0
votes
0answers
14 views

Functional Equations & Integration - Finding the integration of a unknown function

The function f is continuous and has the property $ f(f(x)) = 1 - x $ for all x in $ [0, 1] $, and $ J = \int_{0}^{1} f(x) dx $, then find $f(\frac{1}{4}) + f(\frac{3}{4})$ & the value of J. I ...