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

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0
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1answer
184 views

derivative of sign() as active function in backpropagation

I've got the task that I need to implement the backpropagation algorithm for a neural network. My activation function is just the sign(.). $w^{\prime} = w + \space$learning rate$\space \times \delta ...
-2
votes
2answers
94 views

How to prove a function is concave? (Single Variable)

It has been a while after completing the calculus of single variable. Right now I have a function of single variable $f(x)$, and that $f'(x)=-c$ for all $x$. So $f$ is a decreasing function. Bu, ...
1
vote
1answer
46 views

Distribution function?

Let $F(x) = e^{-1/x}$ for $x>0$ and $F(x)=0$ for $x\leq0$. Now I am investigating if $F$ is a distribution function. Say: \begin{align} \int\limits_0^\infty e^{-1/x} \, dx = \left[ ...
2
votes
1answer
77 views

Inverting the Radial Distortion

Overview The problem is perhaps a very easy one for a trained mathematician. As I am not a mathematician, but instead a researcher in general problem solving, I am reaching out to those who know more ...
3
votes
1answer
54 views

Proof that $f(x)=0 \forall x \in [a,b]$

Lemma: If $f \in C([a,b])$ and $\int_a^b f(x) h(x) dx=0 \ \forall h \in C^2([a,b])$ with $h(a)=h(b)=0$ then $f(x)=0 \ \forall x \in [a,b]$. Proof of lemma: Suppose that there is a $x_0 \in (a,b)$ ...
0
votes
2answers
125 views

If $f$ is continuous on $[a,b]$ then $1/f$ is bounded on $[a,b].$

$f(x) > 0$ is given for all $x\in [a,b]$. I only got to this: Let $c$ belong to $[a,b]$. Then, for all $ε>0$, there exists $δ>0$, such that, $|x-c|<δ\implies|f(x)-f(c)|<ε$.
1
vote
1answer
69 views

Continuity of the maximum of finite continuous functions

Let $(X,\tau)$ be a topological space and let $f_1,\ldots,f_n:X\to\mathbb{R}$ be continuous functions (the topology of $\mathbb{R}$ is the usual one). Define $g:X\to\mathbb{R}$ by ...
2
votes
2answers
32 views

Stuck in expressing factor into a sum of three perfect squares.

Two part question. (i) Consider the function $f(x)=x^3-6kx+k^3+8$. Show that we can write $f(x)$ as $(x+k+2)P(x)$ where $P(x)$ is a quadratic function. (ii) Show that $2P(x)$ can be written as the ...
1
vote
1answer
50 views

Convex function when λ∉[0,1].

f :R→R is convex, Prove, for every x,y∈R, and λ∉[0,1] f(λx+(1−λ)y)≥λf(x)+(1−λ)f(y). In definitoin of convex funcion λ belongs in [0,1], but here not.
0
votes
0answers
27 views

Solving Coupled equations

I need to solve a coupled equation and basically I am completely stuck on how to proceed. The equations are $$ a = u_\pm + \frac{i}{b}\cdot\frac{u_+ - u_-}{\sqrt{u_{\pm}^2-1}} $$ and $$ N_\pm(a) = ...
1
vote
2answers
49 views

Which is greater ? Sum of odd power terms or even power terms in the exponential Taylor series?

I came across this question, in a book. Define $f(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{{(2n+1)}!} $ and $ g(x) = \sum_{n=0}^{\infty} \frac{x^{2n}}{{(2n)}!} $, where x is a real number. Then, ...
21
votes
3answers
897 views

“Least trivial” function preserving rationality

Is there a "non-trivial" function $f(x,y)$ such that $$f(x,y) \in \mathbb{Q} \iff x,y\in \mathbb{Q}?$$ An example of a "trivial" function would be $$f(x,y) = \begin{cases} 0 & x,y\in ...
0
votes
0answers
44 views

sum of zeta function [duplicate]

how do I solve this question? $$\sum_{k\geq2} ( \zeta(k)-1) $$ I know that $\zeta(2)$ is $\frac{\pi^2}{6}$ and $\zeta(k)$ can be represented as $$\sum_{j\geq1} \frac{1}{j^k}$$ Thanks in advance!
6
votes
2answers
116 views

If $f(x)$ is continuous at $a$ and $g(x)$ is not continuous at $a$, then can $(f+g)(x)$ be continuous at $a$?

I know that if both $f(x)$ and $g(x)$ are continuous at $a$, then $(f+g)(x)$ would be continuous at $a$. My first thought here is that $(f+g)(x)$ cannot be continuous at $a$ if $g(x)$ is not ...
1
vote
1answer
60 views

Find period of the Function

The given question $$f(x) = \sqrt{\frac{8}{1+x} + \frac{8}{{1-x}}}$$ $$g(x) = \frac{4}{f(\sin x)}+\frac{4}{f(\cos x)}$$ find period of $g(x)$? What I have done putting $\sin(x)$ and $\cos(x)$ in ...
0
votes
2answers
40 views

Pointwise and uniform convergence of series of funtions.

If I understand it right, uniform convergence by sequence of functions $\{f_n\}$ means, that there is a limit function $F$, and for any $\epsilon > 0$ we can always chose a high enough $n_0$ (the ...
1
vote
3answers
79 views

Find the derivative of $ f(x) = x^9 - x^7$ using limit definition

I can find this without using the limit definition (I think the formula is $\frac{f(x+h) - f(x)}{h}$ My first steps to solving are $f'(x) \lim\limits_{h\rightarrow 0} \frac{(x+h)^9 - (x+h)^7 - x^9 ...
0
votes
1answer
36 views

Finding domain of a function

For example $n(x)=\sqrt{x-2}\sqrt{4-x}$ My attempt, $x-2\ge0$ $x\ge2$ and $4-x\ge0$ $4\ge x$ $x\le4$ So the domain is $2\le x\le4$. Am I correct? How about ...
3
votes
1answer
56 views

Give an example of a continuous function $f: (-1, 1) \rightarrow \mathbb{R}$ which attains a maximum at 0, but is not differentiable at 0

I need a little help with this exercise: Give an example of a continuous function $f: (-1, 1) \rightarrow \mathbb{R}$ which attains a maximum at 0, but is not differentiable at 0 I thought of the ...
0
votes
1answer
112 views

Prove: functions with bounded derivatives are Lipschitz continuous

I'm a 1st year mathematics student, and in my analysis class I'm having trouble with proving the following: Let $M > 0$, $f: [a, b] \rightarrow \mathbb{R}$ be a function which is continuous on ...
0
votes
2answers
25 views

When is the function above its obliques asymptote?

$y = 2x + \frac{3(x − 1)} {x+1}$ How to determine the values for x for which a function such as this one is below its oblique asymptote?
1
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6answers
64 views

Fundamental Theorem of Calculus application for $f(x)\geq 0$

Can anybody help me with how to solve the following question using the fundamental theorem of calculus? I'm a bit confused... If $f$ is a continuous function on $[a, b]$ and $f(x)\geq 0$ for all ...
1
vote
2answers
33 views

does $f(n) \neq O(g(n))$ implies $g(n)=O(f(n))$ [duplicate]

Im pretty sure it doesn't, but how can I be sure? Was thinking by using $$f(x) = \sin(x) + 2$$ and $$g(x) = \cos(x) + 2$$ Thanks!`
3
votes
1answer
109 views

Function defined by integral

this question is driving me nuts, I can't think about an easy solution. Let $F(x)=\int_{0}^{x} \sqrt{1+t^3}\,dt. $ Evaluate $\int_{0}^{2} x\,F(x)\,dx$ in terms of $F(2)$. I know that the derivative ...
1
vote
1answer
29 views

Limit of a Function with Parameter

Given that $a\ne-1, \lim\limits_{x \to 0} f(x) = L$, prove by limit definition ($\epsilon, \delta)$ that $\lim\limits_{x \to \infty} f(\frac{a+1}{2x}) = L$. I would greatly appreciate any thoughts ...
2
votes
1answer
51 views

Can a function which is periodically undefined have a limit as x goes to infinity?

I'm currently preparing for a calculus test. I was trying to solve the exercises of the test of last year, and one of the questions was: Give a full limit research of this function: ...
0
votes
4answers
120 views

$f(x)=x/(x^2+1)$ Deduce that f is not one-to-one. State the range of f。

The function $f$ is defined by $f(x)=x/(x^2+1)$, $x$ is an element of a set of real number. If $a$ is an element of a set of real number and $a$ is not $0$, find the image of $1/a$ under $f$. (This I ...
0
votes
5answers
68 views

Find the inverse function of $g(x)=(x-2)(x-4),\; x≥3$.

Find the inverse of the following function, stating its domain. $$ g(x) = (x-2)(x-4), \quad x≥3. $$ I try to find the inverse function, but I can't eliminate $x$ in my method.
0
votes
1answer
128 views

Continuous function which maps (0,1] to {0}, (0,1), [0,1), [0,1]

Let $f: \mathbb R \to \mathbb R$ be a continuous function. Which one of the following sets cannot be the image of $(0,1]$ under $f$? $\{0\}$ $(0,1)$ $[0,1)$ $[0,1]$. We know that $(0,1]$ is ...
0
votes
0answers
8 views

Solve this equation for implicitly defined $F(x)$

For some constants/parameters $s$, $k$, $c$, $A$, $B$, I have $F(x)$ implicitly defined as $$ \sum_{k=0}^{s-1} G(s,k) (1-F(x))^k = \frac{k-c}{x-c}\frac{x}{k}A + B$$ Where $G(s,k)$ is closely ...
2
votes
1answer
30 views

Existence of a smooth function with given derivative roots

Is there a smooth function $f$ that for all $n\in\mathbb{Z}_+$, $f^{(n)}(n)=0$ i.e. $n$th derivative at the point $n$ is zero and $f^{(n)}(x)\ne 0$ for all $x\in\mathbb R\setminus \{n\}$? If there is ...
5
votes
4answers
87 views

What is an exact characterization for the functions $f$ such that $xf'(x) \leq 2f(x)$?

What is an exact characterization for the functions $f$ such that $xf'(x) \leq 2f(x)$? I know, for instance, that the inequality holds for all functions $f(x) = c_0 + c_1x + c_2x^2$, with $c_0, c_1, ...
3
votes
1answer
116 views

Extreme values of a two-variable polynomial

Is it possible to find a two-variable polynomial which has only two extreme values on the whole plane, one is a local maximum, another is a local minimum, and the local maximum is less than the local ...
1
vote
2answers
57 views

Inverse of a function $xe^x$

How should I proceed about finding the inverse of the function $xe^x$? I have been wondering about it for a long time and can't think of anything to do.
1
vote
0answers
45 views

Question about your function,

I'm Xavier Vigan, a physical oceanographer. I've been using your $f(x)=\dfrac 12 \times \left(X+C-\sqrt{S+(X-C)^2}\right)$ function to calibrate quantile vs quantile plots. Because of the shape of ...
0
votes
1answer
23 views

Searching function starting exactly constant and approaching another constant

for the default of an R API parameter i seek a function that has the property of yielding a good guess. I want the function to be defined for $\mathbb Z^+$ (But no reason not to define it for ...
0
votes
2answers
46 views

Calcultaing function limit of a limit sequence

How do I even start? $$\lim_{x \to 1^+} \lim_{n \to \infty}{x^n \over x^n + 7}$$ I see that it should be $1$ but how do i prove it?
2
votes
2answers
35 views

Basic question on the probability function and the probability distribution function

I have a question on the probability function. In my book it says that if A and B are mutually exclusive events $P(A∪B)=P(A) + P(B)$. Then when it starts talking about the probability distribution ...
0
votes
1answer
45 views

Why can open intervals be used to calculate rate of change?

I was watching this video Question : On which interval does $y(x)$ have an average rate of change of $\frac{1}{2}$? The first option is $-2 < x < 2$ The video narrator just puts in $-2$ and ...
0
votes
0answers
54 views

Holomorphic function in unit disc?

I am stuck on this exercise from Stein and Shakarchi's Real Analysis: suppose $F$ is holomorphic in the unit disc, and $$\sup_{0\leq r < 1} \frac{1}{2\pi} \int_{-\pi}^\pi \log^+ ...
3
votes
3answers
53 views

Show that $f: \mathbb{Z} \to n\mathbb{Z}$ defined as $f(m) = mn$ is bijective

Let $n \in \mathbb{N}$. Show that $f: \mathbb{Z} \to n\mathbb{Z}$ defined as $f(m) = mn$ is bijective such that $f(m_1 + m_2) = f(m_1) + f(m_2)$, $\forall m_1, m_2 \in \mathbb{Z}$ To be bijective, ...
1
vote
1answer
45 views

How do you create the equation for the Cantor Pairing Function?

According to wikipedia, here is the equation: $f(x,y) = \frac{(x+y)(x+y+1)}{2}+y$ How do you go about creating this function? I understand that the X value is found by the corresponding triangle ...
0
votes
0answers
10 views

Defining a function with an arbitrary number of values

In school we learn that a function may have one value at most for a given input. But if we define its range to be the set $\bigcup_{n=1}^{\infty}{{\mathbb{R}}}^n$, then it can have an arbitrary number ...
1
vote
1answer
44 views

For what values of $x\in \Bbb N$ is $\tan^{-1}\left(\frac{360}{x}\right)$ rational?

For what values of $x\in \Bbb N$ is $$\tan^{-1}\left(\frac{360}{x}\right)$$ rational? Just wondering if there is any method to accomplish this.
0
votes
0answers
7 views

On the zero set of a smooth function which is linear on the first variable.

Let $f:C^n\times R^n\rightarrow C$ be a smooth function which is linear in $z\in C^n$. (1) For any $z\in C^n$, $f(z,R^n)$ is compact and there exists $x_z\in R^n$ satisfying $f(z,x_z)=0$. (2) There ...
1
vote
4answers
96 views

Solving the equation $ f^{-1}(x)=f(x)$

I attempted to solve the equation given in the title for the function; $$f: \mathbb R_{++} \to\mathbb R_{++}; \quad f(x)=x^2(x+2)$$ I understand that the problem is equivalent to solving $f(f(x))=x$ ...
4
votes
1answer
45 views

How to determine the generating function?

So I have $$\overset{*}{F} = \overset{*}{F}_{n-1} + \overset{*}{F}_{n-2} + g(n)$$ where $\overset{*}{F}$ is NOT a Fibonacci number for $n \geq 2$. $g(n)$ is any function $g: \mathbb{N} \to ...
1
vote
1answer
14 views

To find, wether '1' lies in the range of f, where $f(x)=[ln(\frac{7x-x^2}{12})]^\frac{3}{2}$?

$f(x)=[ln(\frac{7x-x^2}{12})]^\frac{3}{2}$, For the given function, the question is whether, f(x) can equal 1 for some real value of x?
3
votes
2answers
50 views

How to precisely define $C^\infty$ in $f(x) \in C^\infty$

In single variable calculus, a common way to denote a function that is continuous for all derivatives is to write $f(x) \in C^\infty$ i.e. $f(x) = \exp(x)$ Is there a more rigorous way to define ...
0
votes
1answer
42 views

Exponential type of $\sin z$

An entire function $f$ is of exponential type if $\,\lvert\, f(z)\rvert\le C\mathrm{e}^{\tau\lvert z\rvert},\,$ for all sufficiently large values of $\lvert z\rvert$. The exponential type of $f$ is ...