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

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7
votes
0answers
49 views

What is the precise mathematical definition of what a wavelet is and what is its relation to linear algebra?

I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In ...
3
votes
3answers
47 views

How to find the domain of a function such that it will be all positive numbers?

I've been working on this problem for a while now and I feel like I'm not understanding it: Find all numbers a such that the domain of the function: $$f(x)= {1\over\sqrt{1+2ax-x^2}}$$ Contains all ...
3
votes
1answer
79 views

Non decreasing real function satisfying $f(x)=f(x+1)$ and/or $f(x)=f(x-1)$. [closed]

Let $f:\mathbb R\to\mathbb R$ be a non-decreasing function. For all $x\in\mathbb R$ we have $(f(x)-f(x-1))(f(x+1)-f(x))=0$. What can we say about the function? [EDITED]
0
votes
0answers
16 views

Sequence of continously differentiable extended functions

Suppose a have a sequence of smooth real functions $f_n$ on $[0,1]$, which converges uniformly to a smooth function $f$. Now consider the extended interval $E=[-0.5, 1.5]$. By Whitney's Theorem we ...
1
vote
0answers
22 views

Look for Max in function

I need to show that the follwing function has just got a minimum and no maximum. I know what it looks like and it is pretty obvious but i can't find a way to explain. The question implicates we might ...
-2
votes
1answer
24 views

Identifying a function is even or odd or not even and odd. [closed]

Here I have a very confusing problem. I'm right now solving Fourier transform. In which different formulas has to be applied according to the nature of the function wether it is odd or even or not ...
31
votes
4answers
955 views

Is it true that this function $f(n)=n^{13}$?

Assume strictly monotone increasing function; such that $f:N^{+}\to N^{+}$, $h$ for all $n\in N^{+}$, $$f(f(f(n)))=f(f(n))\cdot f(n)\cdot n^{2015}$$ Prove or disprove:$f(n)=n^{13}$ ...
1
vote
0answers
39 views

Would this derivation of an integral formula be valid?

So, employing the method of synthetic division, I derived that or in much simpler form Given this, my question is, would the following statement be valid?
0
votes
1answer
10 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 ...
-1
votes
2answers
33 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
36 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
40 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
41 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
37 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 ...
0
votes
0answers
39 views

Is this a bijective mapping?

Considering a one-dimensional chaotic map $T$. The domain of the map (the closed interval $[0,\,1]=I_1 \cup I_1$) is partitioned into two regions (i.e. $I_0\cap I_1=\emptyset$): $[0,\,A] = I_0$ and ...
2
votes
2answers
19 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
47 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
23 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
35 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
848 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
42 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!
1
vote
0answers
41 views

Distribution Function and Transformation [on hold]

Mike is randomly shoot an arrow at 1m bar. Jackie will shoot an arrow at randomly the right side of Mike shoot. X is the point, arrow of Mike, Y is the point, arrow of Jackie. X,Y are from the ...
6
votes
2answers
70 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 ...
0
votes
1answer
43 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
35 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 ...
0
votes
3answers
73 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
30 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
33 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
25 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
24 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
vote
6answers
60 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
26 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
58 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
18 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
36 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
102 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
47 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
57 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
27 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
83 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
40 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 ...
0
votes
2answers
42 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
43 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
22 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
36 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?
1
vote
2answers
29 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
34 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
37 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^+ ...