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

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0
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2answers
45 views

Is the maximum of the absolute value of the second derivative always smaller than the maximum of the absolute value of the 1st derivative?

Assume a function $f(x)$ is differentiable, is the maximum of the absolute value of the second derivative always smaller than the maximum of the absolute value of the 1st derivative? $$\max\left(\frac{...
1
vote
1answer
24 views

Uniformly Cauchy sequence of functions

I am trying to show the following: For each $n \in \mathbb N$, let $f_n:X \to Y$, where $(Y,d)$ is a complete metric. Suppose that for every $\epsilon>0$, there exists $n_0 \in \mathbb N$ such ...
0
votes
1answer
45 views

How accurate is the approximation of the number of rough numbers?

A number is called a $y$-rough number, if it has no prime divisor below $y$. The number of rough numbers in an interval, lets say, $[10^{99},10^{100}]$ is approximately the length of the interval ...
3
votes
3answers
69 views

Taylor-polynomial of function $f(x) = e^{x}*\sin(2x)$

This is not homework, I'm asking to learn for an exam which I'll write in 2.5 months. Count the Taylor-polynomial 3th grade of the function $f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = e^{x}*\sin(...
0
votes
0answers
41 views

On the vanishing of integrals involving the $\sinh$ function. [on hold]

Suppose for some positive real $\theta$ that $$\int_1^\infty f(x)\sinh(\theta\log \sqrt x) \mathrm{d}x = 0$$ Where $f(x)$ is a non-constant and continuous function of $x$. What necessary properties ...
0
votes
4answers
52 views

For what $k$ is $f(x) = kx^2-2x+k$ negative for all values of $x$?

What are the values of $k$ for which the quadratic function $f(x) = kx^2-2x+k$ is negative for all values of $x$? The values of $k$ should definitely be negative.
2
votes
1answer
32 views

How is it called if two functions have the same order?

Lets have $f(x_1)>f(x_2)\implies g(x_1)>g(x_2) \forall x_i \in \mathbb{R}$. Is this property between $f$ and $g$ named in some way?
1
vote
2answers
46 views

Function is continous

If functionis continous at x=0 the we have to find the value of k I got a solution , but I am not able to understand what they have done in second step . Can anyone explain me
4
votes
4answers
101 views

Sum to infinity of trignometry inverse: $\sum_{r=1}^\infty\arctan \left(\frac{4}{r^2+3} \right)$

If we have to find the value of the following (1) $$ \sum_{r=1}^\infty\arctan \left(\frac{4}{r^2+3} \right) $$ I know that $$ \arctan \left(\frac{4}{r^2+3} \right)=\arctan \left(\frac{r+1}2 \right)-\...
0
votes
2answers
60 views

Taylor-polynomial of $f(x)=\log(\cos(x))$

$f: (-\frac{\pi}{2}, \frac{\pi}{2}) \rightarrow \mathbb{R}, f(x) = \log(\cos x)$ Count the Taylor-polynomial $T_{2}(f, 0)(x)$ of the second degree of $f$ in $x_{0} = 0$ Alright because it was ...
1
vote
1answer
19 views

Confusion regarding uniform continuity

I was trying to check the validity of the following: If $f:\mathbb R\rightarrow\mathbb R$ and its derivative $f'$ are unbounded, then $f$ is not uniformly continuous on $\mathbb R$. To me,the ...
-1
votes
0answers
18 views

confusion in linear function

let f={(1,1),(2,3),(0,-1),(-1,-3)} be a function described by the formula f(x)=ax+b for some integer a,b .determine a,b(EXACT COPY OF THE BOOK) My solution :question is quite easy in my sense .since ...
0
votes
2answers
20 views

set theoretic equation to evaluate a function

In axiomatic set theory a function $F$ is defined to be a class/set all of whose members are ordered pairs. Given a function $F$ there are set theoretic equations involving union, intersection and ...
0
votes
0answers
29 views

Discontinuous solutions to Cauchy's functional equations.

Does there exists a discontinuous function $f: \mathbb R \rightarrow \mathbb R$ satisfying both the Cauchy's equations simultaneously $f(x+y) = f(x) + f(y) \text { and } f(xy) = f(x)f(y)$for every $x,...
0
votes
2answers
19 views

Domain and range of Composite function.

$f : A \to B$ and $g : B \to C$. So I know that $(g \circ f)(x):A\to C$. What are the domain, codomain and range of $(f \circ g)(x)$?
3
votes
0answers
43 views

Function with $f(f(n))=f(n-1)f(n+1)-f(n)^2$

Let $\mathbb{N}$ denote the set of positive integers. Does there exist a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $$f(f(n))=f(n-1)f(n+1)-f(n)^2$$ for all $n\geq 2$? If $f$ is linear, ...
0
votes
0answers
32 views

Tietze Extension Theorem - How does the induction work?

I am reading a version of the Tietze Extension Theorem here: https://proofwiki.org/wiki/Tietze_Extension_Theorem There was a Lemma that says: And then it was repeatedly applied: How was the ...
10
votes
5answers
857 views

APICS Mathematics Contest 1999: Prove $\sin^2(x+\alpha)+\sin^2(x+\beta)-2\cos(\alpha-\beta)\sin(x+\alpha)\sin(x+\beta)$ is a constant function of $x$

This is question 3 from the APICS Mathematics Competition paper of 1999: Prove that $$\sin^2(x+\alpha)+\sin^2(x+\beta)-2\cos(\alpha-\beta)\sin(x+\alpha)\sin(x+\beta)$$ is a constant function of $x$...
2
votes
2answers
30 views

What is the coordinate of the maximum value of a quadratic function given by two points and axis?

There are only three pieces of information available: the graph passes through (0,0) and (6,0) the symmetry axis is $x$ = 3 the graph is downward My attempt: I've tried to work on ...
2
votes
0answers
50 views

Find all functions such that $f(c+x)+f(c-x)=k$ for some constants $c,k$

The question is inspired by this answer. Find all continuous and differentiable functions $\mathbb{R} \to \mathbb{R}$ such that $f(c+x)+f(c-x)=k$ for some constants $c,k$ in some interval $x \in (a,b)...
0
votes
2answers
29 views

If $f:A\to P(A)$, show that $Z_f := \{x \in A | x \notin f(x)\}$ is not in the Image of $f$

How can I prove that for a function $f: A \to P(A)$, $Z_f := \{x \in A | x \notin f(x)\}$ is not in the Image of f? It can be shown using Russel's Paradox, but i have really no clue on how to start. ...
0
votes
1answer
21 views

Showing properties of a function and its inverse image

I tried proving the following question but did not get too far. Let $\ f:A \to B$ be a function and $\ f^{-1}(Y)$ be the inverse image of $\ Y\subseteq B$ on $\ f$. Consider the following ...
0
votes
0answers
34 views

Alternative derivation of Euler's product formula for sine

Euler's product formula states that: $$\sin(x)=x\prod_{n=1}^{\infty}\left[1-\frac{x^2}{\pi^2n^2} \right].$$ There is also a very simple formula for another product representation for the sine ...
1
vote
1answer
21 views

Vector function of distance traveled.

Let the scenario be the following: We have a driving car whose start velocity is $100\frac{m}{s}$ and it's brakes reduce the velocity by $10\frac{m}{s}$, quite simple. If we were to make a vector ...
0
votes
2answers
25 views

Number of positive integral solutions of a polynomial inequation

The question : Let $f(x)=30-2x-x^3$. Find the number of positive integral values of $x$ which satisfies $f(f(f(x))))>f(f(-x))$. When I looked at this problem I noticed that the question talks ...
0
votes
0answers
23 views

A heuristic explanation of the Curse of Dimensionality

From Principles and Theory for Data Mining and Machine Learning, Clarke et al. (2009): This phrase [the "Curse of Dimensionality"] was first used by Bellman (1961)... The result is that ...
0
votes
3answers
46 views

Proving if $F^{-1} $ is function $\Rightarrow F^{-1}$ is $1-1$?

Let F be a function from set A to set B. If $F^{-1}$ is a function, then $F^{-1}$ is one to one. Prove: If $F: A \rightarrow B $ and $F^{-1}$ is a function, then F is one-to-one. Proof: ...
0
votes
0answers
26 views

What are interesting functions in 2D that vary visually as compositionality increases?

I wanted to create a function that its shape was a function of the depth of the compositionality (on a fixed interval). For example consider some compositional function $$f(x_1, x_2) = g( g( g( h_1(...
0
votes
1answer
28 views

Function Application and its Notation

For years and years and years I've always been taught that in mathematics, functions are applied as $f(x)$. But in my university textbook they also use three other notations: $$f\ x,$$ $$fx,$$ $$\...
-3
votes
1answer
56 views

Prove that f is constant under those conditions [closed]

Let $f:\mathbb{R}\to \mathbb{R}$ be continuous at $0$ and $1$ and assume further that $f$ satisfies the functional equation $$f(x^2)=f(x).$$ Prove that $f$ is constant.
3
votes
2answers
98 views

Very strange - what's the limit of $\lim_{x \rightarrow 0}\frac{sin(x)+cos(x)}{x}$?

What's the limit of: $\lim_{x \rightarrow 0}\frac{sin(x)+cos(x)}{x}$ ? $\lim_{x \rightarrow 0} \left (sin(x) + cos(x) \right) = sin(0)+cos(0) = 1 $ $\lim_{x \rightarrow 0} x = 0$ $\Rightarrow \frac{...
0
votes
2answers
46 views

Proof on surjective functions

I have this assignment: given a function $f:A\longrightarrow B$ between two sets $A,B$ prove that $f$ is surjective if and only if there exists a function $\psi:B\longrightarrow A$ such that $f\circ\...
4
votes
2answers
32 views

Show that the following equation has got exactly one solution for each $C>0$

Show that the equation $$C=\left ( 1+x+\frac{1}{2}x^{2} \right)*e^{-x}$$ has got exactly one solution for each $C>0$. Alright so I did it like that but not sure if it's correct: $0<\left ...
0
votes
2answers
46 views

How to find range of $f(x)=\tan x/\tan 3x$

This is one example I tried to solve when I was preparing for entrance exams. I have given $f(x)=\tan x/\tan 3x$ Then how can I find the range of $f(x)$?
0
votes
2answers
42 views

constructing a specific (real-) analytic function

Im searching for an example of a special-behaving analytic function. Maybe you can beat me to constructing such one. The criterias are $g :\mathbb{R}\rightarrow \mathbb{R^+}$ is analytic $g$ is $\...
1
vote
2answers
45 views

Is $\cos\theta$ or $\sin\theta$ an increasing function in first quadrant?

The question asks whether $\sin\theta$ is increasing function in first quadrant or $\cos\theta$ is increasing function in first quadrant. Other options are $e$ and $e^x$. I think the answer is $\sin\...
6
votes
1answer
42 views

Limit of sequence $\lim_{n\to\infty}\frac{1+(\sqrt{n}+1)^{3}+2\sqrt{n}}{n+\sin(n)}$

This is no homework. It's another task of a sample exam and I'd like to know how to solve it. Find the limit of $$\lim_{n\to \infty}\frac{1+(\sqrt{n}+1)^{3}+2\sqrt{n}}{n+\sin(n)}$$ Both ...
0
votes
1answer
34 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 ...
0
votes
3answers
72 views

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

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

Is it possible / allowed to use L'Hôpitals rule for products?

In our readings, we had L'Hôpitals rule and defined it like that: $\lim_{x\rightarrow x_{0}}\frac{f'(x)}{g'(x)}$ Because we had it in our readings, we are allowed to use this to find limit of ...
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 $$\...
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 ...
-1
votes
0answers
37 views

Find the area of $4x^2-2xy+y^2=1$ [closed]

Any help? Ive tried everything I can think of.
3
votes
2answers
56 views

How to precisely define a function that chooses randomly from a finite set?

Let $A = \{1, 2, \ldots, n\}$. I want to define a function that picks with uniform probability an element in $A$, so that $$f(A) = i \in A.$$ I don't know how to precisely define this mathematical ...
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 ...
5
votes
1answer
129 views

Find all solutions to $f\left(x^2+xf(y)\right)=xf(x+y)$

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^2+xf(y)\right)=xf(x+y)$$ for all $x,y\in\mathbb{R}$. This is somewhat related to this question, but with an $xf(y)$ term instead ...
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 ...
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, ...
1
vote
3answers
85 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\...
6
votes
3answers
316 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} \...