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

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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 ...
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3answers
36 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: ...
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0answers
20 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(...
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1answer
21 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: $$f\ x,$$ $$fx,$$ $$\...
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1answer
51 views

Prove that f is constant under those conditions [on hold]

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.
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2answers
88 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{...
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2answers
38 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\...
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2answers
28 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 ...
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2answers
44 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 $f(x)$?
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2answers
36 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 $\...
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2answers
40 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\...
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1answer
34 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 ...
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1answer
32 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 ...
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3answers
67 views

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

Let there be $$f:(-1,4)→ R$$ $$\text{differentiable on} (-1,4) , f(3)=5 , f'(x)≥-1$$ $$\text{which is the maximum value of}$$$$f(0)$$
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4answers
322 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 ...
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0answers
18 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 $$\...
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1answer
22 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 ...
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0answers
35 views

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

Any help? Ive tried everything I can think of.
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2answers
54 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 ...
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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 ...
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1answer
99 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 ...
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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 ...
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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, ...
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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\...
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3answers
294 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} \...
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4answers
397 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, & \...
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2answers
20 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 (...
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3answers
29 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, ...
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2answers
36 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 ...
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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: ...
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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 ...
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2answers
46 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 $...
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2answers
31 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 ...
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3answers
49 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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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2answers
51 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\...
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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: ...
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1answer
27 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 ...
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0answers
18 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)
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3answers
60 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 ...
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19 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 ...
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3answers
62 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.
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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 ...
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62 views

Is this an Example of a Dual Space? [on hold]

Is the set of possible bases that I describe $∀(e_1,e_2,e_3)$justSlash$∀(e_1,e_2,F(e_1, e_2))$ F defined V, \times. v=e_1 \times e_2*for any linear vector space of dimension 3* and their linear ...
2
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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\...
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0answers
18 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}...
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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$...