# Tagged Questions

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

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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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### 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.
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### 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} \...