# Tagged Questions

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

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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 [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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### 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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### 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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### 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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### 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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### 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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### 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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### 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.