# Tagged Questions

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

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### Show that $\frac{1}{1+k}=\frac{\frac{1}{k}}{1+\frac{1}{k}}\leq \ln(1+\frac{1}{k})\leq\frac{1}{k}$

Prove the following: $$\frac{1}{1+k}=\frac{\frac{1}{k}}{1+\frac{1}{k}}\leq \ln(1+\frac{1}{k})\leq\frac{1}{k}$$ I know I can prove it with induction if the values were naturals. However, the "problem" ...
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### Big-Oh, Big-Omega, Big-Theta determination

I am given a recurrence relation and told to solve it. Once we solve it we are supposed to determine whether it is in $O(f(n)), \Omega(f(n))$, or $\Theta(f(n))$. The relation is $t_n = 2nt_{n-1}$. ...
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### Is there any way to differentiate such function?

Let $S$ be a set. If I had a bijection $f$ mapping each element $n\in \mathbb{N}$ to an element $s \in S$ such that: $$s = f(n) = \sum^{n}_{k=1} {1\over k}$$ Is the function differentiable in ...
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### Has the following concept for integer valued functions been studied

Let $[a] = \{1,2,..a\}$, $[b] = \{1,2,..b\}$ and $F(a,b)$ be the set of all functions from $[a]$ to $[b]$. I define a subset $S \subset F(a,b)$ to be $k$-distinct if each pair of functions in $S$ ...
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### Is there a difference between $f(x,y)$ $f(x;y)$ and $f(x\mid y)$?

While reading I have come across all three of these notations seemingly at random, and as far as I can tell they are all positional arguments to a function, but I can't tell if they mean different ...
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### Can such an equation exist? [closed]

$$y(x)=\lim_{h\to 0}\tan\left(\frac{2\pi}{h}x\right)$$ $$L(z)=y(x)z+c$$ I found such an example in a strange maths book in the dusty section of the library. It said this equation produces rotating ...
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### Functions to represent set operations?!

Assume you have set of real positive numbers $a_1,...,a_n$. And a strictly decreasing convex function $f$. Assume the intervals $A_i = [0,f(a_i)]$ to represent $i^{th}$ set, $i = 1,...,n$. Can we ...
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### Part of a sigmoid function?

I revised a sigmoid function to use in my research. The function looks like this. $$f(x) = 0.4 \cdot \frac{1}{1 + e^{-5x}}+ 0.3$$ where $x \in [-1,1]$. Is there a specific name to refer to this ...
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### Are the following families of sets closed under intersection?

Problem Statement Let $X$ be any set whatsoever, and let $f:X\to X$ be any function. Note that in general, no structure is imposed on $f$ whatsoever (i.e. continuity, linearity, etc). The problem is ...
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### What is the polar formula for $y=x$?

$y=x$ is a basic cartesian equation, but I'm at a loss as to what it is in polar form. It seems the only way I've found to express it is with $r$ on both side of the equation, but is there a way of ...
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### Proving P(x) > 0 given a condition.

$P(x)$ is a polynomial function such that, $P(1) = 0, P′(x) > P(x), ∀ x > 1.$ Prove that $P(x) > 0, ∀ x > 1.$ I was trying to do by taking the P(x) in the denominator and then ...
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### What is the correct method of finding the leading order behavior of a function in a given limit?

I am kind of confused about finding the correct leading order behavior of a function. Example: If I want to understand the behavior of the following function $$f(x)=\coth(x)-\frac{1}{x}$$ I can ...
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### Assume that $f : X \rightarrow Y$ is surjective. Show that $f(A^c) = (f(A))^c \ \forall A\subset X$ iff f is also injective.

Assume that $f : X \rightarrow Y$ is surjective. Show that $f(A^c) = (f(A))^c \ \forall A\subset X$ iff $f$ is also injective. So I tried starting with the right implication $\Rightarrow$ Since ...
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### Continuous non-differentable functions

I'm looking for some examples of everywhere continuous functions which are nowhere differentable. I found already Takagi curve and Weierstrass function. Can you point out some online courses or pdf ...
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### Meanvalue theorem for quadratic arguments

I have trouble proving the following result and I would be happy about any kind of suggestion how the precise argument looks like. Let $f : [0,\infty) \rightarrow \mathbb{R}$ denote a continuously ...
### Is $x\sin x$ Surjective?
I have to determine whether $x\sin x$ Is a surjective function, in $\Bbb R$. My solution: Let $f(x)=x\sin x$. $f(\frac{\pi}{2})=\frac{\pi}{2}$ $f(-\frac{\pi}{2})=-\frac{\pi}{2}$ Therefore in ...
### $C ([1,2] \times [0,1] \to \mathbb R)$ dense in $C ( [1,2] \rightarrow L^{2} ([0,1] \to \mathbb R))$?
Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense ...