# Tagged Questions

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### How to find an example

I want to find a function $f\in C^1([0,+\infty)\times\mathbb{R},\mathbb{R})$ such that $f(t,0)=0$ $f(t,u)\leq \alpha u+\beta$, $\alpha<\lambda_1,\beta\geq 0$ $f(t,u)\geq C_1 |u|^{\sigma}$ where ...
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### Composition of injections (proof)

I'm trying to prove that composition of injections is an injection. I want to know if this is a good proof: Composition of injections is an injection. Let $f:S_1\rightarrow S_2$ and ...
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### If $f(g(x))=\sqrt {x^2-2x+8}$ and $f(x)=\sqrt x,$ find $g(x)$.

If $f(g(x))=\sqrt {x^2-2x+8}$ and $f(x)=\sqrt x,$ find $g(x)$. There is no example like this in my math book.
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### problem on continuity [on hold]

For $x>0$, let $[x]$ denote the largest integer less than or equal to $x$. Let $f:[0,\infty)\rightarrow\mathbb{R}$ be given by $f(x)=[x^2+[x^2]]\sin(2\pi x)$. Then $f$ is continuous at $2$ or ...
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### Sketch $y=2x^3/(x^2-2)$ [closed]

Sketch the curve $$y=\frac{2x^3}{x^2-2}.$$ Can someone answer this for me as basic as possible. Year 11 extension if possible. Thanks
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### Asymptotic Notation of polynomials [closed]

If ${P(n)=\sum_{i = 0}^{i=d} a_in^i}$ , where $a_d>0$ be degree-d polynomial in n and let k be a constant. How to proof: If $k>=d$ then $p(n)=O(n^k)$ also $k=<d$ then $p(n)=\Omega(n^k)$ ...
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### Hypergeometric Distribution Function?

I'm looking for a function that I can use in excel to calculate the probabilities of having certain cards in an opening hand. For example a function that will calculate the probability to get AT ...
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### Surjectivity of composition

I know that this question has been posted few times, but I want to check MY proof, because this is my first time trying to prove anything in mathematics. (I'm afraid if I just copy paste their proofs ...
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### Getting to answer on difference quotient/function problem

Q: Find the difference quotient $\dfrac{f(x) - f(3)}{x - 3}$ for $f(x) = \dfrac{1}{x}$ Ans a: $\dfrac{1}{3x}$ Haven't been able to get to that answer. I got the bottom $3x$ right once but the top ...
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### Existence of injective function in a manifold with special atlas

I am trying do the following question: Let $M$ be a $n$-dimensional smooth manifold that admits an atlas with only two charts. Show that there exists an injective smooth map ...
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### Finding the range and domain of $f(x)=\tan (x)$

I am attempting to find the range and domain of $f(x)=\tan(x)$ and show why this is the case. I can seem to find the domain relatively well, however I run into problems with the range. Here's what I ...
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### Finding the range and domain of $h(x) = \sec (x)$

I am attempting to show how to find the range and domain of $h(x) = \sec (x)$. Here's my working so far. Consider $h(x) = \sec (x)$, which is defined as $h(x) = \sec (x)=\frac{1}{\cos(x)}$. We know ...
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### Functions and Relations - Help!

Given that : \begin{align} &f: D_1 \rightarrow \mathbb{R} \\ & g: D_2 \rightarrow \mathbb{R} \end{align} Find, $f + g : D_1 \cap D_2 \rightarrow \mathbb{R}$.
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### What is integral of $x^x$?

I have no idea on how to approach this problem. I tried solving it by taking logarithm and then evaluating, but that won't serve the purpose I guess. Can someone please help?
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### What is the linear analog of cusp? And difference between cusp and pole?

say some function has a singular line. Is that pole? If yes then what is the difference but cusp and pole besides the former is a point and the later is line?
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### Transformations order?

If I have the function $$f(x)$$ would I do e.g. a stretch scale factor $1/a$ parallel to the $x$-axis followed by a translation of $b$ units to the left like this $$stretch: f_1(x)=f(ax)$$ and then do ...
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### For which sets, $X$ the relation is a partial function

Given $T=\left\{\ \left<A,B\right> \in (P(X))^2 | A\subseteq B \right\}$ For which sets, $X$, the relation $(P(X))^2-T \cap (P(X))^2-T^{-1}$ is a partial function? Here's my solution: ...
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### Modulus function (working out coordinates)

Lets say you have $y = -|3x - 1|$ when working out where it cuts the axis, particularly the x-coordinate you do the following when $y = 0, 3x - 1 = 0$ therefore $x = 1/3$ the modulus and the ...
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### How to prove that a function is continuous?

Could you give me some hint how to solve this question: Suppose $f$ is a differentiable function for all $0<x<1$,$f(0)=1,f'(x)>0$ in the given interval. It is obvious that $f$ is continuous ...
Does anyone know how to calculate integral of $\sqrt{ 1-\cos (x)}$ ? I tried several methods resulting in $-2\sqrt2 \cos (x/2) + c$, but this is wrong in accordance with the text book, so i dont know ...