Rational functions are ratios of two polynomials, for example $(x+5)/(x^2+3)$.

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Automorphism of $\mathbb C(x,y)$ and its order in $\mathrm{Aut}(\mathbb C(x,y))$

In the following problem Let $M=\bigg (\begin{matrix} a& b\\ c& d\end{matrix}\bigg )$ be a nonsingular matrix with integer coefficients and $L=\mathbb C(x,y)$. (i) Show that $\phi(x)=...
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Plot of a function

What is the plot of: $$y=\frac{\beta(1-\alpha)x}{\alpha(1-\beta)+(\beta-\alpha)x}$$ with $0<\alpha<\beta<1$. How do I handle the parameters? How do I compute the derivatives to check for ...
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1answer
97 views

Luröth's Theorem

I've been struggling trying to understand the Jacobson's Basic Algebra vol. II proof of the Luröth's theorem. Let $K$ be a field, $K(X)$ the field of rational fonctions and take $L$ to be a sub-...
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Why do some rational functions approach the asymptote from the other side?

There are some rational functions that do a switcheroo when $x \to -\infty$ where (when graphed) they cross the horizontal asymptote in order to approach from the other side. It only seems to happen ...
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Simplify $f(x)=\Gamma(n/2)/(\Gamma(1) \Gamma(n/2-1))$… a Rational Expression using the Gamma Function.

I was reviewing a document about an algorithm wherein it is stated that $f(x)$ is a probability density function: (1)$$ f(x)=\frac{\Gamma(\frac{n}{2})}{\Gamma(1)\Gamma(\frac{n}{2}-1)}\frac{2}{n-2}\...
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Fixed Point in the Space of Rational Functions

Let $\mathcal R$ be the space of rational functions and $F: \mathcal R \to \mathcal R $ be a function that transforms a rational function into another rational function. Is there a fixed point ...
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Subtracting $\frac{(x+3)}{(x^2-1)} - \frac{(x-2)}{(x^2+2x+1)}$

$\frac{(x+3)}{(x^2-1)} - \frac{(x-2)}{(x^2+2x+1)}$ To solve the problem I first dissembled the equation on the denominator $ \frac{(x+3)}{(x-1)*(x+1)} - \frac{(x-2)}{(x+1)^2}$ I multiplied the ...
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Showing that the slope of a complicate function is greater than 1

I have a $V$ function: $$V_n(x, y) = -\frac{1+2x}{7+5x+n+2nx} + \frac{-1+2x}{2-n+x(5+2n)} - \frac{(1-2y)^2}{(2-n+y(5+2n))^2} + \frac{4(2+y)^2}{(7+n+y(5+2n))^2}.$$ Here's the input form in case you ...
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34 views

Prove the inequality involving exponential function in form of $\exp( \frac{1}{x} )$

For $\nu > 0$, $0 < x \leq \nu $, and a positive integer $S$, (we think) following an inequality always holds $1- \left( \frac{1}{x+1} \right)^S \geq \exp \left( -\frac{1}{Sx} \right) $ Does ...
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Constructing a rational function from its asymptotes

Question: Give an example of a rational function that has vertical asymptote $x=3$ now give an example of one that has vertical asymptote $x=3$ and horizontal asymptote $y=2$. Now give an example of a ...
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Why is $y=\frac{x+y}{x}$ the same as $y=\frac{x}{x-1}$?

Why is $y=\frac{x+y}{x}$ the same as $y=\frac{x}{x-1}$? It seems to be the same on graphing calculators, but I don't get why. To generalize my statement, I would like to know how to simplify ...
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1answer
22 views

Three-dimensional curve whose coordinates are rational functions

Are there three real rational functions $f,g,h$ with no poles in $[0,1]$, such that $f\geq 0,g\geq 0,f+g \geq h \geq 0$ on $[0,1]$ and the curve $\gamma(t)=(f(t),g(t),h(t)) (t\in [0,1])$ passes ...
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35 views

A function that is locally a quotient of polynomials but not globally [duplicate]

Let $X =\{ x_1x_4=x_2x_3\;, (x_2,x_4) \neq (0,0)\} \subset \mathbb{C^4}$, i.e. not both of $x_2,x_4$ are zero. Define a function $\phi$ on $X$ by $\phi(x)=\left\{\begin{matrix} \frac{x_1}{x_2} & ,...
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Approximating the normal CDF for $0 \leq x \leq 7$

In answer http://stackoverflow.com/a/23119456/2421256, an approximation of the complementary normal CDF (ie $\frac{1}{\sqrt{2 \pi}} \int_x^{+\infty} e^{-\frac{t^2}{2}} dt$) was given for $0 \leq x \...
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Why is a polynomial also a rational function?

In a recent question, I asked about non-standard-looking rational functions, i.e., something that was not in the classic numerator-denominator form. I was told that all polynomials are rational ...
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1answer
14 views

Creating a rational function with specific parameters

For a game I made, tetris, the blocks must go faster and faster every level, I want the speed to be $500$ at level $1$, and $+-250$ at level $6$ ($500$ means, $1$ block is moving down per $500$ $ms$)...
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Evaluating the rational integral $\int \frac{x^2+3}{x^6(x^2+1)}dx $

Evaluate $$\int \frac{x^2+3}{x^6(x^2+1)}dx .$$ I am unable to break into partial fractions so I don't think it is the way to go. Neither is $x=\tan \theta$ substitution. Please give some hints. ...
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Basic equation solving $t/(1+t)=1-1/(1+t)$

In found this equality in my math book, could anyone explain to me why it is equal? $t/(1+t)=1-1/(1+t)$
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1answer
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Must a rational function always be in numerator-denominator form?

This may seem trivial, but I'm looking at two examples from high school math books and wondering if they are really examples of rational functions. The first is a line $\overline{DT}$ made up of two ...
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If f/g is symmetric (resp homogeneous), must f and g be as well?

Suppose we have two polynomials $f$, $g$ in $k[X_1, ..., X_n]$ over some field $k$, and they have no factor in common. Suppose that $f/g$ is symmetric. Must than $f$ and $g$ also both be symmetric? ...
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When a Rational function becomes a line?

Why is it that when $AD = BC$, this equation becomes a horizontal line? $$y = \frac {Ax+B}{Cx+D} $$ For any other values where $AD$ isn't equal to $BC$ it is a rational function.
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Cancelling variable factors in a rational function

Consider the function $\displaystyle\frac{2x−1}{x+5}$. The domain of this function is all real numbers except $x = -5$. Now consider that I do this: $\displaystyle\frac{2x−1}{x+5}⋅\frac{x}{x}$. This ...
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64 views

Why do rational functions like $\frac{1}{x}$ have weird shapes? [closed]

As many know, rational functions have weird shapes like hyperbolas placed diagonally. They are in couple parts, for example $\frac{1}{x}$. Why does that happen?
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How to show that $X^p-t\in\mathbb{F}_p(t)[x]$ is irreducible? [duplicate]

This question is previously asked here, but there is no complete solution of it. I understand that the root $\alpha$ exist in the algebraic closure of $\mathbb{F}_p(t)[x]$, and it is the only root ...
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28 views

Does every non-Archimedean ordered field contain $\mathbb Q (x)$ as an ordered subfield?

Every ordered field $\mathbb F$ contains $\mathbb Q$ in a canonical way. If the field is not Archimedean there exists an $x>n$ for all $n \in \mathbb N$. Since we are dealing with a field any ...
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Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$

Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$. How come the answer is left as $\frac{5x}{(x+2)(x-3)} + \frac{4}{(x+2)^2}$. Why don't we go any further?
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Continuity of solutions for systems of rational ODEs

If you have a system of ODEs of the from $\frac{d\mathbf{x}}{dt}=\mathbf{f(x,p)}$ where $\mathbf{x}$ is a vector valued variable, and $\mathbf{p}$ is a vector of parameters, and $\mathbf{f(x,p)}$ is a ...
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Showing that the field of rational functions is not dense

I am going through Counter Examples of Analysis but I am having trouble understanding a claim it makes. The book establishes that the set of rational functions defines an ordered field where the "...
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How would I divide $\frac{4x^2-2x}{x^2+5x+4}\div \frac{2x}{x^2+2x+1}$? [closed]

How would I divide $\frac{4x^2-2x}{x^2+5x+4}\div \frac{2x}{x^2+2x+1}$? Work would be appreciated. What would I do first?
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How do I multiply $\frac{8y-4}{10y-5}\times \frac{5y-15}{3y-9}$?

I am not sure how to multiply these fractions. Do I cross multiply or multiply the numerators together and multiply the denominators together? How do I multiply $\frac{8y-4}{10y-5}\times \frac{5y-15}{...
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The best way of integrating irrational functions

Ok, so, here is the example integral: $$I=\int\frac{x-2-\sqrt{-x^2-4x+4}}{x^2-\sqrt{-x^2-4x+4}}dx$$ I always solve these types of integrals using Euler's substitutions, but, recently, I came across ...
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Why is this a correct way to multiply 2 terms together?

In a problem, I noticed the author did this: $$\frac{1}{(a+2)+(z-2)} = \frac{1}{(a+2)}\cdot \frac{1}{1+\frac{z-2}{a+2}}$$ What he is saying is to take the entire $(a+2)$ term and multiply it by $1$ ...
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What is known about this group reminiscent of the anharmonic group?

The anharmonic group is this nonabelian group of six rational functions with the operation of composition of functions: \begin{align} t & \mapsto t & & \text{order 1} \\[8pt] t & \...
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Has this chaotic map been studied?

I have recently been playing around with the discrete map $$z_{n+1} = z_n - \frac{1}{z_n}$$ That is, repeatedly mapping each number to the difference between itself and its reciprocal. It shows some ...
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$\{a x^{z}: a\in \Bbb{Q}, z \in \Bbb{Z}\} \approx \Bbb{Q}^{\times} \otimes_{\Bbb{Z}} \Bbb{Z}^+ \implies$? what about $\Bbb{Q}$-linear sums?

Consider all functions $f: \Bbb{Q} \to \Bbb{Q}$ of the form $f(x) = a x^z$ where $a \in \Bbb{Q}, z \in \Bbb{Z}$, call it $G$. It forms an abelian group under usual multiplication. I think it's ...
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Rational function not in lowest term with same zeros in top and bottom always has hole in the graph or not?

I'm studying about rational function and I have watch this video: https://www.youtube.com/watch?v=sRgmtojZGFg&list=PLDE28CF08BD313B2A&index=17 in the minute (1:09:50) the instructor talks ...
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Quickest way to factorize $\frac{w^2 + 5kw + 4k^2}{w^2+kw}$

I would say 90% of polynomials in my textbook are factorable e.g. $$\frac{w^2 + 5kw + 4k^2}{w^2+kw}$$ This gives $$\frac{(w+k)(w+4k)}{(w+k)w}$$ $$\frac{w+4k}{w}$$ This took me far too long to ...
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Does $f(x) = 1 = \frac{x-1}{x-1}$ have a hole at $x=1$?

While learning about asymptotes and holes in rational functions in Precalculus, I came across a problem that shouldn't happen, but I don't understand. Something just doesn't add up in my head... Here ...
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Does this equation has a closed form solution?

We have $K$ non-negative coefficients: $a_1,a_2,\dots,a_K,A_1,A_2,\dots,A_K$, where $A_i\geq0,\;a_i\in(0,1),\;\sum A_i<T$. The equation is: $$\sum_{i=1}^K\frac{A_i}{1-a_ix}=T,\quad x\in(0,1)$$
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integrate $\int \frac{x^2+x+1}{3x^2-2x-5}$

$$\int \frac{x^2+x+1}{3x^2-2x-5}$$ I can see where t start, maybe $$\int \frac{x^2+x+1}{3x^2-2x-5}=\int \frac{6x-2}{3x^2-2x-5}+\frac{x^2-5x+3}{3x^2-2x-5}=\ln|3x^2-2x-5|+\int \frac{x^2-5x+3}{3x^2-2x-...
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Indefinite integral of rational functions

This question seemed very trivial at the first glance, but evidently it is a non-trivial problem, at least, for me. Let $p, q$ be polynomials in $x$, and $\deg q > \deg p$, with $q (x)$ being ...
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How to solve integrals where you can't factor a polynomial?

Hi there guys I don't know if the title of the question should be the one for this but the thing is that I'm trying to solve this integral $\int \frac {\frac 12-u^2}{2u^4-2u^2+1}$$du$ and I have this ...
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What does it mean that “$f = g$ in $k(h(t))$?”

Let $k$ be a field and consider the rational function field $k(t)$. I was just reading that if $f(t),g(t),h(t)\in k(t)$ are such that $f(h(t)) = g(h(t))$, then "$f = g$ in $k(h(t))$." What does that ...
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Does this method of finding the range of rational functions always work?

Consider the irreducible rational function in $\mathbb{R}^2$. $$y=\frac{A(x)}{B(x)}$$ where at least one term is quadratic and the other term has degree either 0, 1 or 2. The classic way of ...
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Degree of Rational Function

This might sound like a very trivial question but I found different answers on the web. Assume one has a rational function $$\frac{f(x)}{g(x)} ,$$ where $f(x)$ and $g(x)$ are polynomials. What is ...
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48 views

Is there a rational function $f$ satisfying $f(x) =f\left( \frac{1}{1-x} \right)$ for all $x$?

I would like to find a pair of relatively prime polynomials $p,q \in k[x]$ (where $k$ is a field) such that $$\frac{p(x)}{q(x)} = \frac{p \left( \frac{1}{1-x} \right)} {q\left( \frac{1}{1-x} \right)} ....
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How do I perform u-substitution on this problem?

I am having trouble with this problem: $$\int {\frac{3x + 5}{5x^2 - 4x - 1}} dx$$ I can't seem to find a u where the du exists in the numerator so that it will cancel. If I choose: $$u = 5x^2 - 4x - ...
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200 views

How to integrate $\frac{1}{(1 + x^5)(1 + x^7)}$

My cousin who is in high school asked me if it is possible to integrate $$ \int \frac{1}{(1 + x^5)(1 + x^7)} \, dx $$ I checked the list of integrals of rational functions on Wikipedia link and it ...
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why does the equation $(-x^2 + 2x)/(5x - 4) = 6$ have 2 solutions?

Hmm, I have been wondering about this when I went to solve the following equation: $$\frac{-x^2+2x}{5x-4} = 6$$ How come the above equation has two solutions, $-14 + 2\sqrt{55}$ and $-14 - 2\sqrt{55}$...
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148 views

Is there a rational surjection $\Bbb N\to\Bbb Q$?

The question is in the title. Is there a one-dimensional rational function $f\in\Bbb R(X)$ which restricts to $\Bbb N\to\Bbb Q$, which is a surjection onto $\Bbb Q$? My guess is no. Expanding the ...