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

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Find the sum of the maximum and minimum

For a real number $x$ find the sum of the maximum and minimum. $$y=\frac{x^2-2x-3}{2x^2+2x+1}$$ This is a sample question for a math competition. It seems like calculus would be used to solve this, ...
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63 views

Geometry aspect of a extreme value problem

In a plain with orthogonal coordinate $XOY$, set point $A(a,a)$, and $P$ is a point in function $y=\frac{1}{x}$,where $x>0$. If the distance between $P$ and $A$ is $2\sqrt{2}$.Find all $a$ ...
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How prove this equation roots $\frac{1}{x}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+2\sqrt{\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}}$

Question: Let $a,b,c>0$ are give numbers and $x>0$, such that $$ \sqrt{\dfrac{a+b+c}{x}}=\sqrt{\dfrac{b+c+x}{a}}+\sqrt{\dfrac{c+a+x}{b}}+\sqrt{\dfrac{a+b+x}{c}} $$ show that $$ ...
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Solve this simple polynomial

$$\text{Problem: }{x^3-x^2-x-2 \over x^2 + x - 6}$$ My textbook was able to come up with $(x-2)(x^2+x+1)$ $$\text{Textbook: }{(x-2)(x^2+x+1) \over (x-2)(x+3)}$$ I've tried grouping and using the ...
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Conditions under which asymptotes can be intersected/crossed?

What are the conditions under which some horizontal asymptotes can be crossed, and why is this acceptable? I'm speaking specifically to rational functions, such as in the case of the function $$ f(x) ...
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19 views

Notational issue

Let $K = F(t)$. If $r \in K: (\nexists c \in F: r(t) = c \forall t)$ is a rational function and $L = F(r(t))$, then what form does $f \in L$ have? Is it a rational function where the coefficients are ...
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131 views

How to evaluate $\int_0^1\frac{1+x^4}{1+x^6}\,dx$

$$\int_0^1\frac{1+x^4}{1+x^6}\,dx$$ Can anyone help me solve the question? I am struggling with this.
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How do i define 'complex rational function'?

http://en.wikipedia.org/wiki/Rational_function I don't get the definition in wikipedia. It would be great to define "complex rational function" with the domain $\overline{\mathbb{C}}$, namely ...
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I am not sure if the answer for the dividing rational expressions problem (attached) should be simplified

$$ \frac{x}{x+2} \div \frac{1}{x^2 - 4} $$ Original image: http://i.stack.imgur.com/EEqd4.jpg -I am not sure if the answer for the problem (which is attached) would be $\frac{x^3-4x}{x+2}$ or if it ...
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Looking for methods/results for explicitly bounding iterations of rational functions

This is a cross-post of http://mathoverflow.net/questions/155775/looking-for-methods-results-for-explicitly-bounding-iterations-of-rational-funct But I received no answer there to the actual ...
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39 views

What is the common denominator

Help me find the common denominator please! Thank you Here it is : $$\dfrac{5}{3x+2}-\dfrac{3}{3x-2}=\dfrac{7}{6x-4}+\dfrac{x+4}{18x^2-8}. 1$$
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PSD Rational function to power series

I have a Power Spectral Density given as a Rational function that is: \begin{equation} \phi(e^{i\omega}) = 1/(1-1.7464e^{-i\omega}+1.2602e^{-2i\omega}-0.4366e^{-3i\omega}+0.625e^{-4i\omega})^2 ...
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How does $\frac{x^2}{ x^2 + 1}$ simplify to $1-\frac{1}{1+x^2}$

How does $$\frac{x^2}{ x^2 + 1}\quad\text{ simplify to }\quad 1-\frac{1}{1+x^2}\;?$$ Can someone explain the steps of how to get to that alternate form?
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Laplace transform to describe a bounded function

It is easy to show that if a real function $f:\mathbb{R}\rightarrow\mathbb{R}$ is contained in a strip $[a,b]$, that is if $\forall_{x}\, a\le f(x)\le b$, then its Laplace transform is bouned by ...
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$\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$

Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1,\, n\in \mathbb{N}$ For example. For $n=2$, we have $\lfloor ...
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17 views

How can I simplify this rational function for graphing by translation?

I've actually struggled with how to properly simplify this rational function, and I'm hoping someone can point me in the right direction. It's part of my precalculus class/section on graphing rational ...
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41 views

What does the notation $O(x^n)$ mean?

I am reading a book about Padé Approximations, and I am trying to understand the following line: We denote the $[L/M]$ Padé approximant to $A(x)$ by $A(x) - P_L(x)/Q_M(x) = O(x^{L+M+1})$ where ...
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how to find the integral of a rational logarithmic function

I can't seem to figure this one out, it is: $$\int\frac{\ln(x)}xdx $$ I substituted $u$ for $\ln(x)$, so $u = \ln(x)$ and $du = \frac1x dx$ then to find $x$ in terms of $u$: $e^u = x$ so ...
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Prove that there is no rational function in $F(x)$ such that its square is $x$

I came across this homework problem and honestly have no idea where to begin. Prove that there is no rational function in $F(x)$ such that its square is $x$. First of all, I am confused about ...
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51 views

When the system of equations below had a solution?

The system of equations is $$\begin{cases} \frac{c_1}{1-x_1}+\frac{c_2}{1-x_2}+\frac{c_3}{1-x_3}=0\\ \frac{c_1}{k-x_1}+\frac{c_2}{k-x_2}+\frac{c_3}{k-x_3}=0\\ ...
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118 views

I wonder whether the system of equations and inequations below have a solution.

I wonder whether the system of equations and inequations below have a solution. If there are solutions, what are they? A numerical solution is also desired. $$\begin{cases} ...
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Connection between rational functions and matrices

$$ y = \frac{ax+b}{cx+d} \Longleftrightarrow x = \frac{dy-b}{-cy+a}, ad-bc \ne 0 $$ on the other hand $$ \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)^{-1} = ...
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$f$ differentiable, $f(x)$ rational if $x$ rational; $f(x)$ irrational if $x$ irrational. Is $f$ a linear function?

Let $f$ be an everywhere differentiable function whose domain consists of all real numbers. Assume that $f(x)$ is rational for rational $x$ and irrational for irrational $x$. Can we conclude that $f$ ...
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Is there an accepted definition of the “coefficients” of a multivariate rational function?

I'm reading Complexity and Real Computation by Blum, Cucker, Shub, and Smale. In in defining "machine constants" of a Blum-Shub-Smale machine, they talk about the "coefficient" of a multivariate ...
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21 views

Correspondence between an order of rational function field and a Dedekind cut

In the first page of Real Algebraic Geometry by Jacek Bochnak, Michel Coste, Marie-Francoise Roy, they briefly connect an order of the rational function field $R(X)$ with a Dedekind cut ...
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360 views

Solving a distance/time/speed problem using the quadratic formula.

"The distance between Toronto and Ottawa is 352.72 km. The speed on a road trip from Ottawa to Toronto was double of the return, and therefore the drive took 2 hours less. What was the speed on the ...
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Build the function by its values. Only combination of +, -, *, abs() allowed for this function.

I've decided to open a new, more common question about the simplest function f(1)=-1; f(2)=0; f(3)=1; f(4)=0.. So, here is the question. Let's say we have some function $y=f(x)$ we'd like to find by ...
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Ring of rational functions ideal generators

There is an affine variety $X\subset \mathbb{A}^n$ with its ring of rational functions which is the quotient ring of $\mathbb{k}[X]$ (each $f\in \mathbb{k}(X)$ has a form $\frac{p}{q}$ where $q$ does ...
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Integrating a complicated function

After spending a couple of weeks, I was able to find the solution to a certain differential equation, given below (Well they are the eigenfunctions to be exact): $$y_n(x) = ...
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Rational locus of a function defined on $x^2+x^3=y^2$

We have a curve $X$ on $\mathbb{A}^2$ given by $y^2=x^2+x^3$. Consider the rational function $f$ on $X$ which maps $(x,y)\in X$ to $\frac{y}{x}$. There is a nice geometric interpretation of $f$: if we ...
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Graphs of Rational Functions of higher degree

A rational function $f(x)=\frac{kx^n+lx^{n−1}+⋯}{jx^m+ix^{m−1}+⋯}$'s asymptotes are as follows: If $n<m$, then $f(x)$ will approach the line $y=0$ as $x \to \infty$ If $n=m$, then $f(x)$ will ...
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General identity for a double summation theorem

I've been reading a research paper, and I'm interested in generalizing a certain theorem but I can't seem to understand how the following results are derived: $$\sum_{i=1}^{n}\sum_{j\neq i}^n ...
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26 views

How to do this transformation of complex rational function?

Here is a description of constructing Möbius map sending $1$, $i$, $-1$ to 0, 1, $\infty$, respectively $m(z) = \frac{(z-1)}{(z+1)} \frac{(i+1)}{(i-1)}$ I do not know how it was converted to : ...
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Help explain method to solve two equations in two unknowns where one of the variables has a square term

Given: $-\frac{96}{x^2y}+1+y=0$ $-\frac{96}{xy^2}+2+x=0$ Solve for $x$ and $y$ How should I find $x$ and $y$? I thought of using the methods I learned in linear algebra but then I noticed that ...
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Determining the positioning of rational functions without plotting points

When graphing rational functions, how do I determine the orientation of a rational function around the asymptotes without plotting points? For example, is it possible to determine which one of these ...
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Need help simplifiying a rational expression

There's a math question on an online test which asks the following Multiply the following expression, and simplify: $\frac{x^2-16y^2}{x} * \frac{x^2+4xy}{x-4y}$ But no matter how I try I keep ...
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55 views

Predicting what the graph of a function looks like?

How can you predict what the function $$f(x) = \frac{(x - 5)(x + 4)(x - 3)^2(x)}{(x-5)(x)}$$ looks like before you plot it?
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meromorphic functions on the extended complex plane

I'm highly confused by one theorem. Every meromorphic function on the extended complex plane is rational. But $e^z$ is analytic everywhere in the plane, and since it is analytic outside a bounded set, ...
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When does a rational function pass through the horizontal asymptote/how do I know how it behaves?

Sorry if this is a bit simple compared to everything here, but I can't really seem to find an answer. If I have $$f(x) = \frac{(x-2)(x-4)}{x(x-1)}$$ 1) When is the horizontal asymptote is crossed? ...
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If two elements a and b in a field extension L/K are algebraic in K and have the same minimal polynomial, prove that $K(a)=K(b)$

I have a field extension $L/K$ and two elements $a,b \in K$, which are algebraic over $K$ and both have the same minimal polynomial, and need to prove that then $K(a)=K(b)$. I can see how this can be ...
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Prove that $\mathbb Q(\sqrt 2) \neq \mathbb Q(\sqrt 3)$

I've tried writing out the contents of each and attempting to get a contradiction by equating arbitrary elements but can't get this to work. I can't think of any counterexamples as everything I come ...
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Given $K = \mathbb Q[x]/(x^2 - 5)$, a field extension of $\mathbb Q$, prove that $K \simeq \mathbb Q(\sqrt 5)$

My reasoning so far is that $(x^2 - 5) \cong 0$ in K, so we can set $x^2-5=0$ and get $x=\pm \sqrt 5$, which to me suggests $K \simeq \mathbb Q[x]$ but I'm having trouble jumping to the field of ...
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Is every injective rational function $f:\mathbb Q\to\mathbb Q$ a polynomial?

I thought this might be quite easy to show, and then realized that the tools I know from real analysis aren't going to help here. Suppose we have a rational function: $$ f(X)=\frac{P(X)}{Q(X)} $$ ...
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Finding a Rational Function

Find a rational function $f:\mathbb{R} \to\mathbb{R}$ with range $f(\mathbb{R})=[0,1]$ (Thus $f(x)=\dfrac{P(x)}{Q(x)}$ for all $x\in \mathbb{R}$ for suitable polynomials $P$ and $Q$, where $Q$ has no ...
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$\arctan$ of a square root as a rational multiple of $\pi$

I know that if $x$ is a rational multiple of $\pi$, then $\tan(x)$ is algebraic. Is there a fairly simple way to express $x$ as $\pi\frac{m}{n}$, if $\tan(x)$ is given as a square root of a rational? ...
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Quick way to determine if a rational function has a hole on the $x$-axis…

To sketch a rational function's graph, one step is to determine the sign $(+/-)$ of various intervals. I create intervals separated by the vertical asymptote (VA) and $x$-ints on a number line (since ...
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Technique to solve this equation of 2 unkowns

I was solving a problem of single phase eletrical circuits where I had to find the inductor $L$ and resistance $R$. I managed to get two equations containing the two unknowns. ...
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Limit approaching 2a

When I substitute 2a into x, I get 0 but the answer should be 2.. Any help will be appreciated!
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How to convert a rational function into a power series with variable as a function?

I have a rational function $f(t)$ and a polynomial $g(t)$ e.g. $$f(t)=\frac{0.42(t^3+20.0 t^2+105.0 t+138.0)^3}{t^9+79.53 t^8+2728.0 ...
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How to compute $\frac{t}{t+1}$ to the form $1-\frac{1}{t+1}$?

How to compute $\frac{t}{t+1}$ to the form $1-\frac{1}{t+1}$? What else? Well. Well can you use long division?