Tagged Questions

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

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Key Features of This Rational Function

$$-2x^2-15x-25\over x^2-x-5$$ I'm not sure how you would find the Intercepts or the Asymptote of this function. I've tried factoring the equation but it leaves me with $$-1(2x+5)(x+5)\over ...
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1answer
23 views

Bivariate rational (quadratic over linear) model fitting by least squares

I am trying to fit a simple model over 2D data points, in the frame of an image formation model with perspective and optical distortion. My model is the ratio of a second degree polynomial over a ...
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1answer
35 views

Simplify using either Method 1 or 2. [on hold]

I need to simplify $$\dfrac{1 - \dfrac 4{t+5}}{\dfrac{4}{t^2-25} + \dfrac{t}{t-5}}$$ I am posting the methods I can use as images. !
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0answers
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Rational function interpolation?

We know that $n+1$ points is enough to completely determine a polynomial of degree $n$. Are there any techniques which says that a certain number of points is enough to completely determine a rational ...
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1answer
11 views

Prescribing zeros and poles of a rational function on $\bar {\mathbb C}$ at once

I have to show that for any points $P_1$, $\ldots$, $P_n$ and $Q_1$, $\ldots$, $Q_n$ ($P_i \neq Q_j$ for all $i$, $j$) on $\bar{\mathbb C}$ there exists a rational function $f$ with poles at $P_j$, $j ...
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2answers
589 views

Beautiful problem on a progression

$\{x_n\}$ is a sequence defined as follows: $x_1=20,\quad x_2=14,\quad x_{n+2}=x_n - \frac{1}{x_{n+1}}$. Prove that $0$ is among the members of this sequence. Find its number. I tried some stuff ...
2
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0answers
23 views

Finding the singular part of a particular rational function

Find the singular part of $$f(z)= \frac{1}{(1+z^3)^2}$$ at $z=-1$. I tried to compute the Laurent series expansion at $z=-1$, as the term with the negative exponent of $(1+z)$ would be the singular ...
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2answers
16 views

Domain and Range of a rational function

given the rational function $\frac{1}{x^{2} - \frac{x}{2}-3 }$ and asked for the domain and range, I multiplied thru by 2 and got $\frac{2}{2x^{2} - x-6 }$ I understand the domain includes all real ...
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1answer
21 views

What is the possible domain and range for this rational function?

Given a function: y= $\frac{(2x^2+11x+15)^{1/2}}{(x^2-9)^{1/2}}$ What is the domain and range for this function? My attempt: To find domain, I need to find the value of x such that the function ...
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0answers
24 views

Estimating the distance to the Julia set of a rational map

Suppose that $f \colon \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ is a rational map of degree $d \ge 2$. Let $z_0$ be a point in the Fatou set $F(f)$. I'm interested in finding an estimate for the ...
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0answers
31 views

Rational function inequality

I've encountered the following inequality in a proof: $$ \left| \frac{P(Re^{it})}{Q(Re^{it})} \right| \leq \frac{R^k|a_k+..+a_0/R^k|}{R^n|b_n-..-b_0/R^n|}$$ Where $P(z)=a_0+a_1z+..+a_kz^k$ and ...
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2answers
35 views

How to compute the limit of a rational function at infinity?

I am unable to compute the limit, anyone can help ? I only understand some basic knowledge of limit .
4
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3answers
131 views

Limit $\lim_{{x\rightarrow 0}}{\left(\frac{\frac{1}{\sqrt{1+x}}-1}{x}\right)}$

I am a high school student in Calculus, and we are finishing learning basic limits. I am reviewing for a big test tomorrow, and I could do all of the problems correctly except this one. I have no ...
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1answer
32 views

Rational function in both $k(X)[Y]$ and $k(Y)[X]$

If I have a rational function in $X$ and $Y$ and it can be written as both a polynomial in $Y$ with coefficients being rational functions in $X$ (that is, an element of $k(X)[Y]$) and as a polynomial ...
1
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1answer
42 views

Possible for variable to cancel in a product of multivariate rational expressions?

Let $f,g,p_i,q_i$ be polynomials over some field with $\gcd(p_i,q_i)=1$ and $q_i$ are not constants for $i=1,2$. Assume that one or more of $p_i$ or $q_i$ has a term containing a variable $x$ not ...
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0answers
33 views

Rational equality modulo $p$

Assume that we have two rational expressions $f,g\in \mathbb{Z}(x,y_1,\ldots,y_n)$ with the property that the variable $x$ can only appear in the numerator of $f$ while only in the denominator of $g$. ...
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4answers
50 views

Find the domain of the function $f(x) =\frac{x+4}{x^2-9}$

I need to find the domain of the function $\;f(x) =\dfrac{x+4}{x^2-9}.$ My answer was: $(-\infty, -3)\cup(3, \infty)$. The book's answer was: $(-\infty, -3)\cup(-3,-3)\cup(3,+\infty)$ It's question ...
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2answers
114 views

Simplify rational expression

How do I simplfy this expression? $$\dfrac{\frac{x}{2}+\frac{y}{3}}{6x+4y}$$ I tried to use the following rule $\dfrac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b}\cdot \frac{d}{c}$ But I did not get the ...
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3answers
72 views

How to evaluate an integral of the form $\int \frac{dx}{-ax^2 + b}$?

I need to evaluate $\int \frac{dx}{-ax^2 + b}$ while both $a$ and $b$ are positive. I was blocked while I was trying $ x=\tan\theta $ which turned $ dx=\sec^2\theta\, d\theta $ This method didn't ...
4
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4answers
259 views

Limit of a rational function

Calculate the limit $$ \lim_{x \to 0} \frac{3x^{2} - \frac{x^{4}}{6}}{(4x^{2} - 8x^{3} + \frac{64x^{4}}{3} )}$$ I divided by the highest degree of x, which is $x^{4}$, further it gave $$ ...
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2answers
51 views

Integral of $\arcsin$ of a rational function, using integration by parts

I'm a class 12 student and this a question from my textbook: $$I=\int{\arcsin{2x\over 1+x^2}}\mathrm{d}x$$ I did it using integration by parts like this: $$I=\arcsin{\left(2x\over ...
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3answers
52 views

A simple-looking rational limit

Please help me compute: $$ \lim_{z\to 0}\frac{\sqrt{2(z-\log(1+z))}}{z} $$ I know the answer is 1 because I plugged it into Mathematica. Attempts with L'Hopital's Rule didn't work. This a step in an ...
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2answers
76 views

Find all values $c$ such that $(x+1)/(x^2+2cx+4)$ has domain R

Word for word: Find all values of $c$ such that $f(x)=\frac{x+1}{x^2+2cx+4}$ has a domain R I really don't know where exactly to start. I'm not sure what it means by "a domain R"
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1answer
43 views

Why is this rational expressions indeterminate when evaluated?

I have this rational expression to evaluate, $$ {{3a-3}\over {4a(a-1)}} \text { if } a=1. $$ I understand that if you substitute 1, both the numerator and denominator would turn out 0, thus making ...
2
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2answers
52 views

Integration of the rational function $ 4/( 1+4t^2) $

So it's August so my memory of math is a little rough right now. I was wondering if someone could help me with integration with a fraction involved? For example: $$\int_0^{1/2} \frac 4{1+4t^2} \, ...
2
votes
1answer
37 views

Is there any rational map from the nonsingular Segre quadric surface in $\mathbb{CP}^3$ to a nonsingular surface of degree greater or equal to 4?

Is there any rational map from the nonsingular Segre quadric surface in $\mathbb{CP}^3$ to a nonsingular surface in $\mathbb{CP}^3$ of degree greater or equal to 4? Someone told me that the answer is ...
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5answers
151 views

Why is $f(n) =\frac{n(n+1)(n+2)}{(n+3)}$ in $O(n^2)$?

Let: $$f(n) = n(n+1)(n+2)/(n+3)$$ Therefore : $$f∈O(n^2)$$ However, I don't understand how it could be $n^2$, shouldn't it be $n^3$? If I expand the top we get $$n^3 + 3n^2 + 2n$$ and the biggest ...
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2answers
68 views

Neither $\log x$ nor $\exp(x)$ are rational functions [closed]

(a) Prove that $\log x$ cannot be expressed in the form $f(x)/g(x)$ where $f(x)$ and $g(x)$ are polynomials with real coefficients. (b) Prove that $e^x$ cannot be expressed in the form $f(x)/g(x)$ ...
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0answers
63 views

Integral of an exponential of rational function

I have an integral of the form $\int_{a}^{b} \text{exp}\left(\frac{\lambda}{\rho^2 m + \sigma^2_u}\right) \frac{1}{m^2}\text{exp}\left(-\frac{\lambda}{m}\right) dm$. Can this integral be found ...
1
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1answer
80 views

Why does the graph of $x^3/x^3$ not have a horizontal asymptote?

I am a graduate student studying math, and am actually teaching College Algebra right now. But every once in a while, I come upon something new in a subject that I have supposedly mastered. Why does ...
2
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0answers
39 views

Truncation error in Padè approximants

Suppose only the following data are known about a rational function $R(x)=P(x)/Q(x)$ (for $P,Q$ polynomials): (a) the degree of $P$ is $\leq m$ and the degree of $Q$ is $\leq n$; (b) the first $k$ ...
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0answers
37 views

Variation of the argument of a rational function along a circle

Crossposted here on MO. Let $f:\bar{\mathbb C}\to \bar{\mathbb C}$ be a rational function, and take a circle $C$ not crossing the zero- and polar-locus of $f$. The argument principle tells us the ...
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3answers
42 views

Help solve rational expression

I need help solving this rational expression. Divide $$\frac{4x^4 + 6x^3 + 3x - 1}{2x^2 + 1}$$ How do you solve this problem? Where do I start?
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3answers
43 views

Setting up word problem for finding length and width

Word Problem: The length of a rectangular sign is $3$ feet longer than the width. If the sign has space for $54$ square feet of advertising, find its length and width. I have not idea where to start. ...
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3answers
48 views

Rationals over an interval

Suppose $I$ is an interval $[a,b]$. It is noted that $a$ and $b$ are real integers. Divide the interval into $n$ parts with step size $h=(b-a)/n$. Clearly all the points $a$, $a+h$, ...
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1answer
48 views

Solving the polynominal: $s(t) = -16t^2 + 48t + 160$

The height of a ball is thrown directly upward from an initial height of $160$ ft with an initial velocity of $48$ ft per second is given by the function: $s(t) = -16t^2 + 48t + 160$, where $s(t)$ ...
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4answers
76 views

Solving for $x$: $\;\frac8{x-2}-\frac{13}2=\frac3{2x-4}.$

I have solved this sort of problem before, but frustratingly I have forgotten to. The problem is: Solve for $x$: $$\dfrac8{x-2}-\dfrac{13}2=\dfrac3{2x-4}.$$ So hence the title, How should I ...
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0answers
48 views

The value of $a/b+b/a-ab$ under the condition $ab=a-b$ (AMC 10 problem)

I am confused about the problem below. I answered correctly but used substitution to solve instead of changing everything to the same denominator as shown in the solution. The problem: Two non-zero ...
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2answers
37 views

Derivative of rational function help.

consider $$f(x)=\frac{1}{2x-4}$$ The derivative should be $\displaystyle -\frac{1}{2(2x-4)^2}$ However I get $\displaystyle -\frac{2}{(2x-4)^2}$ my workflow: $$\begin{array}{} f'(x)&= ...
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1answer
48 views

Supremum of a rational function

Let $f(z)$ be a rational function in the complex plane such that $f$ does not have any poles in $\{z:\Im z\ge0\}$. Prove that $\sup\{|f(z)|:\Im z\ge0\}=\sup\{|f(z)|:\Im z=0\}$. Let $\Gamma_r$ be a ...
5
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1answer
121 views

There cannot exist a rational function $f: \mathbb{R} \to \mathbb{R}$ injective, not surjective

I was looking for a rational function $f: \mathbb{R} \to \mathbb{R}$ that looks like $\arctan$, in that it is injective not surjective well-defined on all $x\in \mathbb{R}$ (no vertical ...
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2answers
84 views

Where rational functions are undefined

I have another question/comment I'd like a fresh pair of eyes on The question is "A rational function can have infinitely many x-values at which it is not continuous" I know since Q(x) in the ...
0
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0answers
25 views

min max of a rational function

The variable is a vector $x \in \mathbb{R}^n \times\mathbb{R}^m_+ \times \mathcal{E}$ where $\mathcal{E}$ is an ellipsoid of dimension $e$. I would like to find the min and max of the following ...
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0answers
38 views

Rational function on a noetherian scheme

Let $X$ be a noetherian scheme and $f$ a rational function on $X$, so by definition the domain of $X$ includes all associated points of $X$. I think the following is true: $f$ is regular on $X$ if and ...
7
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1answer
361 views

The solutions for the equation $\frac{a}{c-b+1}+\frac{b}{a-c+1}+\frac{c}{b-a+1}=0.$

How can I find the solution for the following equation in $a,b \mbox{ and } c$. $$\frac{a}{c-b+1}+\frac{b}{a-c+1}+\frac{c}{b-a+1}=0.$$ Also $b-c \neq 1$, $c-a \neq 1$ and $a-b \neq 1$. Thanks!
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0answers
52 views

Branch of mathematics that studies groups / rings or rational functions

I'm not really a mathematician, and looking for some literature which could potentially help me in research. Im interested in algebra of rational functions (ratios of polynomials) of finite order. ...
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1answer
23 views

Termwise differentiation of sequence of rational functions when the uniform limit is analytic

Given a sequence $\{f_n(x)\}$ of rational functions which converges uniformly to the analytic function $\{g(x)\}$ on $[a, b]$ ($f_n(x)$ are defined on $[a, b]$ and hence are analytic), what can we say ...
0
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2answers
44 views

Inverse functions problem

There are two functions $f\colon\mathbb Q \to \mathbb Q \setminus \{-1\}$ and $g\colon\mathbb Q \to \mathbb Q \setminus \{1\}$. $$g(x) = \frac{f(x)}{f(x)+1}.$$ Prove that if there is a inverse ...
4
votes
3answers
54 views

Values of $m$ for which $y^2 + 2xy + 2x -my -3$ can be factorised

For what values of $m$, will the expression $y^2 + 2xy + 2x -my -3$ be capable of resolution into two rational linear factor? This is how I did it: $$y^2 + 2xy + 2x -my -3 = y^2+(2x-m)y+2x-3$$ ...
2
votes
1answer
55 views

rational parameterization of quartic

With the curve $x^4 - 6x^2 - y^2 + 1 = 0$ in the range of $x$ inside of $(-1,1)$, I can only identify two rational points $(0,1)$ and $(0,-1)$. Is it possible to determine if there are others?