Questions tagged [rational-functions]

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

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1answer
34 views

How to solve $\int\frac{4y+3}{4y^2-9}dy$? [closed]

For an assignment from my book I have to evaluate: $$ \int\frac{4y+3}{4y^2-9}dy $$ (and therefore actually solve it), but I don't know how to start. Thanks in advance for your help! For your ...
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68 views

Existence of a rational isomorphism between circles

Reference: "The Arithmetic of Elliptic Curves" (Silverman, second edition), Exercise 1.11. (b). Consider two circles $X^2 + Y^2 = p \cdot Z^2$ and $X^2 + Y^2 = q \cdot Z^2$. If $p$ and $q$ ...
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cyclic rational inequalities $\frac{1}{a^2+3}+\frac{1}{b^2+3}+\frac{1}{c^2+3}\leq\frac{27}{28}$ when $a+b+c=1$

I've been practicing for high school olympiads and I see a lot of problems set up like this: let $a,b,c>0$ and $a+b+c=1$. Show that $$\frac{1}{a^2+3}+\frac{1}{b^2+3}+\frac{1}{c^2+3}\leq\frac{27}{28}...
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35 views

Rational Parameterization of Quartic Curve (Variety)

I am trying to find a rational parameterization of the curve (variety) $$x^4+a^2x^2=x^2y^2+(h^2+a^2)y^2$$ If I have done my math correctly, this curve has a singularity of multiplicity 2 at the origin,...
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Unique solution of equation on $Z$

I am a computer science student and in one of our (online, given the circumstances) class, we met an equation that has only 1 solution. I am unable to find it analytically, I was only able to solve it ...
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rational function with distinct roots

Let $r(X)$ be a rational function over $\mathbb{C}$. Is there (always) a complex number $z$ such that the rational function $r(X)-r(X+z)$ has distinct roots?
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Solve $\int \frac{5}{4x^2+3}dx$

I've tried few way to resolve $\int \frac{5}{4x^2+3}dx$ but I think there's somthing I'm missing. As a first step I've took the constant out: $5\int \frac{1}{4x^2+3}dx$. Next I've thought it would be ...
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33 views

Find the limits between which $a$ must lie in order that $\frac{ax^2-7x+5}{5x^2-7x+a}$ can take all real values

Find the limits between which $a$ must lie in order that $$\frac{ax^2-7x+5}{5x^2-7x+a}$$ may be capable of taking all real values for all possible real values that $x$ may take. My Attempt: Let $$y=\...
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1answer
85 views

Determine whether the given is rational functions or rational equation

Is a rational function or a rational equation or none of these? 1.) $y=5x³-2x+1$ 2.) $g(x)=7x³-4√x+1/x²+3$ Hope you'll help me thanks
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For rational functions, is there a way to tell from a graph alone how many powers the denominator is above the numerator? [closed]

For rational functions, is there a way to tell how many powers the denominator is above the numerator only based on the graph? I know that if the denominator has a greater degree than the numerator, ...
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Partial Fractions With Repeated Quadratics

I'm told that given a function $f(x)=\frac{P(x)}{Q(x)}$, if $\deg(P)>\deg(Q)$ then $f$ is improper, which makes sense when I think of real numbers like $5/2$. And in this case we would have to do ...
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Simplifying a rational function on an algebraic curve

Does a rational function $\phi$ on a smooth projective algebraic curve $F$ over a algebraically closed field $K$ always have a representative $\frac{f}{g}$, where $f$ and $g$ are polynomials without ...
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Integration - Evaluate $\int \frac{x^7+2}{(x^2+x+1)^2} \ dx$

Evaluate $$\int \frac{x^7+2}{(x^2+x+1)^2} \ dx$$ This problem is from G N Berman, no. 2056 (integrate using ostrogradsky's method). I referred to this question as well as this article but I could ...
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Prove that for a rational function $f(x)$, $f^{-1}(x) \ne f'(x)$

Is there a way of proving that for a rational function $f(x)$ the inverse $f^{-1}(x)$ can never equal its first derivative $f'(x)$? I have graphed out a rational function of $\frac{\text{linear}}{\...
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1answer
32 views

Proving by $\varepsilon-\delta$ that $1/(x+2)$ is continuous at $x=1$

Work So Far: I know the general definition of $\varepsilon-\delta$ continuity at a point $x_0$: $\forall \varepsilon > 0$ $\exists \delta > 0$ such that $$|x-x_0| < \delta \implies |f(x)-f(...
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Is there meaning in the minimal instances of a variable you need to write a rational expression?

This is something I've never thought about before. Given a rational function $f \in \mathbf{k}(x)$, the minimum number of $x$ you need to write down a formula for $f$ on its domain is well-defined. My ...
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28 views

Formal Proof For Rational Function Limit with a Quadratic in the denominator

I've been trying my hand at a another practice problem for my first-year differential calculus class, but I cannot seem to make it past a certain step. The problem is as follows: Let $f(x)=\frac{-x}{(...
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Rational function simplifying

From the definition of rational function $f(x) = \frac{p(x)}{q(x)}$ , where P(x) and q(x) are polynomials and q(x) ≠ 0 so for the function $f(x) = \frac{x^{-2}+3 }{x-5}$ by the definition f(x) isnt ...
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Formal Proof of a One-Sided Limit of a Rational Function Tending Towards Infinity

I've been trying my hand at a practice problem for my first-year differential calculus class, but I cannot seem to make it past a certain step. The problem is as follows: Let $f(x)=\frac{5x}{x-5}$. ...
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1answer
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Why is $\text{Aut}(\overline{K}(x)\mid K(x))$ equal to $\text{Aut}(\overline{K}\mid K)$?

Let $\overline{K}$ be an algebraic closure of $K$. I saw in a paper that $\text{Aut}(\overline{K}(x)\mid K(x))$ is equal to $\text{Aut}(\overline{K}\mid K)$, but there was no explanation why this ...
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49 views

$\mathbb{R}(x)$ as non o-minimal structure

Let us consider the field of real rational functions in one variable $\mathbb{R}(x)$ as an ordered field with $ x > r$ for all $r \in \mathbb{R}$. have to show that $\mathbb{R}(x)$ is not o-...
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21 views

Rational function and horizontal asymptote

Let $f(x)$ be a rational function of the form $\dfrac{p(x)}{q(x)}$, where $q(x) \neq 0$. Assume that $q(x)$ has two distinct roots $x_{1}$ and $x_{2}$ where $x_{1} < x_{2}$, and the degree of $p(x)$...
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Domain of Rational Functions

I have the following definition for rational functions: "Let $p(x), q(x)$ be polynomials and define $D=\{x\in \mathbb{R} : q(x) \neq 0 \}$ Then, the function $$f:D \rightarrow \mathbb{R}$$ $$f(x) ...
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Is $y=\frac{x^2}{x^3+3}$ has a horizontal asymptote? [duplicate]

Is $y=\frac{x^2}{x^3+3}$ has a horizontal asymptote? Since the graph passes through $(0,0)$, hence $y=0$ is not a horizontal asymptote. Can a rational function may not have a horizontal asymptote?
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Solving system of equations with fraction

I am having difficulties solving the following system : $u \neq t$ and $(t, u) \in \mathbf{R} - \{-1, 1\}$ $\frac{t}{t^2-1}-\frac{u}{u^2-1}=0$ $\:\frac{t^2}{t-1}-\frac{u^2}{u-1}=0$ I tried expanding ...
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how to solve a rational expression problem [closed]

The area is square units of a rectangle can be represented by the expression 12x^(2)+25x-7,and it's length can be represented by (3x+7). Determine the value of "x" if the perimeter of the ...
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64 views

Integrals of the form $\int x^m(a+bx^n)^p dx$

Find $$\int{x^{13/2}(1+x^{5/2})^{1/2}}dx$$ My attempt: Usually when I face the form $\int x^m(a+bx^n)^p dx$, I would factor out $x^n$ and proceed. Example: $$I=\int{x^{-11}(1+x^4)^{-1/2}}dx$$ $$I=\...
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Ratio of polynomials identity

Suppose $a,b,c,d,a',b',c',d',t$ are positive integers such that $\frac{(1-x^{a+t})(1-x^{b+t})}{(1-x^a)(1-x^b)}+\frac{(1-x^{c+t})(1-x^{d+t})}{(1-x^c)(1-x^d)}=\frac{(1-x^{a'+t})(1-x^{b'+t})}{(1-x^{a'})(...
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An interesting conjugate-type fraction

I was playing around with numbers and I appear to have discovered a very fascinating fraction: For all $n>0$, $$\cfrac{\bigg(1-\cfrac 1{3n+2}\bigg)\bigg(1-\cfrac 1{6n+1}\bigg)}{\bigg(1-\cfrac 1{3n+...
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Solve parametric equations satisfying that the equations have positive roots…

Solve parametric equations satisfying that the equations have positive roots: $\left\{\begin{matrix} x_{1}+x_{2}+x_{3}+...+x_{m} &=9 \\ \frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+...+\frac{1}...
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Simplifying $xy^2-\frac{x(-xy-1)^2}{(x-1)^2}+2y\left(\frac{-xy-1}{x-1}-1\right)+\left(\frac{-xy-1}{x-1}-1\right)^2=0$

I have the following equation $$xy^2-\frac{x(-xy-1)^2}{(x-1)^2}+2y\left(\frac{-xy-1}{x-1}-1\right)+\left(\frac{-xy-1}{x-1}-1\right)^2=0$$ which I was trying to simplify. I know the solution is $$\frac{...
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1answer
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Why is the $y$ intercept for this equation positive?

I am tasked with sketching the following rational function$$\frac{\left(x+1\right)\left(3x+2\right)}{\left(x+1\right)^{2}}$$ I have found that the two asymptotes are: $y=3$ and $x=-1$ and there are ...
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Finding values of $a$, $b$, $c$, $d$ such that a $f(x)=\frac{ax+d}{cx+b}$ is self inverse

I am trying to solve the following problem: (screenshot) For which numbers $a$, $b$, $c$, and $d$ will the function$$f(x)=\frac{ax+b}{cx+d}$$ satisfy $f(f(x))=x$ for all $x$? I have a solution, but ...
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Evaluating contour integrals over the unit circle of rational functions.

Let $p(z):= a_0 + a_1z + \cdots + a_nz^n$ be a degree $n$ polynomial, let $m$ be a large integer (which we may assume much larger than $n$), and let $k$ be some integer in the range $m+1, \cdots, m+n$....
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Rational functions with rational function antiderivatives

Which rational functions over $\mathbb{R}$ have rational functions as antiderivatives? I can think of a few trivial examples. Polynomials over $\mathbb{R}$ are rational functions and of course have ...
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A faster way to solve $\frac1{x^2+6x-15}+\frac2{x^2+6x+3}=\frac3{x^2+6x+1}$ in a timed test?

What is a faster way to do this problem? Solve the equation $$ \frac{1}{x^2+6x-15} + \frac{2}{x^2+6x+3} = \frac{3}{x^2+6x+1} $$ This is from a timed test, so the fastest answers would be the best. I ...
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Why is -2 an answer for $\frac{|x+3|+x}{x+2} > 1$?

In the inequality $\frac{|x+3|+x}{x+2} > 1$, by doing sign chart method the answer I am getting is $x \in (−5,−2)\cup(−1,\infty)$ But the graph in Desmos shows that it should be $x \in (−5,−2]\cup(...
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1answer
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How to simplify this fraction $\frac{x^2+7x+1}{x^2-4}$? [closed]

I'm familiar with polynomial long division but I keep getting stuck when I try to answer this question. Online math calculations also say there's no way to simplify this (but it's a homework question ...
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Taylor series of degenerate rational function

I have a rational function $f$ which is well-defined on $(0, 1]$, degenerate at zero. I can find the limits as $x \rightarrow 0$ of $f$ and its derivatives, but find that the third and higher ...
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1answer
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Rational Function Approximation of the Maximum Function [closed]

Is is possible to create a rational function $R(x)$ that is an approximation of $\max(x,0)$ (that is, as $x \to -\infty, \, R(x) \to 0$ and as $x \to \infty, \, R(x) \to x$)?
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Find minimum value of $f(x)=\frac{(x^2-x+1)^3}{x^6-x^3+1}$ [closed]

Find minimum value of $f(x) $ where $$f(x)=\frac{(x^2-x+1)^3}{x^6-x^3+1}$$ On differentiating I got $$f'(x)=\frac{3(x^2-x+1)^2\left(x^6-2x^5-x^4+x^2+2x-1\right)}{(x^6-x^3+1)^2}$$ which doesn't help ...
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What would the picture for partial fractions look like?

As how integration by parts has the picture below, what would the picture for partial fractions look like? Although there's probably no way to escape the heavy algebra necessary for partial fractions,...
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44 views

How to approximate $f(x)=\frac{a_1x^2+a_2x+a_3}{\sqrt{a_4x^2+a_5x+a_6}} + \frac{b_1x^2+b_2x+b_3}{\sqrt{b_4x^2+b_5x+b_6}}$?

How to approximate the following function? $$f(x) = \frac{a_1x^2+a_2x+a_3}{\sqrt{a_4x^2+a_5x+a_6}} + \frac{b_1x^2+b_2x+b_3}{\sqrt{b_4x^2+b_5x+b_6}}$$ where $a_i$ and $b_i$ are constants. I thought ...
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151 views

Calculus of $ \lim_{(x,y)\to (0,0)} \frac{8 x^2 y^3 }{x^9+y^3} $

By Wolfram Alpha I know that the limit $$ \lim_{(x,y)\to (0,0)} \dfrac{8 x^2 y^3 }{x^9+y^3}=0. $$ I have tried to prove that this limit is $0$, by using polar coordinate, the AM–GM inequality and ...
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2answers
57 views

Bounding coefficients of the polynomial $(1-x^2)^n (1-x)^{-m}$.

I have been trying to get some upper bound on the coefficient of $x^k$ in the polynomial $$(1-x^2)^n (1-x)^{-m}, \text{ $m \le n$}.$$ A straightforward calculation shows that for even $k$, the ...
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0answers
31 views

Rational Function Is Eventually Greater Than 0

Let $m, n \in \mathbb{Z}^{\geq 0}$. Let $a_1, ..., a_n \in \mathbb{R}$. Let $b_1, ..., b_m \in \mathbb{R}$. Assume $\displaystyle\frac{a_n}{b_m} > 0.$ Let $f(x) = \displaystyle\frac{a_n x^n + a_{n -...
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1answer
46 views

Is $y=\frac{x^3+0.3x-7}{2x-1}$ Rational or Quotient Function?

I understand this is an easy one question however, I just wanted to clarify which it would be considered as a rational function or a quotient function. I understand that a rational function has $y=\...
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1answer
108 views

If $a_i\in\mathbb{R}$, $\omega^2+\omega+1=0$, and $\sum_{i=1}^n\frac{1}{a_i+\omega^k} =2\omega^{2k}$ for $k=1,2$, find $\sum_{i=1}^n\frac{1}{a_i+1}$.

In this question, $\omega$ is the complex cube root of $1$ and $a_i \in \mathbb R$. If $$\sum_{i=1}^n \frac{1}{a_i + \omega} =2\omega ^2$$ and $$\sum_{i=1}^n \frac{1}{a_i + \omega ^2} =2\omega\,,$$ ...
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1answer
29 views

Intuition and Visualization of Limits of Complex Rational Functions

I am having some difficulty evaluating and visualizing limits of complex rational functions. For example, $$ \lim_{z \to z_0 } f(z),\ where\ Im(z_0)\ne 0,\ Re(z_0)\ne 0$$ $$f(z):=\frac{L(z)}{M(z)}, ...
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0answers
19 views

Computing sum absolute value of residues of rational function

Suppose I have a rational function $R(X) = P(X)/Q(X)$. For simplicity, let's assume that the roots of $Q$ are all real and simple, and $\deg P < \deg Q = n$. Then I can decompose $R$ using partial ...

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