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

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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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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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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 - ...
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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 ...
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Piecewise from Rational Absolute Value Function

How would one separate a function like the following into piecewise? $$f(x)={\left|4-x\right|\over{\left|x-4\right|}}$$ I've been taught that with a rational function with an absolute value in the ...
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23 views

coefficients of Laurent series of rational function

Let $F(z)$ be a rational function $\frac{P(z)}{Q(z)}$ such that the degree of $P(z)$ is less than the degree of $Q(z)$ and suppose that all the zeros of $Q(z)$ are contained in the open disk $|z| < ...
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Finding the value of a parameter for which a rational function is tangent to x-axis.

I'm stuck on this problem and hope someone can help: The curve $C$ has equation $$ y= \frac{px^2+4x+1}{x+1}$$ where $p$ is a positive constant and $p\neq 3$. Find the values of $p$ for which ...
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1answer
100 views

A rational orbit that's provably dense in the reals?

Iterating the map $\ \ x\ \mapsto\ x-\frac{1}{x},\ \ $ the orbit of initial point $2$ is "probably" dense in $\mathbb{R}$. Is there an explicit rational mapping together with an initial rational ...
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How do I solve this rational function?

Determine the zeros and the location and type of all singularities of the following rational functions $R(x)$. Determine the partial fraction of $R(x)$. Make each sample. Sketch the graph of the ...
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1answer
40 views

How can I integrate rational functions with denominator with just quadratics as factors?

I am told about integration by partial fraction method.I usually guess to decompose a fraction into partial fractions and then I solve the constants.In this problem: ...
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1answer
36 views

$T: \mathbb{Q}[X]\mapsto\mathbb{Q}[X]$. Proving $T$ is injective and not surjective.

Given $T: \mathbb{Q}[X]\mapsto\mathbb{Q}[X]$, $T(f)=X\cdot X\cdot f - (X+1) \cdot f'$. Prove $T$ is injective and not surjective. I try to prove this by applying the definition of injectiveness but I ...
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Can $\frac{P(x)s(x)+Q(x)}{R(x)}$ be a polynomial for some polynomial $s(x)$?

Given some polynomials $P(x), Q(x), R(x), \operatorname{gcd}(P(x), R(x)) = 1$ does there exist some polynomial $s(x)$ so $$ \frac{P(x)s(x) + Q(x)}{R(x)} $$ is also a polynomial? If so how do I ...
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Construct a rational function whose graph in the xy-plane has a vertical asymptotes lines x = 3 and x = 5 oblique asymptote the lines y = 2x -3

I'm studying for a test and I came across this example problem and the oblique part is throwing me off. How do i go about solving this?
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1answer
22 views

Is it possible to represent a root as a simple rational function with an exponent?

Using the following function:$$y=\frac{mx^p+b}{d}$$... where $m$, $p$, and $b$ may be any integer ... where $d$ may be any integer $\gt0$ ... and where $x$ may be any rational number $\ge0$ Is it ...
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Genus of exponential algebraic curve

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$When it comes to algebraic curves and genus, I get confused. In particular, the curve $\:1-e^y-e^xe^{-y}=0\:$ holds me in doubt. But, I ...
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1answer
36 views

I can't understand to solve the question (Khan Academy Algebra Basics - Exponent Properties)

I was learning (practicing to solve) simplifying the rational expressions. I know how to simplify the rational expressions... but I can't understand some part of the questions. The question that I ...
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$\mathbb{Z_p}(x^p,y^p,x-y^n) \neq \mathbb{Z_p}(x^p,y^p,x-y^m)$ for $n \neq m$

I am trying to find an easy proof for the following result: Let $p$ be a prime number. Show that for $n \not\equiv m \mod p$ $$\mathbb{Z_p}(x^p,y^p,x-y^n) \neq \mathbb{Z_p}(x^p,y^p,x-y^m).$$ The ...
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Solving a system of 4 equations

why gives me that system of equations: $-\tfrac{1}{x} +2ux =0$ $-\tfrac{1}{y}+2uy=0$ $1+2uz=0$ a)--> $x^2 = y^2= -z $ ? And using the equation: $x^2+y^2+z^2 = 1$ gives b)$z = 1 (+,-) \sqrt{2}$ I ...
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How to calculate the local minimum of a hyperbola without using derivatives?

I've got the following rational function, which is a hyperbola. $f(x) = \frac{2\cdot\pi (x+ 4)^2}{x}$$\quad$on WolframAlpha There is a minimum in the first quadrant and a maximum in the third ...
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37 views

Inverses of Rational Functions

Consider the function $g$, where $$g(x)=\frac{3x}{5+x^2}$$ (a) Given that the domain of $g$ is $x\ge a$, find the least value of $a$ such that $g$ has an inverse function. I know that $g(x)$ ...
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82 views

Why does the extra $.1$ make $\int \frac{\left(x^2+2.1x\right)}{\left(x^3+3x+12\right)^6}dx$ much harder?

I'm a first-time Calc I student currently struggling in class. Yesterday we started on Substitution and Integration with integrals. One problem our professor put on the board was: $$\int ...
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How can I show that this meromorphic function is a rational function of two polynomials?

Here's my updated attempt: Write$$f(z) = \sum_{n=-1}^{\infty} a_n(z-z_1)^n + ...+\sum_{n=-1}^{\infty} m_n(z-z_m)^n+\sum_{n=+1}^{-\infty} \psi_n(z)^n$$ with the last series being an expansion about ...
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40 views

A simple complex rational function.

Let $$f(z)=\dfrac{z-a}{z-b},\,\,\,\,\,\,z\not=b\not=a$$ be a complex valued rational function. How can I show that, if $|a|,|b|\lt1,$ then there is a complex number $z_0$ satisfying $|z_0|=1$ and ...
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28 views

Rational Multivariable limit

I am having some issues with the following multivariable limit: $$\lim_{x,y\to0,0} \frac{x^2+y^2}{x+y}$$ I am trying to show whether it exists and is equal to 0, or whether it does not exist. What I ...
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If a rational function is even then the numerator and the denominator have same parity

Let $K$ be a field and let $F={P\over Q}\in K(X)$ be a rational fraction, for simplicity we denote also by $F$ the rational function associated to the rational fraction $F$. It is clear that if $P$ ...
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19 views

howto find where a slant aymptote crosses a function

i have the function $f(x) = \frac {(x+1)(x-1)^2}{x^2}$ and i want to sketch it so i found the following: Vertical Asymptote: is x=0 X-intercepts: x=1, x=-1, x=1 Slant Asymptote: $y = x-1 $ then i ...
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Can a function ever cross a vertical asymptote?

I have the function $f(x) = \frac {x^2-1}{x^2-4}$ that I need to graph. This is what I found: Vertical asymptotes: $x=2$ and $x=-2$ Horizontal asymptote: $y = 1$ $x$-intercepts: $x = 1$ and $x = ...
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52 views

Solve the following relation.

Find the values for $a$, $b$, $c$ and $d$ in the following equation $$\frac{2a}{2m} = \frac{2b}{a+m} = \frac{2c}{a+b} = \frac{2d}{c+m} = \frac{2M}{c+d}.$$ (Note: $m$ and $M$ are different. In fact, ...
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Sum of finite number of terms of the series $\sum\limits_{L=2}^{L_{max}}\frac{1}{e^{i \phi / L} -1}$

Good day everyone. Are there any chances to get a compact formula for the following sum of finite number of terms? $$\sum\limits_{L=2}^{N} \dfrac{1}{e^{ \frac{i \varphi}{L}} -1}$$ N and $\varphi$ ...
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Assistance on beginning the integral $\int\frac{dx}{(x+1)(n-x)}$ [duplicate]

This is the integral $$\int\frac{dx}{(x+1)(n-x)}=\int kdt$$ I just need some assistance on how to begin the left side integral and I will most likely be able to continue it from there thank you.
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Why are removable discontinuities even discontinuities at all?

If I have, for example, the function $$f(x)=\frac{x^2+x-6}{x-2}$$ there will be a removable discontinuity at $x=2$, yes? Why does this discontinuity exist at all if the function can be simplified to ...
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Integration ambiguity

I am trying to Evaluate $$I=\int\frac{x^4+1}{x^6+1}dx$$ Now $I$ can be written as $$I=\int\frac{x^4(1+\frac{1}{x^4})}{x^6(1+\frac{1}{x^6})}dx=\int\frac{(1+\frac{1}{x^4})}{x^2(1+\frac{1}{x^6})}dx$$ ...
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infinity over infinity and zero multiplied infinity in a calculation which gives (correctly) 1

I hope you don't put as a duplicate my question as it is thought for the specific case I am going to show you now, which is about a calculation. I read about the topic of infinity over infinity, ...
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59 views

Find the number of students, given the total bill to which four did not contribute

There is a lunch bill of $\$239.25$. $4$ students are not paying because it is their birthday. All the other students pay $1$ extra dollar to cover the bill. How many students had lunch? I assume ...
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62 views

Rational functions that are positive on the unit circle

I am studying chapter 14 in Rudin "real and complex analysis" and I am trying to solve number 4 page 293, in which he asks to find the form of rational functions that are positive on the unit circle. ...
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Calculate using integration by parts $\int \frac{x^2}{(x^2+1)^2}\,dx$

Calculate using integration by parts $$\int \dfrac{x^2}{(x^2+1)^2}$$ I'm looking through some working for this question and it gives $u=x/2, u'=1/2$ $v'=\dfrac{2x}{(x^2+1)^2}, ...
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42 views

What is an easy way to find the range for a rational function?

I sometimes have some trouble finding the range of a rational function. What is an easy or good way to do this?
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Solving the logarithmic rational equation

I'm wondering there exist the way to solve the equation form of: $$ \log f(x) + g(x) = c $$ where $f(x)$ and $g(x)$ are rational functions, $c$ is a constant. Is there any general(in closed form) ...
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Distributing multiplication of rational functions

I am having trouble distributing with fractions. This $$ \left(\frac{1}{(x + 3)} + \frac{(x + 3)}{(x - 3)}\right)\, (9 - x^2) = -\frac{(x^2-3)}{9-x^2}(9-x^2) $$ has the answer $\left\{\left[x = - ...
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51 views

How is this is an identity?

So we have started studying partial fractions.The book teaches two methods: By equating coefficients By utilizing the fact that when a rational fraction is decomposed to partial fractions it is an ...
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Solving a particular kind of equation

Suppose that we have a rational function of the form $$\displaystyle f(x) = \frac{P(x)}{Q(x)},$$ where $P,Q$ are polynomials of equal degree with rational coefficients, for which there exists two ...
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39 views

Vertical asymtotes where denominator = 0

I have 2 problems as stated below: (a).LeAnn claims that when p(x) and q(x) are polynomials the graph of y = $\left(\frac{p(x)}{q(x)}\right)$ has vertical asymptotes at all the points where q(x) is ...
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The rule for evaluating limits of rational functions by dividing the coefficients of highest powers

I have a Limit problem as below: Connor claims: " $\lim_{x\to \infty} \left(\frac{6x^2 + 7x +3}{2x^3 + x^2 -2x -1}\right) = 3$ because my high school calculus teacher told us the limit of ratio of ...
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32 views

Reciprocal Rational Function

Suppose we have a rational function: $$f(x) = \dfrac{a+x}{b + cx}$$ And our task is to draw the reciprocal function of $f(x)$, or in other words, $\frac{1}{f(x)}$. My teacher argues that because ...
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18 views

Getting the rational function with given characteristics

The curve C has an equation $$y = \frac{ax^2+bx+c}{x+d},$$ where $a$, $b$, $c$, and $d$ are constants. The curve cuts the $y$-axis at $(0,-2)$ and has asymptotes $x=2$ and $y = x + 1$. From a ...
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2answers
120 views

rational number solutions to $\frac{a}{a^2+1} + \frac{b}{b^2+1} = \frac{c}{c^2+1}$ with $abc\ne 0$

This question concerns the equation $$\frac{a}{a^2+1} + \frac{b}{b^2+1} = \frac{c}{c^2+1}$$ and the possibility of rational number solutions with $abc \ne 0$. In comments arising from: Using ...