Tagged Questions

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

28 views

System of (quite similar) two equations

Given the real, natural and binary successions $\{t_1,...,t_N\} \in \mathbb{R}$, $\{n_1,...,n_N\} \in \mathbb{N}$ and $\{E_1,...,E_N\} \in \{0,1\}$ we would like to find the $(x,y)$ that satisfies the ...
26 views

Simplfying complex rational expression

I'm trying to simplify $$\frac {\dfrac {x}{y} - \dfrac {y}{x}}{y}.$$ My method of trying to solve this is try to simplify the numerator $\frac {x}{y}-\frac{y}{x}$ Then I find the GCD: $xy$, multiply, ...
75 views

23 views

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 ...
76 views

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 ...
12 views

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 ...
37 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 ...
22 views

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 ...
67 views

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 ...
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 ...
35 views

46 views

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 ...
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 $500$ $ms$)...
766 views

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. ...
39 views

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? $t/(1+t)=1-1/(1+t)$
31 views

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 ...
46 views

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? ...
34 views

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.
28 views

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 ...
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?
43 views

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 ...
30 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 ...
35 views

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?
11 views

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 ...
49 views

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 "...
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?