Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [rational-functions]

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

In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field $K$. In this case, one speaks of a rational function and a rational fraction over $K$. The values of the variables may be taken in any field $L$ containing $K$. Then the domain of the function is the set of the values of the variables for which the denominator is not zero and the codomain is $L$.

The set of rational functions over a field $K$ is a field, the field of fractions of the ring of the polynomial functions over $K$.

1246 questions
3
votes
2 answers

Dividing zeros out of the denominator

Take the equation $\displaystyle y=\frac{x^2+x-6}{x^2-4}$, which is formed from simplifying $\displaystyle y= \frac{(x+3)(x-2)}{(x+2)(x-2)}$. If we divide out $(x-2)$, from the numerator, we get the function $\displaystyle y = \frac{x+3}{x+2}$ which…
waiwai933
  • 355
3
votes
2 answers

A question about the domain of definition of rational functions

Is the function $f(x) = \frac{x^2}{x}$ defined for every $x \in \mathbb R$, or only defined on $\mathbb R \setminus \{0 \}$? Background: Say we are given $P(x) = x^2 - 4x + 3$ and $Q(x) = (x - 1)(2x + 3) - (2x - 2)(3x + 5)$. The question asks to…
user230734
2
votes
0 answers

Is there a special name for symbolic ratios that share no common symbol in numerator and denominator

I am wondering if there is a name for the operation $f$ that separates a multivariate rational function into fractions, wherein neither numerator nor denominator share common symbols: given $y=\frac{x^2+2x+y}{(y-1)(x-1)z}$ the result would be $f(y)…
smichr
  • 359
2
votes
1 answer

range of all rational functions

I want to find all subsets of $\Bbb R$ that is the range of some function$$f:Q^{-1}(\Bbb R\setminus\{0\})\to\Bbb R,f(x)=\frac{P(x)}{Q(x)}$$where $P,Q$ are coprime real polynomials. From a discussion, the conclusion is below ($a,b$ are any real…
hbghlyj
  • 2,115
2
votes
0 answers

Invariant function of polynomial / rational function

I am interested in "invariants" of polynomials or rational functions (over $\mathbb C)$. Consider for instance the polynomial $P(x)=x^3+1$. It has a a "symmetry" $r(x)=e^{2\pi i/3}x$, meaning that $P(r(x))=P(x)$. This means we can find an…
ModularGymnastics
  • 69
2
votes
4 answers

How do I solve this rational function in terms of $x$?

How would I solve for $x$ where $y=(1-x)/(1+x)$?. I have tried multiplying both sides by $(1+x)$, but I couldn't get far with that, as I had a pesky $xy$ on one side, which I couldn't figure out how to take out. How would I solve this?
scrblnrd3
  • 488
2
votes
1 answer

prove equation has no rational root.

Prove For all $n>1$, equation $\sum _{k=1}^n \frac{x^k}{k!}+1=0$ has no rational root. I'm not sure whether there are two questions,for without brace after Sigma. My thought is to prove it is not reducible on rational field.
HyperGroups
  • 1,393
2
votes
1 answer

Questions about ratios

At a school dance, each boy danced with exactly three girls and each girl danced with exactly two boys. if 100 boys attended the school dance, how many girls attended?
Kate
  • 51
  • 2
2
votes
1 answer

$\sqrt{x} \notin F(x)$

I wanted to prove that if $F$ is a field and we consider the fraction field $F(x)$, then $\sqrt{x} \notin F(x)$. I said that if $\sqrt{x} \in F(x)$ then there are $f,g \in F[x]$ such that $\bigl(\frac{f(x)}{g(x)}\bigr)^2=x$ so $f(x)^2=xg(x)^2$ so…
roi_saumon
  • 4,196
2
votes
4 answers

How to simplify this rational expression?

This expression should be extremely easy to simplify, but for some reason I can't do it. $$\frac{x^4-1}{x-1}$$ I know it simplifies down to this, but I don't know how to get there $$x^3+x^2+x+1$$ This is a very basic question on my calculus…
Brad
  • 307
2
votes
2 answers

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 the point at $x = -2$ is a vertical asymptote, we can…
GoodChessPlayer
  • 254
2
votes
2 answers

Complicated rational equation to solve

I am trying to solve the following equation, but I have no idea where to start. Can somebody point me in the right direction? ...well, if somebody knows how to solve it that would be great, otherwise hints would be…
Adrien Hingert
  • 167
  • 7
1
vote
1 answer

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 the graph of $$y=\frac{x^3}{x^3}$$ not have a…
user85362
1
vote
2 answers

General method to numerically solve for the absolute maximum(s) of an irregular sinusoid

I am trying to find a strictly mathematical (i.e., not manual graphing or plugging) method to solve for the absolute (global) maximum value of an irregular sinusoid. Given a rational function $f(x)$, i.e., $$f(x) = \frac{8480128-33x}{100x+63}$$ and…
J W
  • 13
1
vote
1 answer

Solve for constants in rational model

I have the following rational model equation which fits distance versus time data very well: $$y = \frac{bt}{1+ct+dt^2}$$ The derivative of this equation is : $$\frac{dy}{dt} = \frac{(b-bdt^2)}{(1+ct+dt^2)^2}$$ Now at $t = 0, y = 0$…
rdemyan
  • 97
1
2
3 4 5 6 7