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 $K$.

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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 equations where "$y$" is on both sides, but can't be…
John
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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 $500$ $ms$) So, $f(6) = 250$ and $f(1) = 500$ Keep in mind,…
JonasAnon
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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? $t/(1+t)=1-1/(1+t)$
n0tis
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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.
Ryan
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Why do rational functions like $\frac{1}{x}$ have weird shapes?

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?
Trup Krup Drób Grób
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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 the $x$-axis is tangent to $C$. Here's what I…
H_Hassan
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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?
user299305
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Hole of rational expression: (x-9)/(sqrt(x) - 3)

$$(x-9)/(√x - 3)$$ Note that the $3$ of the denominator is not part of the square root (just $x$). Additionally, the hole of this expression is $(9,6)$. I don't know how to reach this result, aside from the fact that I must first find a common…
Ben
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Second Hole in (3x^2 + 6x) / (x^3 - 4x)

$$(3x^2 + 6x) / (x^3 - 4x)$$ To find the hole(s) of a rational expression, I must first create a common factor in both the numerator and denominator; however, can't come up with anything for the above expression. I should add that I am aware of…
Ben
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Can every rational function be represented in barycentric form?

This article about polynomial interpolation claims that (it is known that) every rational function may be represented in barycentric form: $$r(x)=\frac{\sum_{j=0}^N\frac{w_j}{x-x_j}y_j}{\sum_{j=0}^N\frac{w_j}{x-x_j}}$$ What about the rational…
Thomas Klimpel
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Oblique asymptote question

First part is original equation, second is after it was divided using polynomial division to find equation for oblique asymptote.I don't understand the 'x-1' part being the equation of the asymptote here,and the rest can be ignored because its…
amnna
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How would I make sense of the answer choices for solutions to a rational equation?

I am trying to find all of the asymptotes of a rational function, and the answer choices I am given are confusing. Because the degree of the function in the numerator was greater than the degree of the function in the denominator, I divided the…
RyuJV
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Solving Rational Equations.

Solve $1+\frac{1}{x^2}=\frac{3}{x}$ I tried to cross multiply after combining 1 + $\frac{1}{x^2}$ but it just came out weird. How do I solve this?
QuantumPi
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Determine the exact point(s) of intersection between f(x)=x^2-x-13 and it’s reciprocal function

does anybody know how to solve this? I’m not sure what the reciprocal function means.
James Baxter
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Find a range of $\frac{4x - 1}{2x + 3}$

Given a function $\frac{4x - 1}{2x + 3}$ I know its Domain is $\{x \in \mathbb{R} | x \ne \frac{-3}{2}\}$ But how can I find its range? Any hint is appreciated.
OOD Waterball
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