Mathematics Stack Exchange
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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 aren't equations with rational expressions equivalent to different forms?

When learning about rational equations, we learn that we must consider the domain and/or denominator, to ensure that our solution does not cause a division by 0 error. For example: $$ \frac{x + 1}{9 - x} = \frac{2}{3} $$ Here, we say that $x$ has a…
ybakos
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Should we simplify or not for function domain?

If $f(x) = \frac{x^2}{x}$, what is the domain of $x$? Should we simplify it to $f(x) = x$ so that $x$ is any real number? Or we do not simplify it and that the $x$ is any real number and $x \ne 0$. But depend on the web service such as this example…
Allen
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When are rational functions considered equal? Eg, $\frac{z-1}{z^2-1}$ vs $\frac{1}{z+1}$

I came across an example like this recently. Consider $$ f(z) = \frac{z-1}{z^2 - 1}.$$ I want to determine the singularities of $f$. Since $z^2-1 = (z-1)(z+1)$, algebraically $f(z) = \frac{1}{z+1}$, i.e. $f$ has a simple pole in $-1$. However, when…
tolUene
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$F(X)=F(\frac 1 X) \iff \exists G \in \mathbb{C}(X), G(X+ \frac 1 X)=F$

Let $F \in \mathbb{C}(X)$ be a rational function with complex coefficients (and $\mathbb{C}(X)$ the set of such functions) i.e. $F=\frac P Q$ for $P,Q \in \mathbb{C}[X]$ I need to show that $F(X)=F(\frac 1 X) \iff \exists G \in \mathbb{C}(X), G(X+…
Bastien Tourand
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Rational Equations

So, I got this question: Essentially, a person wants to open an art venue, but the cost is $\$400$ and the foodservice she's working with is charging $\$3.75$ per person for food and drink. The task is to create an equation that evenly splits the…
Alexander The Great Hamilton L
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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
Miss Q
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Number of fixed points of a rational function

We know if $R(z)$ is a rational function, then it is in the form $R(z) = f(z)/g(z)$, where $f$ and $g$ are complex polynomials. If we want to find its fixed points, we can take $R(z) = z$, which gives the equation $f(z) - z\cdot g(z) = 0$. By the…
Atratrana Suna
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Finding the equation of a Rational function with a given slant asymptote

Algebra concepts only: Find the equation of a rational function with zeroes at $2$ and $3$. No horizontal asymptote and a slant asymptote of $y = x$. If I had been given only 1 zero, for example x = 2, then I would use the parent graph form to set…
McMath
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Finding the number of roots which are real

Let $0
The Learner
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Simplification of rational expression gone wrong(high school math)

Currently doing high school math and can't get this one right. I think I'm using entirely incorrect practices and am trying to pinpoint what it is. Could someone tell me where exactly I went wrong? Original…
Zae
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I didn't understand the below from sets and functions algebra caltech pdf. The highlight is my problem. What does it mean and why does say X=0?

I am not English speaking and also highlight is yellow in solution… The graph in the fogure shows an even function $f(x) = \dfrac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are rational quadratic polynomials. Give possible formulas for $p(x)$ and…
Mustafa Asghari
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Question about rational functions and horizontal asymptotes

I am working on a math problem for class and am stumped by the nature of its horizontal asymptote. The equation is f(x) = x / (x+5)(x-2). Based on the rules for horizontal asymptotes given to me in class, when the degree of the leading coefficient…
hallu
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reduce a rational expression to lowest terms

$$\frac{x^2 + 4x + 3}{x^2 - 2x - 3}$$ I'm coming up with $\frac{x + 3}{ x - 3}$ however it seems wrong. $$x^2 + 4x + 3 = (x + 3) ( x + 1)$$ $$x^2 - 2x - 3 = (x - 3 )(x + 1) $$ cancel out $x + 1$ and left with $\frac{x + 3}{x - 3}$
Lukus Lyon
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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 want to copy it: -((1 + 2 x)/(7 + n + 5 x + 2 n x))…
Art
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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 functions, that perhaps I should just imagine them as…
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