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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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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 can't understand If you look at this image, you…
user298530
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Equations and how to do them

Solve $$\frac{1}{3n}=\frac{3}{4(2n-1)}$$ please explain how to do this and why you have to change signs from positive to negative or vise versa.
Anaymous
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Properties of rational polynomials

I have experimental data points that can be modeled by two different rational polynomials. I am wondering if there is a way (e.g. by a transform or integral), to discriminate the following two rational polynomials (defined for…
JFNJr
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How do I rewrite this rational expression?

How do I rewrite the rational expression: $$\frac{x^3+5x^2+3x-10}{x+4}$$ But in the form of: $$q(x) + \frac{r(x)}{b(x)}$$
bryanna
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Does such a polynomial always exist, for any pole of a rational function?

Let $S \subseteq \mathbb{R}$ denote a cofinite subset of $\mathbb{R}$, and suppose $r : S \rightarrow \mathbb{R}$ is a rational function. Suppose $a$ is an element of $S^c$ (i.e. suppose $a$ is a pole of $r$.) Does there necessarily exist a…
goblin GONE
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Key Features of This Rational Function

$$-2x^2-15x-25\over x^2-x-5$$ I'm not sure how you would find the Intercepts or the Asymptote of this function. I've tried factoring the equation but it leaves me with $$-1(2x+5)(x+5)\over x^2-x-5$$ But I'm confused, how would I find the key…
MathisHard
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Domain and Range of a rational function

given the rational function $\frac{1}{x^{2} - \frac{x}{2}-3 }$ and asked for the domain and range, I multiplied thru by 2 and got $\frac{2}{2x^{2} - x-6 }$ I understand the domain includes all real numbers except for (vertical asymptotes) …
user118512
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How does $\frac{x^2}{ x^2 + 1}$ simplify to $1-\frac{1}{1+x^2}$

How does $$\frac{x^2}{ x^2 + 1}\quad\text{ simplify to }\quad 1-\frac{1}{1+x^2}\;?$$ Can someone explain the steps of how to get to that alternate form?
user182036
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Why can we discard remainders when calculating slant asymptotes of rational functions?

I understand how and when to calculate slant asymptotes of rational functions with numerators with one degree higher than the denominator, but I am confused as to why we can disregard the remainder when calculating the slant asymptote of the…
Declan H
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y and x intercepts with holes

I am solving for the intercepts of a rational function. It has a root of (0,0) which would be an x and a y intercept, however there is also a hole at this location. If the function is undefined (because of a hole) at an intercept, does the intercept…
Raquelle
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"Constrained" value of a function of two variables

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a rational function, whose numerator and denominator are (second-degree) polynomials of $x$ and $y$. The problem is to decide whether, for some given $k,k',T\in\mathbb{R}^+$, there exist two values…
Alberto M.
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Why care about excluded values of rational expressions when in lowest terms?

I'm struggling to come up with an explanation for why we specify excluded values for rational expressions that have been simplified to lowest terms, and yet do not exhibit the same division by zero situations found in their original expressions. For…
ybakos
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find all rational functions such that $f(x,y,z)=f(\frac{y+z}{2},\frac{x+z}{2},\frac{x+y}{2})$

Find all rational functions such that $f(x,y,z)=f(\frac{y+z}{2},\frac{x+z}{2},\frac{x+y}{2})$ I'm pretty sure it implies it must be a rational function in $x+y+z$ but I've not been able to prove it.
razivo
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When calculating the functions intersection of a horizontal asymptote, do you use the simplified function or the main function?

let’s say we have a function $$ f(x)=\frac{2x^2-5x+2}{x^2-4} $$ finding the ratio of the leading terms gives you $2x^2/x^2=2$, so we have a horizontal asymptote at $y = 2$. If you factor out the denominator and numerator, you get $$ f(x) =…
Abdulmalek Alem
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How do I write the equation of a rational function given these characteristics

$Y$ intercept at $-5$ No $x$ intercepts Discontinuous points at $(-1,-5)$ and $(3, -5)$ This was on an assignment, please help! Edit: the graph is NOT linear
Harrison
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