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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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Determining X-intercepts of a rational function

I have a rational function: $y = \frac{x^2 - 3x}{x^2 + 2x - 48}$ The explanation I was given to find the x-intercepts was: "let y = 0, and solve for x. Basically you just set the numerator of the fraction equal to 0 and factor." What does…
VerySeriousSoftwareEndeavours
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Domain and Continuity of Rational Functions

Question From the equation $x^{21} - 1 = 0 $ Define 20 complex roots as $\omega_{1},\omega_{2}, \omega_{3},...,\omega_{20}$ What is the value of $(1-\omega_{1})(1-\omega_{2})(1-\omega_{3})...(1-\omega_{20})$? We can rewrite the equation…
VladeKR
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What does this ratio mean?: percent treated or removed / percent remaining

I have found this function in several math textbooks: $$ C(p)=\frac{80000p}{100-p} $$ It represents the cost C(p) to remove the percent p of impurities from coal. The function shows that as we try to remove a higher percentage of impurities, the…
user3050028
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Determine all m parameter values for which the domain of the function are all real numbers

The function is: $$ f(x) = \frac{2x}{mx^2 + 1} $$ The answer sheet says the answer is: $ m \ge 0 $ but I don't get it. My answer was $ m \in \mathbb{R} \setminus \{-1, 1\} $, because then, when $x=1$ or $x=-1$ the denominator may come out as $0$.…
Marek M.
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simplifying polynomial rational expressions

When rational expression $$\frac{x^{2} + 3x}{x^{2} + 5x}$$ is simplified it equals $(x+3)/(x+5)$ where $x\ne 0$. What I don't understand is why are the two functions not equal for the whole domain while we are just doing is dividing for example …
waqar ahmed
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Is a hole at a vertical asymptote recognized?

If I have a hole in a rational function, but the hole is located at a vertical asymptote, is the hole still recognized? For example in the equation (x+3)(x+1)/(x+1)(x+1) I will have a hole at -1, but a vertical asymptote also at -1.
javabrink
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Asymptotes of a General Rational Function

$f(x)$ is a rational function of the form $f(x)=\frac{ax^2+bx+c}{dx+e}$ and the function $g(x)$ is given by $g(x)=\frac{4}{f(x)}$. It is known that $f(x)$ has an oblique asymptote $y=x+1$ and $g(x)$ has vertical asymptotes $x=\frac{1 \pm \sqrt3}{2}…
Inquirer
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Range of a Rational Function with arbitrary constant

Find a condition on $c$ so that the function $f(x) =\frac{x+c}{x^2-3x-c}$ has the whole of the real numbers as its range. I'm not entirely sure how to approach this problem. The answer is $-2
Inquirer
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What type of asymptote is there, if any, in a rational function where the numerator's degree is 2 or more than the denominators?

It is known that there is a slant (oblique) asymptote when the degree of the numerator is 1 more than the denominator in a rational function. To find the equation of this asymptote, you simply divide the denominator into the numerator to yield a…
Beethoven1111
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Find a polynomial that satisfy the given given conditions.... Zeros - 1,1,2^1/2 integer coefficient and constant term 6...

Find a polynomial that satisfy the given given conditions.... Zeros - 1,1,2^1/2 integer coefficient and constant term 6... My question is how do I make them integers... I tried multiplying by square root 2 but I also get square... Any hint would be…
Tarej
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Divisibility of a rational function

Problem Determine coefficients $a$ and $b$ such that $$ \dfrac{x^3+ax^2+bx-6}{bx^2+2x+a} $$ is divisible by $x-2.$ What is actually meant by divisible in this case?
Fatou
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Decomposition of a rational function

I want to decompose the rational function $$ \cfrac{P(s)}{Q(s)}=\cfrac{\prod\limits_{i=1}^m (s+a_i)}{\prod\limits_{i=1}^n (s+b_i)} $$ where $a_i>0$ for every $i=1,\dots,m$, $b_i>0$ for every $i=1,\dots,n$ and $n>m$. In other words, I'm looking for…
Muriel
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Zeroes of rational functions

I am reading upon the rational functions and came across this question. How to prove that a rational function F(s) cannot be zero on any interval on the $j\omega$ axis? By intuition, we can say that a rational function can have only a finite…
Manideep
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Intersection of the domains of rational parametric functions

Let $F(\vec{t}) = R_1(\vec{t}) = 0$ and $G(\vec{s}) = R_2(\vec{s}) = 0$ be two functions where $F,G: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $R_1,R_2$ are rational. Overall, my goal is to find those parameters $\vec{t} = \vec{s}$ where both…
Michael Stachowsky
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How do I graph a transformed rational function?

If the function is $-\frac{5}{x+3} + 2$ how would I graph that without using a table of values? Do I just use the asymptotes $x=-3$, and $y= 2$ and draw the general shape? Do I use "mapping points" like $(x-3, -5y+2)$?
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