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 aren't the functions $f(x) = \frac{x-1}{x-1}$ and $f(x) = 1$ the same?

I understand that division by zero isn't allowed, but we merely just multiplied $f(x) = 1$ by $\frac{x-1}{x-1}$ to get $f(x) = \frac{x-1}{x-1}$ and $a\cdot 1 = 1\cdot a = a$ so they're the same function but with different domain how is this…
Styrix
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$\frac{(x-1)(x+1)}{ (x+1)} \rightarrow (x-1)$ Domain Change

Forgive my ignorance. The below seems 'inconsistent'. If canceling the $(x+1)$ is 'legal', how does the domain change? I realize it does, but would someone be so kind as to provide an explanation? $$ \frac{x^2 - 1}{x + 1} \mbox{ is undefined when }…
John B
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Graph of $\quad\frac{x^3-8}{x^2-4}$.

I was using google graphs to find the graph of $$\frac{x^3-8}{x^2-4}$$ and it gave me: Why is $x=2$ defined as $3$? I know that it is supposed to tend to 3. But where is the asymptote???
Hele
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Questions with respect to rational functions

I am currently studying Hardy's Pure Course of Mathematics and am on chapter 2, section 24: Rational Functions. In this chapter, Hardy defines a rational function as the quotient of two polynomials such that: $R(x)=\dfrac{P(x)}{Q(x)}$ Hardy…
GovEcon
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Finding the range of rational functions

I have a problem that I cannot figure out how to do. The problem is: Suppose $s(x)=\frac{x+2}{x^2+5}$. What is the range of $s$? I know that the range is equivalent to the domain of $s^{-1}(x)$ but that is only true for one-to-one functions. I have…
Kot
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Why does this equation have different number of answers?

I have a simple equation: $$\frac{x}{x-3} - \frac{2}{x-1} = \frac{4}{x^2-4x+3}$$ By looking at it, one can easily see that $x \not= 1$ because that would cause $\frac{2}{x-1} $ to become $\frac{2}{0}$, which is illegal. However, if you do some magic…
Friend of Kim
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Values of $m$ for which $y^2 + 2xy + 2x -my -3$ can be factorised

For what values of $m$, will the expression $y^2 + 2xy + 2x -my -3$ be capable of resolution into two rational linear factor? This is how I did it: $$y^2 + 2xy + 2x -my -3 = y^2+(2x-m)y+2x-3$$ This can always be factorized if $b^2-4ac>0$, so if…
Shaurya Gupta
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Reasoning Behind Holes in Rational Functions

I am having some confusion about holes in rational functions. As I'm aware, a hole is where both the numerator and denominator become zero due to some discontinuity. For example, f(x) = (x+1)(x-1)/(x+1) would have a hole at x = -1. What is the…
foobar512
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Line of Symmetry for Hyperbolas

How might I find the equation for one of the lines of symmetry for the hyperbola $$y= 2 + \frac 6{x-4},\,\text{ where x cannot equal}\; 4.$$ I know that the lines of symmetry for the rational function $y=A/x$ are $y=x$ and $y=-x$...and that to find…
jaykirby
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Finding Slant Asymptotes using synthetic division rather than long division

Is it possible to use repeated synthetic division (rather than long division) to find a slant asymptote for a rational function such as $\displaystyle \frac{2x^3 + 3x^2 + 5x + 7}{(x-1)(x-3)}$? It appears to work, but I am not sure that it is valid…
Becky
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Cancelling variable factors in a rational function

Consider the function $\displaystyle\frac{2x−1}{x+5}$. The domain of this function is all real numbers except $x = -5$. Now consider that I do this: $\displaystyle\frac{2x−1}{x+5}⋅\frac{x}{x}$. This changes the domain of the function to all real…
Wesley
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Why is decomposing rational functions by assigning selected numerical values to x mathematically consistent?

Possible Duplicate: How does partial fraction decomposition avoid division by zero? Say you have the rational function: $\frac{x^2 + 1}{(x-1)(x-2)(x-3)}$ This means that the function is undefined when x is equal to 1, 2, or 3. Then to decompose…
user1086516
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Simple question about the $j$-function

Let $\Lambda(\lambda) := \left\{\lambda, 1 - \lambda, \frac{1}{\lambda}, \frac{1}{1-\lambda}, \frac{\lambda - 1}{\lambda}, \frac{\lambda}{\lambda - 1} \right\}$, and consider the $j$-function $$j(\lambda) = 256\frac{(\lambda ^2 - \lambda +…
Paul
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Predicting what the graph of a function looks like?

How can you predict what the function $$f(x) = \frac{(x - 5)(x + 4)(x - 3)^2(x)}{(x-5)(x)}$$ looks like before you plot it?
user117520
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Why $f(x)=\pi$ is a rational function? Is a constant function a polynomial even though the constant is a transcendental?

Wikipedia said A constant function such as $f(x) = π$ is a rational function since constants are polynomials. The function itself is rational, even though the value of $f(x)$ is irrational for all x. But the definition stated: A…
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