Rational Functions

Introduction

Rational functions are ratios of polynomial functions. This means we have to be worried about points where the denominator is zero. In this section we will focus on the algebraic aspects of rational functions. These aspects include vertical, horizontal, and oblique asymptotes.

Definition

A rational function is a function of the form $R(x)={\frac {P(x)}{Q(x)}}$, where Q(x) is not the zero polynomial. The domain of a rational function is all real numbers except for the zeros of Q(x).

Example:

${\frac {x^{2}+3}{3x^{4}-5x^{3}+9}}$

Asymptotes

An asymptote is a notion that is more widely applicable than to just rational functions.

Let R(x) be a function. If as $x\rightarrow \infty ,{\text{ or }}x\rightarrow -\infty$ R(x) approaches some value L, then the line y = L is a horizontal asymptote of the graph of R. For now just accept this as the definition. We will make this idea of R(x) approaching some value L a bit more concrete later on.

If, as x approaches some number c, the values $\vert R(x)\vert \rightarrow \infty$, then the line x = c is a vertical asymptote of the graph of R. This will get mentioned again later, but for rational functions vertical asymptotes correspond to zeros of the denominator that have higher multiplicity than they do in the numerator, if they are even zeros of the numerator.

Theorem: Locating Vertical Asymptotes

Let $R(x)={\frac {P(x)}{Q(x)}}$ be a rational function in lowest terms, there are no common factors of P(x) and Q(x). Then the vertical asymptotes of R(x) will occur at the zeros of Q(x). So, if r is a zero fo Q(x), then x = r will be a vertical asymptote.

Finding a horizontal aysmptote:

Let $R(x)={\frac {P(x)}{Q(x)}}={\frac {a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots +a_{1}x+a_{0}}{b_{m}x^{m}+b_{m-1}x^{m-1}+\ldots +b_{1}x+b_{0}}}$

1) If $n<m$, then y = 0 is a horizontal asymptote.

2) If $n=m,{\text{ then }}y={\frac {a_{n}}{b_{m}}}$ is a horizontal asymptote

3) No horizontal asymptote.

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