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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R(x) = \frac{P(x)}{Q(x)} = \frac{a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0}{b_mx^m + b_{m-1}x^{m-1} + \ldots + b_1x + b_0}}
1) If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n < m} , then y = 0 is a horizontal asymptote.
2) If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = m, \text{ then } y = \frac{a_n}{b_m}} is a horizontal asymptote
3) No horizontal asymptote.