Rational Functions

The idea of rational functions

“Just like polynomials but division is allowed”

Rational functions are the (smallest) collection of functions satisfying:

  • The constant functions and the identity function $f(x)=x$ are rational functions.
  • If you combine two rational functions using the operations addition, subtraction, multiplication, or division, then the result is a rational function.

The definition of rational functions

Definition: a rational function is any function of the form $f(x)=\frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials. The domain consists of all real numbers $x$ for which $q(x)\neq 0$.

Why rational functions?

The interest in rational functions, as far as school mathematics is concerned, is twofold: they are only one step away from polynomials and are therefore worth knowing, and surprisingly, their graphs display features that are genuinely different from those of polynomials—they have asymptotes—and are therefore instructive for that reason.

Wu, A&G, p131

Question: What are some of the important features of rational functions?

  • roots
  • vertical asymptotes
  • horizontal asymptotes
  • holes in the graph
  • restricted domains (values of x for which the function is undefined)

Examples: For each rational function: Identify the roots, any horizontal and vertical asymptotes, any holes in graph, the domain.

  • $f(x)=\frac{1}{x}$
  • $g(x)=\frac{1}{x^2}$
  • $h(x)=\frac{1}{x^3}$
  • $j(x)=\frac{x+5}{x-3}$
  • $k(x)=\frac{x^2-1}{x}$
  • $m(x)=\frac{x^3-1}{x^2-x}$

Examples: Looking back at each of the rational functions in the previous example, find the roots of the numerator (and multiplicity of each root), and the roots of the denominator (and multiplicity of each root).

Example: Write the formula for a rational function with a single root at $x=5$ of multiplicity 2, a single vertical asymptote at $x=-5$, and a horizontal asymptote at $y=2$.

Questions:

  • Given a rational function $f(x)=\frac{p(x)}{q(x)}$. Suppose $x=3$ is a root of $p(x)$ of multiplicity 3, and $x=3$ is not a root of $q(x)$. What can you tell me about the graph of $f(x)$ at $x=3$?
  • Given a rational function $f(x)=\frac{p(x)}{q(x)}$. Suppose $x=-2$ is not a root of $p(x)$, but $x=-2$ is a root of $q(x)$ of multiplicity 2. What can you tell me about the graph of $f(x)$ at $x=-2$?
  • Given a rational function $f(x)=\frac{p(x)}{q(x)}$. Suppose $x=7$ is a root of $p(x)$ of multiplicity 1, and $x=7$ is also a root of $q(x)$ of multiplicity 1. What can you tell me about $f(x)$ at $x=7$?
  • Given a rational function $f(x)=\frac{p(x)}{q(x)}$. Suppose $x=-4$ is a root of $p(x)$ of multiplicity 3, and $x=-4$ is a root of $q(x)$ of multiplicity 4. What can you tell me about $f(x)$ at $x=-4$?
  • How do you determine the horizontal asymptotes of a rational function?
  • How do you determine the vertical asymptotes of a rational function?

Summary of Rational Functions

Defn. A rational function is a function that can be written in the form $f(x)=\frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials.

Suppose $x=a$ is a real number. Then:

  • If $x=a$ is a root of $f(x)$, then $a$ is a root of $p(x)$ but $a$ is not a root of $q(x)$.
  • If $a$ is not in the domain of $f(x)$ then $a$ is a root of $q(x)$
  • If $x=a$ is a vertical asymptote of $f(x)$, then either:
    • $a$ is a root of $q(x)$ but $a$ is not a root of $p(x)$, or
    • $a$ is a root of both $p(x)$ and $q(x)$, but the multiplicity of $a$ is greater in the denominator (i.e. greater multiplicity in $q(x)$ than in $p(x)$)
  • If $x=a$ is a hole in the graph $f(x)$, then $a$ is a root of both $p(x)$ and $q(x)$, but the multiplicity of $a$ is greater or equal in the numerator (i.e. multiplicity in $p(x)$ is $\geq$ multiplicity in $q(x)$)
  • If $y=b$ is a horizontal asymptote of $f(x)$, then:
    • If $b=0$, then the degree of $p(x)$ is greater than the degree of $q(x)$
    • If $b\neq 0$, then the degree of $p(x)$ equals the degree of $q(x)$, and $b=c/d$ where $c$ is the leading coefficient of $p(x)$ and $d$ is the leading coefficient of $q(x)$