Polynomials and Rational Functions
4.4 Rational Functions
Basic Structure
Rational functions are quotients of polynomials, so denominator behavior drives restrictions.
Definition: Rational Function
A rational function has the form
$$R(x)=\frac{P(x)}{Q(x)},$$
where $P$ and $Q$ are polynomials and $Q(x)\ne0$.
Definition: Vertical Asymptote
A vertical asymptote is a line $x=a$ where $R(x)$ grows without bound near $a$.
Example: Find domain and vertical asymptotes
For
$$R(x)=\frac{x+1}{x^2-9},$$
find domain and vertical asymptotes.
Show Solution
Factor denominator:
$$x^2-9=(x-3)(x+3).$$
So domain excludes $x=\pm3$.
Vertical asymptotes at
$$x=3\quad\text{and}\quad x=-3.$$
MyOpenMath: Practice: rational domains and vertical asymptotes (placeholder)
Horizontal Asymptotes
Compare numerator and denominator degrees for end behavior.
Fact: Horizontal asymptote rules
For $R(x)=\frac{P(x)}{Q(x)}$:
- degree($P$) < degree($Q$): $y=0$
- degree($P$) = degree($Q$): ratio of leading coefficients
- degree($P$) > degree($Q$): no horizontal asymptote
Example: Determine horizontal asymptote
Find the horizontal asymptote of
$$f(x)=\frac{5x^2-1}{2x^2+7x+4}. $$
Show Solution
Degrees are equal (both 2), so asymptote is ratio of leading coefficients:
$$y=\frac{5}{2}. $$
