Basic Facts about Rational Functions

Definition. A rational function is a ratio of polynomials, i.e. a function of the form

Basic algebraic features. There are a few important algebraic aspects to a rational function.

• Zeros. The zeros of a rational function f(x) are the values such that f(x) = 0. Hence, so find them,
set f(x) = 0 and solve using the techniques we've already developed for fractional equations.

• Domain. The domain of a rational function consists of all real numbers except those values of x that
result in a division by 0. Hence, to find the domain, set the denominator(s) equal to 0 and solve.
Exclude these values.

• Simplified form. A rational function is in simplified form if (a) all terms have been brought together
over a single common denominator, and (b) the fraction is reduced, i.e. all common factors in the
numerator and denominator have been canceled.

Two important things to note:

1. Never simplify before finding the domain, as you may lose "bad points" in the process.
2. When factoring the numerator and denominator, you will often have to use factoring techniques
- long division, synthetic division, the rational root test, and so on | but remember that we
don't invoke imaginary numbers when dealing with rational functions.

Asymptotes. Intuitively speaking, asymptotes are "invisible curves" against which a curve appears to line
up with. Rational functions always have these exotic hidden curves, though they may take many different
forms.

• Vertical asymptotes. These occur at the x-values where the simplified denominator equals 0. Never
look for vertical asymptotes until you've simplified the rational function. Remember that the equation
of a vertical line is x = a.

Graphically, the graph of a rational function "breaks" across a vertical asymptote. These are rather
violent "discontinuities" that divide the graph of the function into distinct "pieces."

• Horizontal asymptotes. These occur only if the degree of the numerator is less than or equal to the
degree of the denominator. In the case that the degrees are equal, then vertical asymptote is given by



If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is
y = 0.

Graphically, the graph of a rational function will appear to eventually "lay °at" against the asymptote
to the far left and far right, although in the "middle" it may cross this invisible line any number of
times.

• Other asymptotes. If the degree of the numerator is greater than the degree of the denominator,
then the rational function will not level out to a horizontal asymptote, but it will level out against a
different invisible curve called an asymptotic curve. The equation of this asymptote is y = Q(x), where
Q(x) is the quotient obtained by using long division of the given rational function. (In the special case
that this asymptote is a non-horizontal line, it is called a slant asymptote.)

Graphically, the graph of a rational function will appear to eventually "lay °at" against the asymptotic
curve to the far left and far right, although in the "middle" it may cross this invisible line any number
of times.