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# 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.