Rational Functions
A rational function is a function that can be
written as the quotient of two polynomials. Any rational function r(x)
=
, where q(x) is not the zero
polynomial. Because by definition a rational function may have a variable in
its denominator, the domain and range of rational functions do not usually
contain all the real numbers.
There is special symbolism to describe the
behavior of a function in certain situations, depending on the behavior of the
independent variable. In speaking one might say a function approaches a certain
value as x increases, decreases, or approaches a certain
value. To mathematically say "approaches," an arrow is used. For
example, to say that the function f (x) increases
without bound as x increases without bound, one would
write f (x)âÜ’âàû asxâÜ’âàû . Or to
say the function f decreases without bound as x approaches 0 ,
you would write f (x)âÜ’ - âàû as xâÜ’
0 .
Rational functions often have what are called
asymptotes. Asymptotes are lines that functions approach but never reach. There
are three kinds of asymptotes: vertical, horizontal, and oblique. A vertical
asymptote is a line with the equation x = h if f(x)âÜ’±âàû as xâÜ’h from
either direction. A horizontal asymptote is a line with the equation y = k if f (x)âÜ’k as xâÜ’±âàû .
Oblique asymptotes are linear functions.
Study the graph below of the rational
function f (x) =
.
Figure
%: The graph of f (x) =
.
The line x =
0 is a verical asymptote and y = 0 is a horizontal
asymptote.
A line x = h is
a vertical asymptote of a function f (x) =
if p(h)≠ 0 and q(h)
= 0 . This is the general form of all vertical asymptotes of rational
functions.
Horizontal asymptotes are a little trickier to
understand. Let f (x) =
. If the degree of p is less than that
of q , then y = 0 is a horizontal
asymptote of f . If the degree of p is
greater than that of q , then f does not have
a horizontal asymptote. Ifp and q have the same
degree, then the horizontal asymptote occurs at the line y =
, where candd are the leading
coefficients of p and q , respectively.
An oblique asymptote occurs when the degree of
the numerator function is one greater than the degree of the denominator
function. If this situation arises, dividep(x) by q(x) using
long division. The result will be (
x + k) +
, where r(x) is the remainder. An oblique asymptote
will occur at y =
x + k .
One of the most important parts of working with
rational functions is making sure that the numerator and denominator are
completely factored, and that the common factors are canceled before you try to
calculate any asymptotes. And also keep in mind that not all rational functions
have asymptotes. We only focused on those that do because with long division,
you can calculate which rational functions reduce to simple polynomials, and we
already know how to deal with them.
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