Factorise: x³ + 32x + 20.
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Factorize the denominator:
If , we can cancel the factors of
, which makes
and
removable discontinuities that appear as holes in the plot of
.
We're then left with
which is undefined when , so this is the site of a vertical asymptote.
As gets arbitrarily large in magnitude, we find
since the degree of the denominator (3) is greater than the degree of the numerator (2). So is a horizontal asymptote.
Intercepts occur where (
-intercepts) and the value of
when
(
-intercept). There are no
-intercepts because
is never 0. On the other hand,
so there is one -intercept at (0, 1).
The domain of is the set of values that
can take on for which
exists. We've already shown that
can't be -2, 2, or -1, so the domain is the set