Domain:

\( \displaystyle (\infty, \infty) \)
with undefined points at the roots of
\( \displaystyle 1 + \beta_3x + \beta_4x^2 + \beta_5x^3 \)
There will be 1, 2, or 3 roots, depending on the particular values
of the parameters. Explicit
solutions for the roots of a cubic polynomial are complicated and
are not given here. Many mathematical and statistical software
programs can determine the roots of a polynomial equation numerically,
and it is recommended that you use one of these programs if you need
to know where these roots occur.

Special Features:

Horizontal asymptote at:
\( \displaystyle y=0 \)
and vertical asymptotes at the roots of
\( \displaystyle 1 + \beta_3x + \beta_4x^2 + \beta_5x^3 \)
There will be 1, 2, or 3 roots, depending on the particular values
of the parameters. Explicit
solutions for the roots of a cubic polynomial are complicated and
are not given here. Many mathematical and statistical software
programs can determine the roots of a polynomial equation numerically,
and it is recommended that you use one of these programs if you need
to know where these roots occur.
