4. Process Modeling
4.8. Some Useful Functions for Process Modeling
4.8.1. Univariate Functions
4.8.1.2. Rational Functions

## Cubic / Cubic Rational Function

Function: $$\displaystyle f(x) = \frac{\beta_0 + \beta_1x + \beta_2x^2 + \beta_3x^3} {1 + \beta_4x + \beta_5x^2 + \beta_6x^3}, \ \ \beta_3 \neq 0, \ \beta_6 \neq 0$$
Function
Family:

Rational
Statistical
Type:

Nonlinear
Domain: $$\displaystyle (-\infty, \infty)$$

with undefined points at the roots of

$$\displaystyle 1 + \beta_4x + \beta_5x^2 + \beta_6x^3$$

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.

Range: $$\displaystyle (-\infty, \infty)$$

with the exception that $$\displaystyle y = \beta_3/\beta_6$$ may be excluded.

Special
Features:
Horizontal asymptote at:

$$\displaystyle y = \beta_3 / \beta_6$$

and vertical asymptotes at the roots of

$$\displaystyle 1 + \beta_4x + \beta_5x^2 + \beta_6x^3$$

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.