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

4.8.1.2.8.

Linear / Cubic Rational Function

examples of linear/cubic rational functions
Function: \( \displaystyle f(x) = \frac{\beta_0 + \beta_1x}{1 + \beta_2x + \beta_3x^2 + \beta_4x^3}, \ \ \beta_1 \neq 0, \ \beta_4 \neq 0 \)
Function
Family:

Rational
Statistical
Type:

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

with undefined points at the roots of

\( \displaystyle 1 + \beta_2x + \beta_3x^2 + \beta_4x^3 \)

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 possible exception that zero may be excluded.

Special
Features:
Horizontal asymptote at:

\( \displaystyle y = 0 \)

and vertical asymptotes at the roots of

\( \displaystyle 1 + \beta_2x + \beta_3x^2 + \beta_4x^3 \)

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.

Additional
Examples:
linear/cubic rational function: y = (3 + 5*x)/
(1 + 5*x - 0.5*x**2 + 2*x**3  for -5 to 5
linear/cubic rational function: y = (3 - 5*x)/
(1 - 5*x + 0.5*x**2 - 2*x**3  for -5 to 5
linear/cubic rational function: y = (1 - 2*x)/
(1 + 2*x - x**2 - 0.1*x**3  for -5 to 10
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