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

## Cubic / Quadratic Rational Function

Function: $$\displaystyle f(x) = \frac{\beta_0 + \beta_1x + \beta_2x^2 + \beta_3x^3}{1 + \beta_4x + \beta_5x^2}, \ \ \beta_3 \neq 0, \ \beta_5 \neq 0$$
Function
Family:

Rational
Statistical
Type:

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

with undefined points at

$$\displaystyle x = \frac{-\beta_4 \pm \sqrt{\beta_4^2 - 4\beta_5}} {2\beta_5}$$

There will be 0, 1, or 2 real solutions to this equation corresponding to whether

$$\displaystyle \beta_4^2 - 4\beta_5$$

is negative, zero, or positive.

Range: $$\displaystyle (-\infty, \infty)$$
Special
Features:
Vertical asymptotes at:

$$\displaystyle x = \frac{-\beta_4 \pm \sqrt{\beta_4^2 - 4\beta_5}} {2\beta_5}$$

There will be 0, 1, or 2 real solutions to this equation corresponding to whether

$$\displaystyle \beta_4^2 - 4\beta_5$$

is negative, zero, or positive.