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

## 4.8.1.2.5.

Function: $$\displaystyle f(x) = \frac{\beta_0 + \beta_1x + \beta_2x^2}{1 + \beta_3x + \beta_4x^2}, \ \ \beta_2 \neq 0, \ \beta_4 \neq 0$$
Function
Family:
Rational
Statistical
Type:
Nonlinear
Domain: $$\displaystyle (-\infty, \infty)$$

with undefined points at

$$\displaystyle x = \frac{-\beta_3 \pm \sqrt{\beta_3^2 - 4\beta_4}} {2\beta_4}$$

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

$$\displaystyle \beta_3^2 - 4\beta_4$$

is negative, zero, or positive.

Range: The range is complicated and depends on the specific values of $$\beta_1, \, \ldots , \, \beta_5$$.
Special
Features:
Horizontal asymptotes at:

$$\displaystyle y = \frac{\beta_2}{\beta_4}$$

and vertical asymptotes at:

$$\displaystyle x = \frac{-\beta_3 \pm \sqrt{\beta_3^2 - 4\beta_4}} {2\beta_4}$$

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

$$\displaystyle \beta_3^2 - 4\beta_4$$

is negative, zero, or positive.