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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

4.8.1.2.3.

Linear / Quadratic Rational Function

examples of linear/quadratic rational functions
Function: \( \displaystyle f(x) = \frac{\beta_0 + \beta_1x}{1 + \beta_2x + \beta_3x^2}, \ \ \beta_1 \neq 0, \ \beta_3 \neq 0 \)
Function
Family:
Rational
Statistical
Type:
Nonlinear
Domain: \( \displaystyle (-\infty, \infty) \)

with undefined points at

\( \displaystyle x = \frac{-\beta_2 \pm \sqrt{\beta_2^2 - 4\beta_3}} {2\beta_3} \)

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

\( \displaystyle \beta_2^2 - 4\beta_3 \)

is negative, zero, or positive.

Range: \( \displaystyle (-\infty, \infty) \)
Special
Features:
Horizontal asymptote at:

\( \displaystyle y = 0 \)

and vertical asymptotes at:

\( \displaystyle x = \frac{-\beta_2 \pm \sqrt{\beta_2^2 - 4\beta_3}} {2\beta_3} \)

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

\( \displaystyle \beta_2^2 - 4\beta_3 \)

is negative, zero, or positive.

Additional
Examples:
linear/quadratic rational function example 1:
 y = (1+2x)/(1 - 0.5x - 0.5x^2; 1 < x < 10
linear/quadratic rational function example 2:
 y = (1+2x)/(1 - 0.5x - 0.5x^2; -10 < x < -2
linear/quadratic rational function example 3:
 y = (1+2x)/(1 - 5x + 2x^2; 2 < x < 10
linear/quadratic rational function example 4:
 y = (1-0.5x)/(1 - 0.5x - 0.5x^2; 1 < x < 10
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