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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

4.8.1.2.6.

Cubic / Linear Rational Function

examples of cubic/linear rational functions
Function: \( \displaystyle f(x) = \frac{\beta_0 + \beta_1x + \beta_2x^2 + \beta_3x^3}{1 + \beta_4x}, \ \ \beta_3 \neq 0, \ \beta_4 \neq 0 \)
Function
Family:

Rational
Statistical
Type:

Nonlinear
Domain: \( \displaystyle \left( -\infty, \ -\frac{1}{\beta_4} \right) \ \cup \ \left( -\frac{1}{\beta_4}, \ \infty \right) \)
Range: \( \displaystyle (-\infty, \infty) \)
Special
Features:

Vertical asymptote at:

\( \displaystyle x = -\frac{1}{\beta_4} \)
Additional
Examples:
cubic/linear rational function: y = (4+2*x+7*x**2-2*x**3)/(1-x)
  for x = 1 to 10
cubic/linear rational function: y = (4+2*x+7*x**2-2*x**3)/(1-x)
  for x = -10 to 1
cubic/linear rational function: y = (4-2*x-7*x**2+2*x**3)/(1-x)
  for x = 1 to 10
cubic/linear rational function: y = (4-2*x-7*x**2+2*x**3)/(1-x)
  for x = -10 to 1
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