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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

4.8.1.2.1.

Constant / Linear Rational Function

examples of constant/linear rational function
Function: \( \displaystyle f(x) = \frac{\beta_0}{1 + \beta_1x}, \ \ \beta_0 \neq 0, \ \beta_1 \neq 0 \)
Function
Family:

Rational
Statistical
Type:

Nonlinear
Domain: \( \displaystyle (-\infty, \ -\frac{1}{\beta_1}) \ \cup \ (-\frac{1}{\beta_1}, \ \infty) \)
Range: \( \displaystyle (-\infty, \ 0) \ \cup \ (0, \ \infty) \)
Special
Features:

Horizontal asymptote at:

\( \displaystyle y=0 \)

and vertical asymptote at:

\( \displaystyle x = -\frac{1}{\beta_1} \)

Additional
Examples:
constant/linear rational function example 1: y = 1/(10-7*x);
 -30 < x < -5
constant/linear rational function example 2: y = 1/(0.2 + 3*x);
 4 < x < 5
constant/linear rational function example 3: y = 1/(-50 - 3*x);
 0.5 < x < 1.1
constant/linear rational function example 4: y = 1/(-400 + 0.04*x)
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