LOGISTIC FUNCTIONS AND THEIR GRAPHS
What is logistic function ?
Let a, b, c and k be positive constants, with b > 1.
A logistic growth function in x is a function that can be written in the form
f(x) = c / (1 + a⋅bx) (or) f(x) = c / (1 + a⋅e-kx)
where the constant c is the limit of growth.
If b > 1 or k < 0, these formulas yield logistic decay functions.
By setting a = c = k = 1, we obtain logistic function.
f(x) = 1/(1 + e-x)
This function, though related to exponential function ex, cannot obtained from ex by translation, reflection and vertical stretches and shrinks.
Graph of Logistic Function
f(x) = 1/(1 + e-x)
Set of all real values.
- The function is continuous and increasing for values of x.
- Symmetric about (0, 1/2), but neither even or odd.
- It is bounded below and above.
- There is no maximum and minimum.
Horizontal asymptotes :
y = 0 and y = 1
- There is no vertical asymptotes.
End behavior :
lim x-> -∞ f(x) = 0 and x-> ∞ f(x) = 1
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