Logistic function

Definition

The logistic function is a function with domain $\mathbb{R}$ and range the open interval $(0,1)$, defined as:

$x\mapsto \frac{1}{1+e^{-x}}$

Equivalently, it can be written as:

$x\mapsto \frac{e^x}{e^x+1}$

For this page, we will denote the function by the letter $g$.

Functional equation

The logistic function $g$ has the property that its graph $y=g(x)$ has symmetry about the point $(0,1/2)$. Explicitly, it satisfies the functional equation:

$g(x)+g(-x)=1$

Key data

Item	Value
default domain	all of $\mathbb{R}$, i.e., all reals
range	the open interval $(0,1)$, i.e., the set $\{x\mid 0\le x\le 1\}$
derivative	the derivative is $\frac{-e^{-x}}{(1+e^{-x}{)}^2}$. If we denote the logistic function by the letter $g$, then we can also write the derivative as $g^{\prime }(x)=g(x)g(-x)=g(x)(1-g(x))$
second derivative	If we denote the logistic function by the letter $g$, then we can also write the derivative as $g^{\prime }(x)=g(x)g(-x)=g(x)(1-g(x))(1-2g(x))$
logarithmic derivative	the logarithmic derivative is $\frac{e^{-x}}{1+e^{-x}}$ If we denote the logistic function by $g$, the logarithmic derivative is $g(-x)$
critical points	none
local maximal values and points of attainment	none
local minimum values and points of attainment	none
intervals of interest	increasing and concave up on $(-\mathrm{\infty },0)$ increasing and concave down on $(0,\mathrm{\infty })$
horizontal asymptotes	asymptote at $y=0$ corresponding to the limit for $x\to -\mathrm{\infty }$ asymptote at $y=1$ corresponding to the limit for $x\to \mathrm{\infty }$