The Logistic (Sigmoid) Function $y = \dfrac{1}{1+e^{-x}}$

The logistic function $y = \dfrac{1}{1+e^{-x}}$, also called the standard sigmoid, is the archetypal S-shaped curve. It squeezes any real input into an output between $0$ and $1$, making it ideal for representing probabilities or a smooth "on/off" transition.

Domain and range It is defined for all real $x$. The denominator $1 + e^{-x}$ is always positive and tends to $1$ as $x \to +\infty$, so the range is the open interval $(0, 1)$; the endpoints are never reached.

Symmetry Since $f(-x) = 1 - f(x)$, the graph is point-symmetric about $\left(0, \dfrac{1}{2}\right)$. For instance $f(2) \approx 0.881$ and $f(-2) \approx 0.119$, and their sum is $1$.

Monotonicity The derivative $f'(x) = f(x)\bigl(1 - f(x)\bigr) = \dfrac{e^{-x}}{(1+e^{-x})^2}$ is always positive, so the function is strictly increasing everywhere. The slope is largest at the center $x = 0$, where it equals $\dfrac{1}{4}$.

Asymptotes and limits As $x \to -\infty$ we have $e^{-x} \to +\infty$ so $y \to 0$, and as $x \to +\infty$ we have $e^{-x} \to 0$ so $y \to 1$. Thus there are two horizontal asymptotes, $y = 0$ and $y = 1$.

Inflection point The second derivative $f'' = f(1-f)(1-2f)$ changes sign where $f = \dfrac{1}{2}$, that is at $x = 0$. The point $\left(0, \dfrac{1}{2}\right)$ is the unique inflection point, where the curve switches from concave up to concave down.

Relation to other functions It can be written as $f(x) = \dfrac{1}{2}\left(1 + \tanh\dfrac{x}{2}\right)$, a shifted and scaled hyperbolic tangent. A remarkable feature is that its derivative takes the compact form $y(1-y)$.

Applications and history In the 19th century the Belgian mathematician Verhulst introduced it as the solution of the logistic equation modeling population growth under limited resources. Today it appears wherever growth or probability is involved: logistic regression, activation functions in neural networks, and the progress of chemical reactions.