all real numbers

A logistic function or curve is a mathematical function having an S shape, known as sigmoid curve. The name was given by Pierre Francois Verhulst in either the year of 1844 or 1845.

f(x)=1/1+e^-x

Yuo cannot include a graphical illustration here. Take a look at the Wikipedia, under "exponential function" and "logistic function".

Basically, the exponential function increases faster and faster over time. The logistics function initially increases similarly to an exponential function, but then eventually flattens out, tending toward a horizontal asymptote.

Logistic is the correct spelling.