A function $f(x)$ that increases from $0$ to $1$ when $x$ increases from $-\infty$ to $\infty$.

Question

I am looking for a function $f(x) \in [0,1]$ when $x \in (-\infty, +\infty)$.

$f(x)$ increases very fast when $x$ is small starting from $-\infty$, and then very slow and eventually approach $1$ when $x$ is infinity.

What about the continuity and differentiability of $f(x)$? Any restrictions? — SchrodingersCat, Dec 24 '15 at 12:52

score 8 · Answer 1 · answered Dec 24 '15 at 12:48

8

$$f(x) = \dfrac1{1+e^{-x}}$$ is one possible candidate.

answered Dec 24 '15 at 12:48

Adhvaitha

score 3 · Answer 2 · answered Dec 24 '15 at 13:03

3

Consider the function $f(x) = \frac{\arctan(x)}{\pi} + \frac{1}{2}$

answered Dec 24 '15 at 13:03

SchrodingersCat · Answer 3 · 2015-12-24T13:06:54.997

1

The function $$f(x)=\frac{1}{1+a^{-x}}$$ has such a property $\forall \, a>1 , a \in \mathbb{R^+}$.

The function by Leg where $a=e$ is a special case of this type of functions.

edited Dec 24 '15 at 13:06

answered Dec 24 '15 at 12:58

SchrodingersCat

3 Answers3