Function which gradually rises until some point and then quickly “falls”

Question

Could someone point me to any function ${ f(x) }$ which is continuous at some interval ${ x \in [x_0; x_1] }$ and can be represented by formula, so that it rises until some point and then quickly "falls" like on image below?

What may cause such behavior?

Answer 1

How about the function $f(x)=e^{-|x|}$, which has the following graph:

If you don't want it to be symmetric, you could use $f(x)=e^{-|1-e^x|}$, which has the following graph:

If you need the base line to be the same (the left asymptote of the above graph is $\frac{1}{e}$), you could use $$f(x)=e^{-|e^{-ax}-e^x|}$$ where $0<a<1$ is a constant you can vary. For example, with $a=1/10$, we get

Answer 2

How quickly? Piecewise linear functions satisfy this. In general, for real valued functions you can essentially get as nice bumps as necessary by considering $e^{-x^n}$, where the speed of decay can be acquired by increasing $n$. You can also construct bumps which are supported on compact sets and satisfy the above by playing with functions of the form $e^{-\frac{1}{x^2}}$.

Answer 3

Try something like a log-normal, $$f(x) = \frac{1}{\sqrt{2\pi}\sigma x}\exp\bigg({-\frac{(\log(x) - \mu)^2}{2\sigma^2}\bigg)}$$ For example,