one function for 3 scenarios

Question

Just wondering, is there a function that depends and a small number of parameters and can model these 3 scenarios (depending on the chosen parameters):

The first is a constant, the second a log-normal (?) and the 3rd a decay function. Thanks!

Answer 1

Let's say you have two functions $f_0, f_1: \mathbb{R} \rightarrow \mathbb{R}$. Then you have a continuous deformation ("homotopy") between those two functions given by $$ f: \mathbb{R} \times [0,1] \rightarrow \mathbb{R}, f(x, t) := (1-t) f_0(x) + t f_1(x). $$ It has the property $f(x, 0) = f_0(x)$ and $f(x, 1) = f_1(x)$. This generalizes easily to any number of functions.

Edit: I thought of another way without case distintion to generalize this to any number of cases: Define the standard-n-simplex $\Delta^n$ by $$ \{ (t_0, t_1, ..., t_n) \in \mathbb{R}^{n+1} \mid t_i \geq 0, \sum t_i = 1 \}.$$ Then you can define your "homotopy" function $f$ by

$$f: \mathbb{R} \times \Delta^n \rightarrow \mathbb{R}, f(x, (t_0, t_1, ..., t_n)) := \sum\limits_i t_i f_i(x). $$ This looks kind of complicated, but you can imagine it like this: The closer you go to a vertex of your simplex (e.g. your triangle), the closer you go to one of your function $f_i$.