Imagine we have two functions for calculating the rate (say the internet for investment) as
$$y = 1 + \operatorname{erf}(x-2); 0 < x < 4\\ y = 1 + 1.5 + \operatorname{erf} (x-6); 4 < x < 8 $$
Is it possible to mix these two function into one to be valid for $0<x<8$?
One can write a piecewise function using a unit step function. The unit step function is itself a piecewise function but it allows one to write other piecewise functions as a single expression.
$$ U(x)=\begin{cases}0\text{ for }x<0\\1\text{ for }x\ge0\end{cases} $$
Then given
$$ f(x)=\begin{cases} y_1\text{ for } x< x_1\\ y_2\text{ for }x_1\le x<x_2\\ y_3\text{ for }x_2\le x<x_3\\ \vdots\\ y_{n}\text{ for }x_{n-1}\le x<x_n\\ y_{n+1}\text{ for }x\ge x_n \end{cases} $$
one may write
$$ f(x)=y_1+(y_2-y_1)U(x-x_1)+(y_3-y_2)U(x-x_2)+\cdots+(y_{n+1}-y_n)U(x-x_n) $$
In your example, define $y$ for $0<x<8$ as
$$ y=1+\operatorname{erf}(x-2)+\left[1.5+\operatorname{erf}(x-6)-\operatorname{erf}(x-2)\right]U(x-4) $$