There are of course other alternatives, such as:
f(x) = \cases{ a x & $x < \tau$ \\
1+ (a \tau-1)e^{(ax-a\tau)/(a\tau-1)} & $x \ge \tau$ }
where a > 0 and 0 \le \tau < 1/a. This gives you
This has the benefit of having a maximum value of 1.0.
There are of course other alternatives, such as:
where a > 0 and 0 \le \tau < 1/a. This gives you
This has the benefit of having a maximum value of 1.0.