If I put bounds on a parameter, say, $s \in [0, \infty)$, then is that parameter still a free parameter, or is it no longer a free parameter?

Question

I usually parameterise a curve as $\Gamma = \Gamma(s) = (x_0(s)), y_0(s), u_0(s)) = (s, 0, \sin(s))$, where $s$ is a free parameter.

I now have the following parameterised curve:

$$ \Gamma(s) = \begin{cases} (s,0, \color{black}{\sin(s)}) & s\in [0,\infty) \\ (0,-s, \color{black}{\sin(-s)}) & s\in (-\infty,0] \end{cases}$$

If I put bounds on a parameter, as was done here, is it still a free parameter?