Is this also a homotopy If $F$ is a path at a point $x$ then the following defines a homotopy from the path $FF^{-1}$ to the constant path $e$:
$$ \begin{array}{cc}
H(t,s) = F(2t) & s \ge 2t \\
H(t,s) = F(s) & s \le 2t \land s \le -2t + 2 \\
H(t,s) = F(2-2t) & s \ge -2t + 2
\end{array}$$
Is it possible that the following is also a homotopy from $e$ to $FF^{-1}$?
$$ \begin{array}{cc}
H(t,s) = F(0) & s =0 \\
H(t,s) = F({2t \over s}) & 0 \le t \le s/2 \\
H(t,s) = F^{-1}({2t - 1 \over s}) & s/2 \le t \le s
\end{array}$$

As far as I can see, you haven't described what happens when $t\geq s$ for instance what is $H(\frac{3}{4},\frac{1}{2})$? And so no, this does not define a homotopy as it is not even a well defined function.
