What Do Functions Do for Tiny X?

Congratulations Andrei for another very good solution.

The 4 functions are:

$$A(x) = \sin x,\quad B(x) = 1 - \cos x,\quad C(x) = \log (1+x), \quad D(x) = 1- {1\over (1-x)}.$$ I consider the logarithm in base e. First I plotted the 4 functions using Graphmatica. In all figures $A(x)$ is violet, $B(x)$ is white, $C(x)$ is red and $D(x)$ is cyan. For $[-1, 1]$ I obtain:

For [-0.1, 0.1]

And for [-0.01, 0.01] I have: