Can you please help with the construction of the piecewise function specified in the details and its plotting?

edit

You can get the documentation about piecewise functions by typing Piecewise? In your particular case, you can do:

sage: f1(x) = 0
sage: f2(x) = exp(-x^2/(1-x^2))
sage: f = Piecewise([[(-2,-1),f1],[(-1,1),f2],[(1,2),f1]])
sage: f.plot()

EDIT : i did not answer the second part of your question. Indeed, if you do:

sage: g(x) = x
sage: h(x) = x^2
sage: parametric_plot3d([f, g, h], (x, -2, 2))

Then you get the following error:

AttributeError: PiecewisePolynomial instance has no attribute '__float__'

This is because parametric_plot3d needs to evaluate the function f with the __float__ method that does not exists for piecewise functions. However, piecewise finctions are able to evaluate on floating points (with the __call__ method):

sage: f(0.1)
0.989949833766045

So, a possible wotrkaround is to redirect the __float__ method for piecewise functions to the __call__ method that currently works:

sage: f.__float__ = f.__call__

Now, the following works:

sage: parametric_plot3d([f, g, h], (x, -2, 2))

I am afraid you can't do a 3D parametric plot with Piecewise.

I suggest to use a python dict to define the functions, for example:

F = {(-2,-1):[t,e^-1,0], (-1,1):[t,e^-t^2,0], (1,2):[t,e^-1,0]}
G = Graphics()
for k in F.keys():
    G += parametric_plot3d(F[k],k)
G.show()