plot issue with a self-defined piecewise function

edit

The problem in your first example is that when you do f(x), it is actually calling f and returning a value. However, you want to plot the function, not the return value from one call. So you should do this:

def f(x):
   if x>3:
       return(x^2)
   if x<=3:
       return(3*x)

plot(f,(x,0,5))

Notice that now, I haven't called the function. Instead, I'm just passing the function into plot, and plot will call the function with different values.

Your lambda function trick works because it is passing a function into plot. plot then calls the lambda function, which in turn calls the f function. But I think it's easier to what I did above.

Another option is to use the (not very powerful) Piecewise class. You could at least do some basic calculus with it as well, then. However, you'd have to define endpoints.

sage: g = Piecewise([[(-10,1),x], [(1,10),x^2]], x)
sage: derivative(g)
Piecewise defined function with 2 parts, [[(-10, 1), x |--> 1], [(1, 10), x |--> 2*x]]
sage: integrate(g)
Piecewise defined function with 2 parts, [[(-10, 1), x |--> 1/2*x^2 - 50], [(1, 10), x |--> 1/3*x^3 - 299/6]]
sage: plot(g)

So it works. But see ticket 11225 for a nice list of ways in which piecewise plotting could be improved. For instance, with infinite endpoints, although it does math, it won't plot. Perhaps that makes sense, since we wouldn't see infinity anyway...

sage: f = Piecewise([[(-oo,1),x], [(1,oo),x^2]], x)
sage: f(-10^6)
-1000000
sage: plot(f)
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
ValueError: cannot convert float NaN to integer

To answer the 3d plot question, again, you need to make the functions calculate their values when the plotting routine is actually getting points, not before the plotting routine is called:

Y=parametric_plot3d([lambda s,t: (2+sin(s))*sin(t*f(s)),lambda s,t: cos(s),lambda s,t: (2+sin(s))*cos(t*f(s))],(s,0,2*pi),(t,-1,1))

Toward a related issue - how do I incorporate such a conditional in a 3d plot? Consider the following.

s,t=var('s,t')
def f(x):
    if x<pi:
        return(arccos(-2/(2+sin(x))))
    if x>=pi:
        return(pi)

plot(f,(x,0,2*pi)) --- to check that the function is properly defined

X=parametric_plot3d([(2+sin(s))*sin(t),cos(s),(2+sin(s))*cos(t)],(s,0,2*pi),(t,0,2*pi))  --- to produce a torus
X.show(aspect_ratio=[1,1,1])

Y=parametric_plot3d([(2+sin(s))*sin(t*f(s)),cos(s),(2+sin(s))*cos(t*f(s))],(s,0,2*pi),(t,-1,1))  --- to produce the part of the torus above the z=-2 plane
Y.show(aspect_ratio=[1,1,1])

This fails. Any hints?