It can be useful to sometimes see the "overall" behavior of certain functions, in order to gain some insight about how the function behaves at extreme points or to see how many roots it has.
On 20/1/2003 I posted this question to the newsgroup sci.math. David W. Cantrell partially answered the question in 2007, with this post.
In essence we are looking for a transformation T that maps the graph of an arbitrary function into some finite interval, such as [-1,1]. If we call our function f(x), then translating David's Mathematica code to Maple, we get:
> f:=x->(some function of x);
> g:=x->x/(1-abs(x));
> ginv:=x->x/(1+abs(x));
> T:=x->ginv(f(g(x)));
> with(plots):
In other words, one such transform is T(x)=g-1(f(g(x))), with g(x)=x/(1-|x|).
Let's see some examples of how the transform T matches Maple's built in overall graphing:
> f:=x->x^3-x^2+1;
> plot(T(x),x=-1..1); plot(f(x),x=-infinity..infinity);
Not bad. The transform looks very close to what Maple uses for its x=-infinity..infinity graph. A second example, shows that this transform can fail however:
> f:=x->x^12-2^x;
> plot(T(x),x=-1..1); plot(f(x),x=-infinity..infinity);
Both T and Maple V's internal transform fail, because f has three roots, given by: {-12*W(log(2)/12)/log(2), -12*W(-log(2)/12)/log(2),-12*W(-1,-1/12*log(2))/log(2)}, where W is the Lambert W function. Maple V misses the third root given by the non-principal branch of W. Maple 9 graphs it correctly:
The fact that Maple 9 produces correct graphs, means that both the T transform and the internal transform Maple V uses for overall graphing are not handled correctly in version V, hence it's a bug.