Determine domain and range of linear equation?

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Though it is not created for that purpose, you can use interval arithmetics:

sage: f(x) = abs(x) - 4
sage: a = RIF(-infinity,infinity) ; a
[-infinity .. +infinity]
sage: a.parent()
Real Interval Field with 53 bits of precision
sage: f(a)
[-4.0000000000000000 .. +infinity]

Here you get the exact result, but note that in general, you will only get an interval that contains the image of the function.

Whoops! Stupid answer deleted. :-/