Question 403290
The domain of f(2(x-3))can be determined in all generality, not only using discrete values.
Let g(x) = 2(x-3)  = 2x - 6.
Then f(2(x-3)) = (f o g)(x) = f(g(x)). Let the domain of g be {{{D[g]}}}, its range {{{R[g]}}}.  Similarly let the domain of f be {{{D[f]}}}, its range {{{R[f]}}}.
Then to find the domain of f o g we must find all x values in {{{D[g]}}} (which is the set of all real numbers), that are the pullback of {{{R[g]}}}intersection{{{D[f]}}}.  Since the range of g is represented by 2(x-3), and the domain of f is the closed interval [1,6], we must then have 
{{{1 <= 2(x - 3) <= 6}}}
<==> {{{1/2 <= x 
- 3 <= 3}}}
<==> {{{7/2 <= x <= 6}}}.
Thus the domain of (f o g)(x) = f(2(x-3)) is the closed interval [7/2, 6].