What is the smallest real number $x$ in the domain of the function
$$g(x) = \sqrt{(x-3)^2-(x-8)^2}~?$$
Simplifying under the radical we have
(x -3)^2 - (x -8)^2 =
x^2 - 6x + 9 - ( x^2 - 16x + 64) =
x^2 -6x + 9 -x^2 + 16x - 64
10x - 55 since we can't have a negative under the radical, this must be ≥ 0
So
10x - 55 ≥ 0
10 x ≥ 55
x ≥ 55/10
x ≥ 11/2
+193 
