What is the smallest real number in the domain of the function g(x) = sqrt((x - 3)^2 - (x - 18)^2)?
+592 If we want a real number, then ${(x-3)^2-(x-18)^2}$ has to be greater than or equal to $0$, or $(x-18)^2$ has to be less than or equal to $(x-3)^2$
when $(x-3)^2=(x-18)^2$, x has the smallest possible value with real solutions. So we have:
$(x-3)^2=(x-18)^2=x^2+9-6x=x^2+324-36x$
$30x=315$
$x=10.5$
so if x is any less than $\boxed{10.5}$, then the equation $\sqrt{(x - 3)^2 - (x - 18)^2}$ would have no real solutions.
