+1242 What is the smallest real number $x$ in the domain of the function $$\(g(x) = \sqrt{(x-3)^2-(x-8)^2}~?\)$$
+128474 √ [ (x - 3)^2 - (x - 8)^2 ]
As small as the expression under the radical can be is 0
So....setting it to 0, we have
(x - 3)^2 - (x - 8)^2 = 0
Factor as a difference of squares
[ (x - 3) + ( x - 8)] [ (x - 3) - ( x - 8) ] = 0
Settting the second factor to 0 and solving for x will not produce a solution
Setting the first factor to 0 and solving for x produces
( x - 3) + ( x - 8) = 0
2x - 11 = 0
x = 11/2
So....the smallest x value in the domain is x = 11/2
Here's a graph that confirms this : https://www.desmos.com/calculator/qegganaegs
