What is the smallest real number $x$ in the domain of the function $$\(g(x) = \sqrt{(x-3)^2-(x-8)^2}~?\)$$

Lightning
Jul 24, 2018

#1**+1 **

√ [ (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

CPhill
Jul 24, 2018