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# Domain and Range!

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

Jul 24, 2018

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√ [ (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

Jul 24, 2018