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What is the smallest real number x in the domain of the function g(x) = sqrt(x^2 - (x - 15)^2)?

 May 23, 2021
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What is the smallest real number x in the domain of the function \(g(x) = \sqrt{(x^2 - (x - 15)^2)}\)?
 

Two reasons why a number might not be included in the domain is that it makes an expression in the denominator of a fraction equal to 0, or it makes the expression be the square root of a negative number. These are big no-nos.

 

Keeping this in mind, let's find what values of x don't work, so then we can find the ones (the smallest one in particular) that do. For x not to work, we would have to have:

 

\(\sqrt{x^{2}-(x-15)^{2}} < 0\).

 

Simplifying, we get:

 

\(-30x+225<0\)

\(30x<225\).

 

I'll let you finish!

 May 23, 2021

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