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