For the pair of functions, find the indicated domain. 12) For \( f(x)=\sqrt{x-6} \) and \( g(x)=\frac{1}{x-8} \) Find the domain of \( f g \).

To find the domain of the product of the functions \( f(x) = \sqrt{x - 6} \) and \( g(x) = \frac{1}{x - 8} \), we need to determine the individual domains of \( f(x) \) and \( g(x) \) and then find the intersection of these domains. 1. **Domain of \( f(x) = \sqrt{x - 6} \)**: - The expression under the square root must be non-negative: \[ x - 6 \geq 0 \implies x \geq 6 \] - Therefore, the domain of \( f(x) \) is: \[ [6, \infty) \] 2. **Domain of \( g(x) = \frac{1}{x - 8} \)**: - The denominator cannot be zero: \[ x - 8 \neq 0 \implies x \neq 8 \] - Therefore, the domain of \( g(x) \) is: \[ (-\infty, 8) \cup (8, \infty) \] 3. **Finding the domain of \( f g \)**: - The product \( f g \) is defined where both \( f(x) \) and \( g(x) \) are defined. - We need to find the intersection of the domains: - The domain of \( f(x) \) is \( [6, \infty) \). - The domain of \( g(x) \) is \( (-\infty, 8) \cup (8, \infty) \). - The intersection of \( [6, \infty) \) and \( (-\infty, 8) \cup (8, \infty) \) is: - From \( [6, \infty) \), we consider the interval \( [6, 8) \) (since \( g(x) \) is not defined at \( x = 8 \)). - Therefore, the intersection is: \[ [6, 8) \] Thus, the domain of \( f g \) is: \[ \boxed{[6, 8)} \]