What is the domain of the function \( g(x)=\sqrt{\frac{x-7}{x+9}} \) ? The domain is \( \square \). (Type your answer in interval notation. Simplify your answer. Use integers or fractions for any numbers in the expression.)

To find the domain of g(x) = √((x - 7)/(x + 9)), we need the expression inside the square root to be nonnegative and the denominator to be nonzero. Step 1: Set the radicand nonnegative:   (x - 7)/(x + 9) ≥ 0 Step 2: Identify where the numerator or denominator equals zero:   Numerator: x - 7 = 0 ⟹ x = 7   Denominator: x + 9 = 0 ⟹ x = -9 (this value must be excluded) Step 3: Analyze the sign of (x - 7)/(x + 9) using a sign chart. - For x < -9:   Pick a test value (e.g., x = -10)   Numerator: -10 - 7 = -17 (negative)   Denominator: -10 + 9 = -1 (negative)   The fraction is positive (negative/negative > 0). - For -9 < x < 7:   Pick a test value (e.g., x = 0)   Numerator: 0 - 7 = -7 (negative)   Denominator: 0 + 9 = 9 (positive)   The fraction is negative (negative/positive < 0). - For x > 7:   Pick a test value (e.g., x = 8)   Numerator: 8 - 7 = 1 (positive)   Denominator: 8 + 9 = 17 (positive)   The fraction is positive. Step 4: Include endpoints where the inequality holds:   At x = 7, the numerator is 0, making the fraction 0, which is allowed.   x = -9 is excluded because it makes the denominator zero. Thus, the function is defined for x in (–∞, –9) (where the expression is positive) and for x in [7, ∞). The domain in interval notation is: (–∞, –9) ∪ [7, ∞).