Final answer:
The domain of the function f/g(x) is all real numbers except x = 4 and x = -4, because these are the values for which the denominator g(x) is zero. The domain in interval notation is (-∞, -4) ∪ (-4, 4) ∪ (4, +∞).
Step-by-step explanation:
To find the domain of the function f/g(x) given f(x)=x^2+2x-8 and g(x)=x^2-16, we need to determine the set of all possible x-values for which this function is defined. The function f/g(x) is not defined when the denominator g(x) is equal to zero, because division by zero is undefined in mathematics.
First, set the denominator equal to zero and solve for x:
- g(x) = x^2 - 16 = 0
- x^2 = 16
- x = ±4
Therefore, the denominator is equal to zero when x = 4 or x = -4. These values should be excluded from the domain.
Thus, the domain of the function f/g(x) is all real numbers except x = 4 and x = -4. In interval notation, the domain is (-∞, -4) ∪ (-4, 4) ∪ (4, +∞).