Final answer:
The student is finding the combinations of two square root functions and their domains, which results in a domain of -1 ≤ x ≤ 4 for most operations, and -1 < x ≤ 4 specifically for the quotient f/g.
Step-by-step explanation:
The student is working with two functions: f(x) = √(16 - x²), defined for real numbers where the expression under the square root is non-negative, and g(x) = √(x + 1), defined where x + 1 is non-negative. The task is to find the sum (f + g), difference (f - g), product (fg), and quotient (f/g) of these functions and determine the domains where these operations are defined.
The domain for f(x) is defined by the inequality 16 - x² ≥ 0, which gives -4 ≤ x ≤ 4. The domain for g(x) is x ≥ -1. When these two domains are combined, the common domain for all four operations, based on the constraints of both f(x) and g(x), is -1 ≤ x ≤ 4.
However, for the quotient f/g, we additionally need to ensure that g(x) ≠ 0. Since g(x) is always non-negative for x ≥ -1, the only additional restriction is that g(x) ≠ 0, so x cannot be -1. Hence, the domain for f/g is -1 < x ≤ 4.