Final answer:
The domain of the function f(x) = √x+3(x+8)(x-2) is all real numbers such that x is greater than or equal to -3, but x cannot be 2, to avoid a zero in the denominator.
Step-by-step explanation:
The domain of a function includes all the values for which the function is defined. For the given function f(x) = √ x+3(x+8)(x-2), we need to look for the values of x that would not cause an issue in the calculation, particularly values that do not result in a negative number under the square root, since the square root of a negative number is not defined in the set of real numbers. To ensure the expression under the square root is non-negative, we set up the inequality x + 3 ≥ 0, which simplifies to x ≥ -3. Thus, the domain of the function must have x values greater than or equal to -3. However, since there is also a factor of (x-2) in the function, we have another point to consider: x cannot be equal to 2, as this would result in the function having a zero in the denominator, which is undefined. Therefore, the correct answer is that the domain is (x ≥ -3, x ≠ 2).