Final answer:
The vertical asymptote of the function f(x) is at x = -1, and since the degree of the numerator is greater than the denominator, the function does not have a horizontal asymptote in the form y = b, but resembles the growth of y = 3x. The correct choice is b) Vertical: x = -1, Horizontal: y = 3.
Step-by-step explanation:
The function in question is f(x) = \(\frac{3x^2 - 2x - 4}{x + 1}\). To find its vertical and horizontal asymptotes, we examine the behavior of the function as x approaches certain critical values and as x goes to infinity.
The vertical asymptote occurs where the denominator equals zero and the function is undefined. Setting the denominator to zero gives us x + 1 = 0, which solves to x = -1. Hence, there is a vertical asymptote at x = -1.
The horizontal asymptote is determined by the behavior of the function as x approaches infinity. Since the degree of the polynomial in the numerator (2) is greater than the degree of the polynomial in the denominator (1), we know that the function grows without bound and there is no horizontal asymptote of the form y = b. However, considering the leading coefficients, as x becomes very large, the dominant term 3x^2 will make the behavior of the function resemble that of y = 3x, but since it's not a constant value, we interpret this as the function not having a horizontal asymptote.
Therefore, the correct answer is b) Vertical: x = -1, Horizontal: y = 3.