Final answer:
The rational function that meets all the provided characteristics, including x-intercepts at -4 and 2, vertical asymptotes at x = 1 and x = -1, and a horizontal asymptote at y = -3, is represented by Option C: f(x) = (x + 4)(x - 2) / [(x + 1)(x - 1)].
Step-by-step explanation:
The equation that describes a rational function with the given characteristics must have factors in the numerator that correspond to the x-intercepts and factors in the denominator that correspond to the vertical asymptotes. Additionally, the degree of the polynomial in the numerator and the denominator helps determine the horizontal asymptote. Since the horizontal asymptote is at y = -3, this suggests that the degrees of the numerator and the denominator polynomials are the same, and the leading coefficients must result in -3 when the numerator is divided by the denominator.
- x-intercepts at -4 and 2 suggest factors of (x + 4) and (x - 2) in the numerator.
- Vertical asymptotes at x = 1 and x = -1 suggest factors of (x - 1) and (x + 1) in the denominator.
With these considerations, Option C f(x) = (x + 4)(x - 2) / [(x + 1)(x - 1)] is the correct equation. Both the x-intercepts (-4, 2) and the vertical asymptotes (x = 1 and x = -1) are correctly placed in the equation. The same highest degree in both the numerator and denominator with no leading coefficient reflects the horizontal asymptote at y = -3.