Final answer:
A rational function f(x) with the given conditions can be represented as f(x) = 6(x + 1)(x - 2) / (x + 3)(x - 1)(x - 2), considering the vertical and horizontal asymptotes, x-intercept, hole, and function value at f(0).
Step-by-step explanation:
To find an equation of a rational function f(x) that satisfies the given conditions, we need to take into account the vertical asymptotes, horizontal asymptote, x-intercept, hole, and the function value at f(0).
The vertical asymptotes at x = -3 and x = 1 indicate that the factors (x + 3) and (x - 1) are in the denominator. Since there's an x-intercept at x = -1, the factor (x + 1) should be in the numerator. The hole at x = 2 suggests that (x - 2) should be a factor in both the numerator and denominator, canceling out. The horizontal asymptote at y = 0 denotes that the degree of the numerator should be less than the degree of the denominator. Finally, to ensure f(0) = -2, we will multiply the fraction by a constant to adjust the value.
Considering all the above, the function can be written as:
f(x) = k * (x + 1)(x - 2) / (x + 3)(x - 1)(x - 2)
To find k, we plug in x = 0:
f(0) = k * (0 + 1)(0 - 2) / (0 + 3)(0 - 1)(0 - 2)
f(0) = -2k / 6
-2 = -2k / 6
k = 6
Thus the final equation of the function is:
f(x) = 6(x + 1)(x - 2) / (x + 3)(x - 1)(x - 2).