Final answer:
The equation of the rational function is f(x) = 0(x+4)(x-1)/(x-2) = 0.
Step-by-step explanation:
To find the equation of the rational function with a vertical asymptote at x=2, a horizontal asymptote at y=3, and horizontal intercepts at (-4,0) and (1,0), we can start by considering the asymptotes. The vertical asymptote occurs when the denominator of the rational function is equal to zero. Since the vertical asymptote is at x=2, the denominator must be (x-2). The equation of the rational function can then be written as f(x) = A(x+4)(x-1)/(x-2), where A is a constant.
The horizontal asymptote occurs when the degree of the numerator is less than or equal to the degree of the denominator. Since the degrees of the numerator and denominator are both 1, the horizontal asymptote is y=3.
Next, we can use the given horizontal intercepts to determine the value of A. When x=-4, the function f(x) becomes 0, so we can substitute these values into the equation and solve for A. Plugging in x=-4 and f(x)=0, we get 0 = A(-4+4)(-4-1)/(-4-2). Simplifying this equation gives us A = 0.
Therefore, the equation of the rational function is f(x) = 0(x+4)(x-1)/(x-2) = 0.