Determining the Equations of Rational Functions

Question

determine the equation of a rational function, in factored form, with the following characteristics:

1. as x →∞ , y → -0.5
2. domain: x ∈ (-∞, -2) or (-2, 3) or (3, ∞)
3. zeros; -3 (order 1) and 2 (order 2)

Answer 1

Using the information that the zeros are -3 (order 1) and 2 (order 2), the numerator must be a(x+3)(x-2)²

Using the information that the domain is (-∞, -2) or (-2, 3) or (3, ∞) indicates the denominator could be "(x+2)(x-3)" however, the fact that the horizontal asymptote is -0.5 means that the numerator and denominator must be the same order (degree). Since the numerator is a cubic, the denominator must also be a cubic. I'll select a duplicate of the (x+2) making the denominator "(x+2)²(x-3)". Additionally, the "a" value in the numerator must be -0.5 so: