How do I determine the asymptotes and end behavior for the function f(x) = (3x^3 - 2x + 1) / (2x^2 - x - 3)?
Answer 1
Devin Smith
To determine the asymptotes and end behavior for the function f(x) = (3x^3 – 2x + 1) / (2x^2 – x – 3), we analyze both vertical and horizontal asymptotes. Vertical asymptotes occur where the denominator equals zero, i.e., 2x^2 – x – 3 = 0. Solving this quadratic yields x = 3/2 and x = -1. Horizontal asymptotes depend on the degrees of the numerator and denominator. Here, the degree of the numerator (3) is greater than the degree of the denominator (2), indicating no horizontal asymptote. Instead, we find an oblique asymptote by performing polynomial long division, resulting in y = (3/2)x. The end behavior of the function follows the leading term of the polynomial division, meaning f(x) behaves like (3/2)x as x approaches ±∞.
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