Final answer:
The function f(x) = (4x³ - 24x² + 32x) / (x² - 1) will have a slant (oblique) asymptote due to the numerator having a degree one higher than the denominator. To find the slant asymptote, one must perform polynomial long division or synthetic division, and the quotient (excluding any remainder) will provide the equation of the slant asymptote.
Step-by-step explanation:
To determine the horizontal or slant asymptote of the function f(x) = (4x³ - 24x² + 32x) / (x² - 1), we look at the degrees of the numerator and the denominator polynomials. Since the degree of the numerator is one higher than the degree of the denominator, we can expect a slant (oblique) asymptote rather than a horizontal asymptote. To find the equation of the slant asymptote, perform polynomial long division or synthetic division.
After dividing 4x³ - 24x² + 32x by x² - 1, the quotient, which will not include the remainder, gives the equation of the slant asymptote. In this case, the quotient will be a first degree polynomial in the form of ax + b, which represents our slant asymptote.