Final answer:
The function f(x) = (3x+1)/(x-2) has a vertical asymptote at x = 2 and a horizontal asymptote at y = 3, found by setting the denominator equal to zero and comparing the leading coefficients of the numerator and denominator, respectively.
Step-by-step explanation:
To find the asymptotes of the function f(x) = (3x+1)/(x-2), we need to identify two types: vertical and horizontal asymptotes.
Vertical asymptotes occur where the function is undefined, which is at values of x that make the denominator zero. In this case, setting x-2 = 0 gives us x = 2. So, there is a vertical asymptote at x = 2.
For the horizontal asymptote, we look at the behavior of the function as x approaches infinity. Since the degrees of the numerator and denominator are the same, the horizontal asymptote is determined by the ratio of the leading coefficients. The horizontal asymptote is y = 3 because the coefficient of x in the numerator is 3 and the coefficient of x in the denominator is 1.