Final answer:
The function f(x) has a horizontal asymptote at y = 1/2 and two vertical asymptotes at x = -2 and x = 2.
Step-by-step explanation:
To find the asymptotes of a rational function like f(x) = \frac{x^2-7x-8}{2x^2-8}, we look at the behavior of the function as x approaches certain values. Horizontal asymptotes are determined by the degrees of the numerator and denominator polynomials. If the degrees are equal, the horizontal asymptote is y = the ratio of the leading coefficients.
For the function f(x), the degree of both the numerator and the denominator is 2. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 2. Therefore, our horizontal asymptote is y = \frac{1}{2}.
Vertical asymptotes occur where the function is undefined, typically where the denominator is zero. To find these, we need to solve the equation 2x^2 - 8 = 0. Factoring out a 2, we get x^2 - 4 = 0, which further factors into (x + 2)(x - 2) = 0. This gives us two vertical asymptotes at x = -2 and x = 2.
Thus, the function f(x) has one horizontal asymptote at y = \frac{1}{2} and two vertical asymptotes at x = -2 and x = 2.