Final answer:
To find the horizontal and vertical asymptotes of the given function, we compare the degrees and coefficients of the highest degree terms. The horizontal asymptote is y = 2/3. The vertical asymptotes are x = -1 and x = 1/3.
Step-by-step explanation:
To find the horizontal and vertical asymptotes of the function y = ( 2x2+7) / (3x2-x-1), we need to analyze the behavior of the function as x approaches positive and negative infinity.
Horizontal asymptote: As x approaches positive or negative infinity, the degree of the numerator and denominator functions is the same (2), so the horizontal asymptote can be found by comparing the coefficients of the highest degree terms. In this case, the horizontal asymptote is y = 2/3.
Vertical asymptote: To find the vertical asymptote, we need to determine the values of x for which the denominator function is equal to zero. So, we solve the equation 3x2-x-1 = 0. The two solutions are x = -1 and x = 1/3. Therefore, the vertical asymptotes are x = -1 and x = 1/3.