Question

Find the equation of the vertical asymptote and the equation of the slant asymptote of the rational function f(x) = (8x^2 - 10x - 1) / (2x - 3).

Options:
a. Vertical Asymptote: x = 3/2, Slant Asymptote: y = 4x - 5
b. Vertical Asymptote: x = 3/2, Slant Asymptote: y = 4x + 1
c. Vertical Asymptote: x = 3, Slant Asymptote: y = 4x - 5
d. Vertical Asymptote: x = 3, Slant Asymptote: y = 4x + 1

Answer 1

The equation of the vertical asymptote is x = 3/2, and the equation of the slant asymptote is y = 4x - 5.

Step-by-step explanation:

To find the equation of the vertical asymptote and the equation of the slant asymptote of the rational function f(x) = (8x^2 - 10x - 1) / (2x - 3), we need to determine the behavior of the function as x approaches positive and negative infinity.

1. Vertical Asymptote: We set the denominator equal to zero and solve for x: 2x - 3 = 0 ➡ x = 3/2. Therefore, the vertical asymptote is x = 3/2.

2. Slant Asymptote: If the degree of the numerator is larger than the degree of the denominator by exactly one, there is a slant asymptote. In this case, the degree of the numerator is 2, and the degree of the denominator is 1. We can use long division to divide the numerator by the denominator and find that the quotient is 4x - 5. Therefore, the equation of the slant asymptote is y = 4x - 5.