Final answer:
To identify the asymptotes and zeros of the function f(x) = (7x^2 + 5x - 2) / (2x^2 - x - 3), we check for vertical and horizontal asymptotes and solve for the zeros of the function.
Step-by-step explanation:
To identify the asymptotes of the function f(x) = (7x^2 + 5x - 2) / (2x^2 - x - 3), we need to check the vertical asymptotes and the horizontal asymptote.
Vertical Asymptotes:
To find the vertical asymptotes, we set the denominator equal to zero and solve for x.
- 2x^2 - x - 3 = 0
- Using factoring or the quadratic formula, we find x = -1 and x = 3/2 as the values for the vertical asymptotes.
Horizontal Asymptote:
To find the horizontal asymptote, we compare the degrees of the numerator and the denominator. Since both have the same degree (2 in this case), the horizontal asymptote is determined by dividing the leading coefficients of both polynomials. In this case, the horizontal asymptote is y = 7/2.
The zeros of the function can be found by setting the numerator equal to zero and solving for x.
- 7x^2 + 5x - 2 = 0
- Using factoring or the quadratic formula, we find x = 1/7 and x = -2/7 as the zeros of the function.