Final answer:
The equations of the asymptotes for the function g(x) = (2x² + 1) / (x + 3) are x = -3 (vertical asymptote) and y = 2x (oblique asymptote).
Step-by-step explanation:
To determine the equations of the asymptotes for the function g(x) = (2x² + 1) / (x + 3), we need to consider both vertical and horizontal asymptotes. A vertical asymptote occurs where the denominator is zero and the function is undefined. In this case, the vertical asymptote is x = -3 since if x equals -3, the denominator becomes zero.
For the horizontal asymptote, since the degree of the numerator is one higher than the degree of the denominator, there is no horizontal asymptote. However, there is an oblique (slant) asymptote because the degrees of the numerator and denominator differ by one. We can find the equation of this oblique asymptote by dividing 2x² by x, which gives us an oblique asymptote at y = 2x.
Therefore, the equations of the asymptotes for g(x) are x = -3 and y = 2x.