Chapter 1.2, Problem 66E

Solution

To find: match the function with the corresponding graph by considering end behavior and asymptotes.

  

The correct match is Graph(d).

Given information: 

The function $y=\frac{x^3+2}{2x^2+1}$ .

Calculation:

Consider the function

  $y=\frac{x^3+2}{2x^2+1}$

The denominator is $2x^2+1$ . There are no real $x$ for which the denominator is zero.

Thus, the function has no vertical asymptote.

The function is anticipated to continue growing to positive infinity or negative infinity since the numerator is of a higher degree than the denominator.

Also note that the $y-$ intercept is $y=2$ .

This closely matches the behavior of graph(d).

Hence, the match graph is graph(d).