From the image provided, the function is given as:

$f(x) = \frac{3x^2 + 8}{2x - 1}$

Steps to Analyze and Solve:

1. Determine Vertical Asymptotes:

Vertical asymptotes occur when the denominator equals zero. Solve: $2x - 1 = 0 \implies x = \frac{1}{2}$ Thus, the vertical asymptote is at $x = \frac{1}{2}$.

2. Check for Holes:

Holes occur when there is a common factor in the numerator and denominator, which cancels out. In this case, there is no such factor, so there are no holes.

3. Find the Horizontal Asymptote:

The degrees of the numerator ($n = 2$) and the denominator ($m = 1$) are different. Since $n > m$, there is no horizontal asymptote. Instead, we look for an oblique asymptote.

4. Determine the Oblique Asymptote:

Perform polynomial long division: $\frac{3x^2 + 8}{2x - 1}$ Divide $3x^2$ by $2x$ to get $\frac{3}{2}x$. Multiply and subtract, and continue until you reach the remainder. The oblique asymptote is given by the quotient: $y = \frac{3}{2}x + \frac{11}{4}$

5. Domain and Range:

• The domain is all real numbers except where the denominator is zero: $(-\infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty)$.
• The range is all real numbers except at $y = \frac{3}{2}x + \frac{11}{4}$, as $f(x)$ approaches this line asymptotically.

6. Determine if the Graph Crosses the Horizontal/Oblique Asymptote:

To test if it crosses the oblique asymptote, set $f(x) = \frac{3}{2}x + \frac{11}{4}$ and solve for $x$. Substituting this back into $f(x)$ reveals if there are solutions.

Would you like me to calculate points for plotting or solve specific parts further? Here are a few questions to deepen your understanding:

1. How does the oblique asymptote affect the behavior of the graph for large values of $x$?
2. Can you confirm the behavior of $f(x)$ near $x = \frac{1}{2}$?
3. Would you like detailed steps for the long division used to find the oblique asymptote?
4. Do you need the graph plotted with specific points?
5. Would you like help completing the table of values for better visualization?

Tip: For functions with higher-degree numerators, always check for oblique or other asymptotes carefully!