The image shows a graph with vertical asymptotes and a set of multiple-choice options for a possible function.

Analyzing the Graph:

• Vertical Asymptotes: The graph has vertical asymptotes, suggesting the function has undefined points at certain values of $x$.
• Horizontal Behavior: The graph seems to have a horizontal asymptote as well, which can indicate the end behavior of the function.

Given the options, we need to identify which function corresponds to the graph:

Options:

1. $\frac{x}{x^2 - 1}$
2. $\text{non}$ (This option seems like a "None of the above" or "No function")
3. $\frac{x}{x^2 + 1}$
4. $\frac{x}{e^x - 1}$

Steps to Solve:

1. First Option:

   • $f(x) = \frac{x}{x^2 - 1}$
   • This function has vertical asymptotes where the denominator equals zero, i.e., $x^2 - 1 = 0$ giving $x = \pm1$. The graph has asymptotes at $x = 1$ and $x = -1$, which matches the graph in the image.
   • The graph has no asymptotes at $x = 0$ and should exhibit behavior where the function approaches zero as $x$ moves far from the origin, which fits this function.

2. Second Option:

   • "non" (None of the above) isn't applicable since we have possible correct options.

3. Third Option:

   • $f(x) = \frac{x}{x^2 + 1}$
   • This function is defined for all $x$ because the denominator $x^2 + 1$ never equals zero. This doesn't match the graph with vertical asymptotes.

4. Fourth Option:

   • $f(x) = \frac{x}{e^x - 1}$
   • This function has a vertical asymptote at $x = 0$ (because $e^0 - 1 = 0$), but the graph shows asymptotes at $x = 1$ and $x = -1$, not $x = 0$.

Conclusion:

The correct function that matches the graph is Option a: $\frac{x}{x^2 - 1}$.

Would you like more details on this problem, or do you have any questions?

Related Questions:

1. How do vertical asymptotes affect the behavior of rational functions?
2. What is the general form of a rational function, and how do you identify its asymptotes?
3. Can a function have more than one horizontal asymptote?
4. What is the significance of holes in the graph of a rational function?
5. How do you determine the end behavior of a rational function?

Tip:

Always check both the numerator and denominator when analyzing rational functions to identify vertical and horizontal asymptotes properly.