https://file.aiquickdraw.com/mathbot/file/1724179582137zp5d96hs.jpg

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.

Explore the concept of vertical asymptotes in rational functions through a detailed analysis of a graph. Learn how to identify the correct function from multiple-choice options based on its asymptotic behavior. Suitable for advanced high school students studying calculus and functions.

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