To find the values of $x$ in the open interval $-9 < x < 9$ where the function $f(x)$ has infinite discontinuities, we need to analyze the graph for points where:

1. The function tends toward $\pm \infty$ on one or both sides of a specific $x$-value.
2. The graph shows vertical asymptotes, indicating that the function is undefined at that $x$-value.

Analysis:

From the graph:

1. There is an infinite discontinuity at $x = -4$, as the graph approaches $-\infty$ and $\infty$ on either side of this point.
2. There is another infinite discontinuity at $x = 1$, as the graph shoots to $\infty$ and $-\infty$ on either side.

Conclusion:

The function $f(x)$ has infinite discontinuities at: $x = -4 \quad \text{and} \quad x = 1$

Would you like a detailed explanation of how infinite discontinuities are determined or how they differ from removable discontinuities?

Here are 5 questions to expand on this:

1. What is the difference between infinite discontinuities and removable discontinuities?
2. How can vertical asymptotes be identified from a graph?
3. Can you describe the behavior of the function near $x = -4$ and $x = 1$?
4. What are other types of discontinuities in a function?
5. How can these discontinuities be expressed mathematically in the function's definition?

Tip: To confirm infinite discontinuities analytically, you can use limits to check if $\lim_{x \to c^-} f(x)$ or $\lim_{x \to c^+} f(x)$ tends toward infinity or negative infinity.