Let's analyze the graph and verify each of the four given statements:

1. The domain of $f(x)$ is all real numbers.

   • From the graph, there are vertical asymptotes and open circles at certain points, which suggest that the function is not defined at these specific values.
   • Observing the graph, there seems to be a discontinuity at $x = 1$ and $x = 3$. Therefore, the domain of $f(x)$ excludes these points, meaning the domain is not all real numbers.
   • This statement is false.

2. $f(x)$ has a removable discontinuity at $x = -3$.

   • At $x = -3$, there is an open circle on the graph (indicating a discontinuity) and a filled point below the open circle, suggesting that the function could be redefined to fill this gap.
   • This type of discontinuity is called a removable discontinuity.
   • This statement is true.

3. $f(x)$ has a jump discontinuity at $x = 0$.

   • At $x = 0$, the graph jumps from one value to another, with a gap between them. This is characteristic of a jump discontinuity.
   • This statement is true.

4. $f(x)$ has an infinite discontinuity at $x = 1$.

   • At $x = 1$, the graph appears to approach infinity, as indicated by the vertical asymptote.
   • This is an infinite discontinuity.
   • This statement is true.

Conclusion:

• Statements II, III, and IV are true, while statement I is false. Therefore, 3 out of 4 statements are true.

Would you like further clarification on any part of this problem?

Here are some related questions:

1. What is the formal definition of a removable discontinuity?
2. How does the domain of a function change when there are vertical asymptotes?
3. Can a function have multiple types of discontinuities, such as both jump and infinite discontinuities?
4. How can you redefine a function at a point of removable discontinuity?
5. What are the implications of discontinuities in real-world applications of functions?

Tip: When identifying the domain of a function from its graph, look for vertical asymptotes and open circles—these indicate where the function is not defined.