https://file.aiquickdraw.com/mathbot/file/1726070226846p1mn6rul.jpg

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.

Consider the graph of the function f(x) below. How many of the following statements are true?

I. The domain of f(x) is all real numbers
II. f(x) has a removable discontinuity at x = -3
III. f(x) has a jump discontinuity at x = 0
IV. f(x) has an infinite discontinuity at x = 1

Explore how to analyze a function's discontinuities and domain by examining its graph. This problem focuses on removable, jump, and infinite discontinuities, providing insight into the function's behavior at specific points like x = -3, 0, and 1. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation