https://file.aiquickdraw.com/mathbot/file/1725949636085uta7i5km.jpg

We are asked to analyze the new function \( g(x) = (f(x))^2 \), based on the given graph of \( y = f(x) \), and identify the discontinuities of \( g(x) \) in the interval \([-4, 4]\), classifying them as removable, jump, or other.

### Step-by-step analysis:
Let's inspect \( f(x) \) for any discontinuities and how these affect \( g(x) = (f(x))^2 \):

#### 1. **At \( x = -3 \)**:
- The graph shows a jump discontinuity at \( x = -3 \), where \( f(x) \) jumps from 2 (on the left) to 0 (on the right).
- For \( g(x) = (f(x))^2 \):
  - To the left of \( x = -3 \), \( g(x) = 2^2 = 4 \).
  - To the right of \( x = -3 \), \( g(x) = 0^2 = 0 \).
  - Thus, \( g(x) \) has a jump discontinuity at \( x = -3 \).

#### 2. **At \( x = -1 \)**:
- The graph of \( f(x) \) shows a point discontinuity at \( x = -1 \), with \( f(-1) \) defined as 1, while the limit from both sides seems to approach 2.
- For \( g(x) = (f(x))^2 \):
  - As \( x \to -1 \), \( g(x) \) approaches \( 2^2 = 4 \), but at \( x = -1 \), \( g(-1) = (1)^2 = 1 \).
  - This is a removable discontinuity for \( g(x) \), since the function could be redefined to make \( g(x) \) continuous at \( x = -1 \).

#### 3. **At \( x = 2 \)**:
- At \( x = 2 \), there is a jump discontinuity for \( f(x) \), where \( f(x) \) jumps from \( -2 \) to 0.
- For \( g(x) = (f(x))^2 \):
  - To the left of \( x = 2 \), \( g(x) = (-2)^2 = 4 \).
  - To the right of \( x = 2 \), \( g(x) = 0^2 = 0 \).
  - Thus, \( g(x) \) has a jump discontinuity at \( x = 2 \).

#### 4. **At \( x = 3 \)**:
- At \( x = 3 \), there is a point discontinuity for \( f(x) \), with \( f(x) \) undefined.
- For \( g(x) = (f(x))^2 \), since \( f(3) \) is undefined, \( g(3) \) is also undefined.
  - Therefore, \( g(x) \) has an **undefined discontinuity** at \( x = 3 \).

### Summary of Discontinuities for \( g(x) \):
- **At \( x = -3 \)**: Jump discontinuity.
- **At \( x = -1 \)**: Removable discontinuity.
- **At \( x = 2 \)**: Jump discontinuity.
- **At \( x = 3 \)**: Undefined discontinuity.

Would you like further details or clarifications? Below are some questions to deepen your understanding:

1. What happens to discontinuities in \( f(x) \) when squaring the function?
2. Can a removable discontinuity become a jump discontinuity after squaring the function?
3. How would the behavior of \( g(x) \) change if \( f(x) \) had different values at the discontinuities?
4. What types of limits are used to classify these discontinuities (left-hand, right-hand limits)?
5. How would the analysis change if the function were cubed instead of squared?

**Tip:** When squaring functions, pay attention to how zero crossings and discontinuities affect the final shape of the new function!

Consider the graph of y = f(x) presented here. Consider a new function g(x) = (f(x))^2. Find and classify (as removable, jump, or other) the discontinuities of g(x) = (f(x))^2 on the interval [-4, 4]. Justify your answer(s) with appropriate limit computations.

Explore the classification of discontinuities in the function g(x) = (f(x))^2, including jump and removable discontinuities. Detailed step-by-step analysis of limits and function behavior on the interval [-4, 4]. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation