Math Problem Statement
Solution
We are asked to analyze the new function , based on the given graph of , and identify the discontinuities of in the interval , classifying them as removable, jump, or other.
Step-by-step analysis:
Let's inspect for any discontinuities and how these affect :
1. At :
- The graph shows a jump discontinuity at , where jumps from 2 (on the left) to 0 (on the right).
- For :
- To the left of , .
- To the right of , .
- Thus, has a jump discontinuity at .
2. At :
- The graph of shows a point discontinuity at , with defined as 1, while the limit from both sides seems to approach 2.
- For :
- As , approaches , but at , .
- This is a removable discontinuity for , since the function could be redefined to make continuous at .
3. At :
- At , there is a jump discontinuity for , where jumps from to 0.
- For :
- To the left of , .
- To the right of , .
- Thus, has a jump discontinuity at .
4. At :
- At , there is a point discontinuity for , with undefined.
- For , since is undefined, is also undefined.
- Therefore, has an undefined discontinuity at .
Summary of Discontinuities for :
- At : Jump discontinuity.
- At : Removable discontinuity.
- At : Jump discontinuity.
- At : Undefined discontinuity.
Would you like further details or clarifications? Below are some questions to deepen your understanding:
- What happens to discontinuities in when squaring the function?
- Can a removable discontinuity become a jump discontinuity after squaring the function?
- How would the behavior of change if had different values at the discontinuities?
- What types of limits are used to classify these discontinuities (left-hand, right-hand limits)?
- 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!
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Math Problem Analysis
Mathematical Concepts
Discontinuities
Functions
Limits
Piecewise Functions
Formulas
g(x) = (f(x))^2
Theorems
Discontinuity Classification
Limit Theorems
Suitable Grade Level
Grades 11-12
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