Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Get quick and reliable solutions to your questions from knowledgeable professionals on our comprehensive Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To determine all the discontinuities of the given function \( f(x) \), we need to analyze the behavior of the function at the boundaries where the definition changes. The critical points to check are \( x = -4 \), \( x = -2 \), and \( x = 4 \).
1. At \( x = -4 \):
- For \( x < -4 \), \( f(x) = 4 \).
- For \( -4 \leq x \leq -2 \), \( f(x) = (x + 2)^2 \).
To check if \( f(x) \) is continuous at \( x = -4 \), we need to compare the limits approaching from both sides:
- As \( x \) approaches \(-4\) from the right:
[tex]\[ \lim_{x \to -4^+} f(x) = (-4 + 2)^2 = 4 \][/tex]
- As \( x \) approaches \(-4\) from the left:
[tex]\[ \lim_{x \to -4^-} f(x) = 4 \][/tex]
The values match, thus \( f(x) \) is continuous at \( x = -4 \).
2. At \( x = -2 \):
- For \( -4 \leq x \leq -2 \), \( f(x) = (x + 2)^2 \).
- For \( -2 < x < 4 \), \( f(x) = -\frac{1}{2}x + 1 \).
To check if \( f(x) \) is continuous at \( x = -2 \), we need to compare the limits approaching from both sides:
- As \( x \) approaches \(-2\) from the right:
[tex]\[ \lim_{x \to -2^+} f(x) = -\frac{1}{2}(-2) + 1 = 2 \][/tex]
- As \( x \) approaches \(-2\) from the left:
[tex]\[ \lim_{x \to -2^-} f(x) = (-2 + 2)^2 = 0 \][/tex]
Since \( 0 \neq 2 \), \( f(x) \) has a jump discontinuity at \( x = -2 \).
3. At \( x = 4 \):
- For \( -2 < x < 4 \), \( f(x) = -\frac{1}{2}x + 1 \).
- For \( x > 4 \), \( f(x) = -1 \).
To check if \( f(x) \) is continuous at \( x = 4 \), we need to compare the limits approaching from both sides:
- As \( x \) approaches \( 4 \) from the right:
[tex]\[ \lim_{x \to 4^+} f(x) = -1 \][/tex]
- As \( x \) approaches \( 4 \) from the left:
[tex]\[ \lim_{x \to 4^-} f(x) = -\frac{1}{2}(4) + 1 = -1 \][/tex]
The values match, thus \( f(x) \) is continuous at \( x = 4 \).
Considering all necessary points, the function \( f(x) \) has a jump discontinuity at \( x = -2 \) and no discontinuities at \( x = -4 \) and \( x = 4 \).
Thus, the most accurate list of discontinuities for the function \( f(x) \) would be:
- Jump discontinuity at \( x = -2 \).
Therefore, the correct option is:
```
jump discontinuity at x=-2
```
1. At \( x = -4 \):
- For \( x < -4 \), \( f(x) = 4 \).
- For \( -4 \leq x \leq -2 \), \( f(x) = (x + 2)^2 \).
To check if \( f(x) \) is continuous at \( x = -4 \), we need to compare the limits approaching from both sides:
- As \( x \) approaches \(-4\) from the right:
[tex]\[ \lim_{x \to -4^+} f(x) = (-4 + 2)^2 = 4 \][/tex]
- As \( x \) approaches \(-4\) from the left:
[tex]\[ \lim_{x \to -4^-} f(x) = 4 \][/tex]
The values match, thus \( f(x) \) is continuous at \( x = -4 \).
2. At \( x = -2 \):
- For \( -4 \leq x \leq -2 \), \( f(x) = (x + 2)^2 \).
- For \( -2 < x < 4 \), \( f(x) = -\frac{1}{2}x + 1 \).
To check if \( f(x) \) is continuous at \( x = -2 \), we need to compare the limits approaching from both sides:
- As \( x \) approaches \(-2\) from the right:
[tex]\[ \lim_{x \to -2^+} f(x) = -\frac{1}{2}(-2) + 1 = 2 \][/tex]
- As \( x \) approaches \(-2\) from the left:
[tex]\[ \lim_{x \to -2^-} f(x) = (-2 + 2)^2 = 0 \][/tex]
Since \( 0 \neq 2 \), \( f(x) \) has a jump discontinuity at \( x = -2 \).
3. At \( x = 4 \):
- For \( -2 < x < 4 \), \( f(x) = -\frac{1}{2}x + 1 \).
- For \( x > 4 \), \( f(x) = -1 \).
To check if \( f(x) \) is continuous at \( x = 4 \), we need to compare the limits approaching from both sides:
- As \( x \) approaches \( 4 \) from the right:
[tex]\[ \lim_{x \to 4^+} f(x) = -1 \][/tex]
- As \( x \) approaches \( 4 \) from the left:
[tex]\[ \lim_{x \to 4^-} f(x) = -\frac{1}{2}(4) + 1 = -1 \][/tex]
The values match, thus \( f(x) \) is continuous at \( x = 4 \).
Considering all necessary points, the function \( f(x) \) has a jump discontinuity at \( x = -2 \) and no discontinuities at \( x = -4 \) and \( x = 4 \).
Thus, the most accurate list of discontinuities for the function \( f(x) \) would be:
- Jump discontinuity at \( x = -2 \).
Therefore, the correct option is:
```
jump discontinuity at x=-2
```
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
