Question
Analyze the graphs f(x)=|x|
to g(x)=3.2|x|
. Which statement correctly describes the domain and range from f(x)
to g(x)
?(1 point)
Responses
The domain stays the same, and the range changes to [3.2,∞).
The domain stays the same, and the range changes to [3.2,∞).
The domain changes to [3.2,∞).
The domain changes to [3.2,∞).
The domain and range stay the same.
The domain and range stay the same.
The range changes to [3.2,∞).
to g(x)=3.2|x|
. Which statement correctly describes the domain and range from f(x)
to g(x)
?(1 point)
Responses
The domain stays the same, and the range changes to [3.2,∞).
The domain stays the same, and the range changes to [3.2,∞).
The domain changes to [3.2,∞).
The domain changes to [3.2,∞).
The domain and range stay the same.
The domain and range stay the same.
The range changes to [3.2,∞).
Answers
GPT-4o mini
To analyze the functions \( f(x) = |x| \) and \( g(x) = 3.2|x| \), let's first determine their domains and ranges.
### Function \( f(x) = |x| \):
- **Domain**: The domain of \( f(x) \) is all real numbers, written as \( (-\infty, \infty) \).
- **Range**: The range of \( f(x) \) is all non-negative real numbers because the absolute value is always greater than or equal to zero, written as \( [0, \infty) \).
### Function \( g(x) = 3.2|x| \):
- **Domain**: The domain of \( g(x) \) is also all real numbers, written as \( (-\infty, \infty) \).
- **Range**: The range of \( g(x) \) is all non-negative numbers scaled by 3.2. Since \( |x| \) can take any value from \( 0 \) to \( \infty \), multiplying by 3.2 means the minimum value is \( 3.2 \times 0 = 0 \), and it can take any value up to \( \infty \). Therefore, the range of \( g(x) \) is also \( [0, \infty) \).
### Conclusion on Domain and Range:
- **Domain**: The domain for both \( f(x) \) and \( g(x) \) remains the same: \( (-\infty, \infty) \).
- **Range**: The range for \( f(x) \) is \( [0, \infty) \), and for \( g(x) \) it remains \( [0, \infty) \).
### Summary of Statements:
1. **The domain stays the same, and the range changes to [3.2,∞)**.
- Incorrect. The domain stays the same, but the range does not change to \( [3.2, \infty) \).
2. **The domain changes to [3.2,∞)**.
- Incorrect. The domain does not change; it's still \( (-\infty, \infty) \).
3. **The domain and range stay the same.**
- Correct. The domain remains \( (-\infty, \infty) \) and the range remains \( [0, \infty) \).
4. **The range changes to [3.2,∞)**.
- Incorrect. The range does not change to \( [3.2, \infty) \).
The correct choice is: **The domain and range stay the same.**
### Function \( f(x) = |x| \):
- **Domain**: The domain of \( f(x) \) is all real numbers, written as \( (-\infty, \infty) \).
- **Range**: The range of \( f(x) \) is all non-negative real numbers because the absolute value is always greater than or equal to zero, written as \( [0, \infty) \).
### Function \( g(x) = 3.2|x| \):
- **Domain**: The domain of \( g(x) \) is also all real numbers, written as \( (-\infty, \infty) \).
- **Range**: The range of \( g(x) \) is all non-negative numbers scaled by 3.2. Since \( |x| \) can take any value from \( 0 \) to \( \infty \), multiplying by 3.2 means the minimum value is \( 3.2 \times 0 = 0 \), and it can take any value up to \( \infty \). Therefore, the range of \( g(x) \) is also \( [0, \infty) \).
### Conclusion on Domain and Range:
- **Domain**: The domain for both \( f(x) \) and \( g(x) \) remains the same: \( (-\infty, \infty) \).
- **Range**: The range for \( f(x) \) is \( [0, \infty) \), and for \( g(x) \) it remains \( [0, \infty) \).
### Summary of Statements:
1. **The domain stays the same, and the range changes to [3.2,∞)**.
- Incorrect. The domain stays the same, but the range does not change to \( [3.2, \infty) \).
2. **The domain changes to [3.2,∞)**.
- Incorrect. The domain does not change; it's still \( (-\infty, \infty) \).
3. **The domain and range stay the same.**
- Correct. The domain remains \( (-\infty, \infty) \) and the range remains \( [0, \infty) \).
4. **The range changes to [3.2,∞)**.
- Incorrect. The range does not change to \( [3.2, \infty) \).
The correct choice is: **The domain and range stay the same.**