Question
Compare the given graphs of f(x)=2x−−√ and g(x)=2x−−√3 and determine which of the following is true.
(1 point)
Responses
They are both decreasing on their domains.
They are both decreasing on their domains.
Both graphs go through (−1,−1), (0,0), and (1,1).
Both graphs go through left parenthesis negative 1 comma negative 1 right parenthesis , left parenthesis 0 comma 0 right parenthesis , and left parenthesis 1 comma 1 right parenthesis .
The graphs have different domains.
The graphs have different domains.
When x>1, the function g(x)=2x−−√3 is greater than f(x)=2x−−√.
(1 point)
Responses
They are both decreasing on their domains.
They are both decreasing on their domains.
Both graphs go through (−1,−1), (0,0), and (1,1).
Both graphs go through left parenthesis negative 1 comma negative 1 right parenthesis , left parenthesis 0 comma 0 right parenthesis , and left parenthesis 1 comma 1 right parenthesis .
The graphs have different domains.
The graphs have different domains.
When x>1, the function g(x)=2x−−√3 is greater than f(x)=2x−−√.
Answers
GPT-4o mini
Let's analyze each of the given functions and the statements provided:
1. **Function Definitions**
- \( f(x) = \sqrt{2x} \)
- \( g(x) = \sqrt{2x} + 3 \)
2. **Domains**
- For \( f(x) = \sqrt{2x} \), the domain is \( x \geq 0 \) because the expression inside the square root must be non-negative.
- For \( g(x) = \sqrt{2x} + 3 \), the domain is also \( x \geq 0 \) for the same reason.
**Statement:** "The graphs have different domains."
**Conclusion:** This statement is **false** since both functions share the same domain \( [0, \infty) \).
3. **Behavior of the Functions**
- Both functions are increasing for \( x \geq 0 \). The derivative of both functions is positive where defined.
**Statement:** "They are both decreasing on their domains."
**Conclusion:** This statement is **false**; they are both increasing, not decreasing.
4. **Point Intersection**
- Evaluating at specific points:
- At \( x = -1 \): Both functions are not defined (since the domain is \( [0, \infty) \)).
- At \( x = 0 \):
- \( f(0) = \sqrt{0} = 0 \)
- \( g(0) = \sqrt{0} + 3 = 3 \)
- At \( x = 1 \):
- \( f(1) = \sqrt{2} \approx 1.414 \)
- \( g(1) = \sqrt{2} + 3 \approx 4.414 \)
**Statement:** "Both graphs go through (−1,−1), (0,0), and (1,1)."
**Conclusion:** This statement is **false** since none of the functions are defined at \( x = -1 \), and at \( x = 0 \) and \( x = 1 \), the values do not match those specified.
5. **Comparison for \( x > 1 \)**
- Since \( g(x) = \sqrt{2x} + 3 \) adds 3 to the output of \( f(x) = \sqrt{2x} \), it is clear that for all \( x \geq 0 \), \( g(x) \) is always greater than \( f(x) \).
- Therefore, when \( x > 1 \), it also holds true that \( g(x) > f(x) \).
**Statement:** "When x>1, the function g(x)=2x−−√3 is greater than f(x)=2x−−√."
**Conclusion:** This statement is **true**.
### Final Conclusion:
The correct statement among the provided options is:
- **When \( x > 1 \), the function \( g(x) = \sqrt{2x} + 3 \) is greater than \( f(x) = \sqrt{2x} \).**
1. **Function Definitions**
- \( f(x) = \sqrt{2x} \)
- \( g(x) = \sqrt{2x} + 3 \)
2. **Domains**
- For \( f(x) = \sqrt{2x} \), the domain is \( x \geq 0 \) because the expression inside the square root must be non-negative.
- For \( g(x) = \sqrt{2x} + 3 \), the domain is also \( x \geq 0 \) for the same reason.
**Statement:** "The graphs have different domains."
**Conclusion:** This statement is **false** since both functions share the same domain \( [0, \infty) \).
3. **Behavior of the Functions**
- Both functions are increasing for \( x \geq 0 \). The derivative of both functions is positive where defined.
**Statement:** "They are both decreasing on their domains."
**Conclusion:** This statement is **false**; they are both increasing, not decreasing.
4. **Point Intersection**
- Evaluating at specific points:
- At \( x = -1 \): Both functions are not defined (since the domain is \( [0, \infty) \)).
- At \( x = 0 \):
- \( f(0) = \sqrt{0} = 0 \)
- \( g(0) = \sqrt{0} + 3 = 3 \)
- At \( x = 1 \):
- \( f(1) = \sqrt{2} \approx 1.414 \)
- \( g(1) = \sqrt{2} + 3 \approx 4.414 \)
**Statement:** "Both graphs go through (−1,−1), (0,0), and (1,1)."
**Conclusion:** This statement is **false** since none of the functions are defined at \( x = -1 \), and at \( x = 0 \) and \( x = 1 \), the values do not match those specified.
5. **Comparison for \( x > 1 \)**
- Since \( g(x) = \sqrt{2x} + 3 \) adds 3 to the output of \( f(x) = \sqrt{2x} \), it is clear that for all \( x \geq 0 \), \( g(x) \) is always greater than \( f(x) \).
- Therefore, when \( x > 1 \), it also holds true that \( g(x) > f(x) \).
**Statement:** "When x>1, the function g(x)=2x−−√3 is greater than f(x)=2x−−√."
**Conclusion:** This statement is **true**.
### Final Conclusion:
The correct statement among the provided options is:
- **When \( x > 1 \), the function \( g(x) = \sqrt{2x} + 3 \) is greater than \( f(x) = \sqrt{2x} \).**