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Compare the given graphs of f(x)=2x−−√ and g(x)=2x−−√3 and determine which of the following is true. (1 point) Responses Both g...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
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 .
When x>1, the function g(x)=2x−−√3 is greater than f(x)=2x−−√.
When x greater than 1 , the function g left parenthesis x right parenthesis equals 2 root index 3 Start Root x End Root is greater than f left parenthesis x right parenthesis equals 2 Start Root x End Root .
The graphs have different domains.
The graphs have different domains.
They are both decreasing on their domains.
(1 point)
Responses
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 .
When x>1, the function g(x)=2x−−√3 is greater than f(x)=2x−−√.
When x greater than 1 , the function g left parenthesis x right parenthesis equals 2 root index 3 Start Root x End Root is greater than f left parenthesis x right parenthesis equals 2 Start Root x End Root .
The graphs have different domains.
The graphs have different domains.
They are both decreasing on their domains.
Answers
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Answered by
GPT-4o mini
Let's analyze the functions \( f(x) = 2\sqrt{x} \) and \( g(x) = 2\sqrt[3]{x} \) to determine which of the given statements is true.
1. **Both graphs go through (-1,-1), (0,0), and (1,1)**:
- \( f(-1) \) is not defined since square root of a negative number is not a real number. Thus, this statement is false.
- \( f(0) = 2\sqrt{0} = 0 \) and \( g(0) = 2\sqrt[3]{0} = 0 \), so they both go through (0,0).
- \( f(1) = 2\sqrt{1} = 2 \) and \( g(1) = 2\sqrt[3]{1} = 2 \), so they both go through (1,1).
Therefore, this statement is false since both functions do not go through (-1,-1).
2. **When \( x > 1 \), the function \( g(x) = 2\sqrt[3]{x} \) is greater than \( f(x) = 2\sqrt{x} \)**:
- For \( x > 1 \), let's evaluate \( f(2) \) and \( g(2) \):
- \( f(2) = 2\sqrt{2} \approx 2.828 \)
- \( g(2) = 2\sqrt[3]{2} \approx 1.587 \)
Since \( 2\sqrt{2} > 2\sqrt[3]{2} \), this statement is false.
3. **The graphs have different domains**:
- The domain of \( f(x) = 2\sqrt{x} \) is \( x \geq 0 \) (only non-negative values).
- The domain of \( g(x) = 2\sqrt[3]{x} \) is all real numbers (no restrictions).
Therefore, this statement is true.
4. **They are both decreasing on their domains**:
- \( f(x) = 2\sqrt{x} \) is increasing on \( x \geq 0 \).
- \( g(x) = 2\sqrt[3]{x} \) is also increasing on all real numbers.
Therefore, this statement is false.
### Conclusion:
The only true statement is: **The graphs have different domains.**
1. **Both graphs go through (-1,-1), (0,0), and (1,1)**:
- \( f(-1) \) is not defined since square root of a negative number is not a real number. Thus, this statement is false.
- \( f(0) = 2\sqrt{0} = 0 \) and \( g(0) = 2\sqrt[3]{0} = 0 \), so they both go through (0,0).
- \( f(1) = 2\sqrt{1} = 2 \) and \( g(1) = 2\sqrt[3]{1} = 2 \), so they both go through (1,1).
Therefore, this statement is false since both functions do not go through (-1,-1).
2. **When \( x > 1 \), the function \( g(x) = 2\sqrt[3]{x} \) is greater than \( f(x) = 2\sqrt{x} \)**:
- For \( x > 1 \), let's evaluate \( f(2) \) and \( g(2) \):
- \( f(2) = 2\sqrt{2} \approx 2.828 \)
- \( g(2) = 2\sqrt[3]{2} \approx 1.587 \)
Since \( 2\sqrt{2} > 2\sqrt[3]{2} \), this statement is false.
3. **The graphs have different domains**:
- The domain of \( f(x) = 2\sqrt{x} \) is \( x \geq 0 \) (only non-negative values).
- The domain of \( g(x) = 2\sqrt[3]{x} \) is all real numbers (no restrictions).
Therefore, this statement is true.
4. **They are both decreasing on their domains**:
- \( f(x) = 2\sqrt{x} \) is increasing on \( x \geq 0 \).
- \( g(x) = 2\sqrt[3]{x} \) is also increasing on all real numbers.
Therefore, this statement is false.
### Conclusion:
The only true statement is: **The graphs have different domains.**
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