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 The graphs have different domains. The graphs have different 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 . They are both decreasing on their domains. They are both decreasing on their domains. 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 . Skip to navigation
Answers
GPT-4o mini
Let's analyze the functions \( f(x) = 2\sqrt{x} \) and \( g(x) = 2\sqrt[3]{x} \) to determine the validity of the given statements.
1. **Domain**:
- The function \( f(x) = 2\sqrt{x} \) has a domain of \( x \geq 0 \) since the square root function is only defined for non-negative numbers.
- The function \( g(x) = 2\sqrt[3]{x} \) has a domain of all real numbers \( x \) since the cube root function is defined for all \( x \).
- **Conclusion**: The graphs have different domains.
2. **Common Points**:
- For \( f(-1) \): Not defined (since \(\sqrt{-1}\) is not real).
- For \( f(0) = 2\sqrt{0} = 0 \).
- For \( f(1) = 2\sqrt{1} = 2 \).
- For \( g(-1) = 2\sqrt[3]{-1} = 2(-1) = -2 \).
- For \( g(0) = 2\sqrt[3]{0} = 0 \).
- For \( g(1) = 2\sqrt[3]{1} = 2(1) = 2 \).
- **Conclusion**: They do not go through \( (-1,-1) \), but both go through \( (0,0) \) and \( (1,2) \).
3. **Monotonicity**:
- The function \( f(x) = 2\sqrt{x} \) is increasing for \( x \geq 0 \) since its derivative \( f'(x) = \frac{1}{\sqrt{x}} \) is positive for \( x > 0 \).
- The function \( g(x) = 2\sqrt[3]{x} \) is also increasing for all \( x \) since its derivative \( g'(x) = \frac{2}{3\sqrt[3]{x^2}} \) is positive for all \( x \neq 0 \).
- **Conclusion**: They are both increasing on their domains.
4. **Comparison for \( x > 1 \)**:
- For \( x > 1 \), we need to compare \( f(x) \) and \( g(x) \):
- As \( x \) increases beyond 1, both functions grow, but we need to actually compute specific values or evaluate limits to confirm which is greater.
- For example, at \( x = 2 \):
- \( f(2) = 2\sqrt{2} \approx 2.828 \),
- \( g(2) = 2\sqrt[3]{2} \approx 2.519 \).
- As \( x \) increases, the function \( g(x) \) grows slower than \( f(x) \).
- **Conclusion**: For \( x > 1 \), \( f(x) > g(x) \).
Based on this analysis, the correct statement is:
- **The graphs have different domains.**
1. **Domain**:
- The function \( f(x) = 2\sqrt{x} \) has a domain of \( x \geq 0 \) since the square root function is only defined for non-negative numbers.
- The function \( g(x) = 2\sqrt[3]{x} \) has a domain of all real numbers \( x \) since the cube root function is defined for all \( x \).
- **Conclusion**: The graphs have different domains.
2. **Common Points**:
- For \( f(-1) \): Not defined (since \(\sqrt{-1}\) is not real).
- For \( f(0) = 2\sqrt{0} = 0 \).
- For \( f(1) = 2\sqrt{1} = 2 \).
- For \( g(-1) = 2\sqrt[3]{-1} = 2(-1) = -2 \).
- For \( g(0) = 2\sqrt[3]{0} = 0 \).
- For \( g(1) = 2\sqrt[3]{1} = 2(1) = 2 \).
- **Conclusion**: They do not go through \( (-1,-1) \), but both go through \( (0,0) \) and \( (1,2) \).
3. **Monotonicity**:
- The function \( f(x) = 2\sqrt{x} \) is increasing for \( x \geq 0 \) since its derivative \( f'(x) = \frac{1}{\sqrt{x}} \) is positive for \( x > 0 \).
- The function \( g(x) = 2\sqrt[3]{x} \) is also increasing for all \( x \) since its derivative \( g'(x) = \frac{2}{3\sqrt[3]{x^2}} \) is positive for all \( x \neq 0 \).
- **Conclusion**: They are both increasing on their domains.
4. **Comparison for \( x > 1 \)**:
- For \( x > 1 \), we need to compare \( f(x) \) and \( g(x) \):
- As \( x \) increases beyond 1, both functions grow, but we need to actually compute specific values or evaluate limits to confirm which is greater.
- For example, at \( x = 2 \):
- \( f(2) = 2\sqrt{2} \approx 2.828 \),
- \( g(2) = 2\sqrt[3]{2} \approx 2.519 \).
- As \( x \) increases, the function \( g(x) \) grows slower than \( f(x) \).
- **Conclusion**: For \( x > 1 \), \( f(x) > g(x) \).
Based on this analysis, the correct statement is:
- **The graphs have different domains.**