Question
compare the given graphs of f (x) = 2√x and g (x) = 2∛x and determine which of the following is true.
Answers
GPT-4o mini
To compare the graphs of the functions \( f(x) = 2\sqrt{x} \) and \( g(x) = 2\sqrt[3]{x} \), we can analyze the characteristics of both functions.
1. **Domain**:
- \( f(x) = 2\sqrt{x} \) is defined only for \( x \geq 0 \) (the domain is \( [0, \infty) \)).
- \( g(x) = 2\sqrt[3]{x} \) is defined for all real numbers (the domain is \( (-\infty, \infty) \)).
2. **Range**:
- The range of \( f(x) \) is \( [0, \infty) \) since the square root and multiplication by a positive constant keep the output non-negative.
- The range of \( g(x) \) is also \( (-\infty, \infty) \) because the cube root can produce any real number when applied to any real value of \( x \).
3. **Behavior Near Zero**:
- As \( x \) approaches 0, \( f(x) \) approaches 0, and so does \( g(x) \). However, for values of \( x < 0 \), \( f(x) \) is not defined, while \( g(x) \) produces negative values.
4. **End Behavior**:
- As \( x \) approaches infinity, both \( f(x) \) and \( g(x) \) approach infinity. However:
- \( f(x) \) grows at a slower rate compared to polynomial functions.
- \( g(x) \) grows slightly faster than \( f(x) \) for positive values of \( x \).
5. **Intercepts**:
- Both functions have a y-intercept at (0, 0).
- \( f(x) \) only exists in the first quadrant, while \( g(x) \) crosses through all quadrants.
6. **Graph Shape**:
- The graph of \( f(x) = 2\sqrt{x} \) is a half parabola opening upward, starting from the origin.
- The graph of \( g(x) = 2\sqrt[3]{x} \) has an S-shape, smoothly transitioning through the origin, showing how it increases for both negative and positive values of \( x \).
### Conclusion
To summarize:
- Both functions start from the origin, but their domains and ranges differ significantly.
- \( f(x) \) is defined only for non-negative \( x \), while \( g(x) \) is defined for all \( x \) and goes through negative values.
- The shape and hence the behavior for \( x < 0 \) is vastly different for both functions.
If you have specific statements to evaluate, please provide them so we can determine their validity based on this comparison.
1. **Domain**:
- \( f(x) = 2\sqrt{x} \) is defined only for \( x \geq 0 \) (the domain is \( [0, \infty) \)).
- \( g(x) = 2\sqrt[3]{x} \) is defined for all real numbers (the domain is \( (-\infty, \infty) \)).
2. **Range**:
- The range of \( f(x) \) is \( [0, \infty) \) since the square root and multiplication by a positive constant keep the output non-negative.
- The range of \( g(x) \) is also \( (-\infty, \infty) \) because the cube root can produce any real number when applied to any real value of \( x \).
3. **Behavior Near Zero**:
- As \( x \) approaches 0, \( f(x) \) approaches 0, and so does \( g(x) \). However, for values of \( x < 0 \), \( f(x) \) is not defined, while \( g(x) \) produces negative values.
4. **End Behavior**:
- As \( x \) approaches infinity, both \( f(x) \) and \( g(x) \) approach infinity. However:
- \( f(x) \) grows at a slower rate compared to polynomial functions.
- \( g(x) \) grows slightly faster than \( f(x) \) for positive values of \( x \).
5. **Intercepts**:
- Both functions have a y-intercept at (0, 0).
- \( f(x) \) only exists in the first quadrant, while \( g(x) \) crosses through all quadrants.
6. **Graph Shape**:
- The graph of \( f(x) = 2\sqrt{x} \) is a half parabola opening upward, starting from the origin.
- The graph of \( g(x) = 2\sqrt[3]{x} \) has an S-shape, smoothly transitioning through the origin, showing how it increases for both negative and positive values of \( x \).
### Conclusion
To summarize:
- Both functions start from the origin, but their domains and ranges differ significantly.
- \( f(x) \) is defined only for non-negative \( x \), while \( g(x) \) is defined for all \( x \) and goes through negative values.
- The shape and hence the behavior for \( x < 0 \) is vastly different for both functions.
If you have specific statements to evaluate, please provide them so we can determine their validity based on this comparison.