To compare the domain and range of the functions \( f(x) = 2\sqrt{x} \) and \( g(x) = 2\sqrt[3]{x} \), we can analyze each function.
Function \( f(x) = 2\sqrt{x} \)
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Domain: The square root function is only defined for non-negative inputs. Therefore, the domain of \( f(x) \) is: \[ [0, \infty) \]
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Range: The output of the square root function is also non-negative. Thus, the range of \( f(x) \) is: \[ [0, \infty) \]
Function \( g(x) = 2\sqrt[3]{x} \)
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Domain: The cube root function is defined for all real numbers. Therefore, the domain of \( g(x) \) is: \[ (-\infty, \infty) \]
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Range: The cube root function can produce all real numbers as output, thus the range of \( g(x) \) is: \[ (-\infty, \infty) \]
Summary of Domain and Range
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\( f(x) = 2\sqrt{x} \):
- Domain: \([0, \infty)\)
- Range: \([0, \infty)\)
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\( g(x) = 2\sqrt[3]{x} \):
- Domain: \((-\infty, \infty)\)
- Range: \((-\infty, \infty)\)
Conclusion
The correct response is: f(x) Domain is [0, ∞) and Range is [0, ∞). g(x) Domain is (-∞, ∞) and Range is (-∞, ∞).