To analyze the functions \( f(x) = -\sqrt{x} \) and \( g(x) = -\sqrt[3]{x} \), let's examine their features in terms of domain, range, and whether they are increasing or decreasing.
Domains:
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For \( f(x) = -\sqrt{x} \):
- The square root function is defined for \( x \geq 0 \). Thus, the domain of \( f(x) \) is: \[ \text{Domain of } f(x): [0, \infty) \]
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For \( g(x) = -\sqrt[3]{x} \):
- The cube root function is defined for all real numbers. Thus, the domain of \( g(x) \) is: \[ \text{Domain of } g(x): (-\infty, \infty) \]
Ranges:
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For \( f(x) = -\sqrt{x} \):
- As \( x \) increases from \( 0 \) to \( \infty \), \( -\sqrt{x} \) decreases from \( 0 \) to \( -\infty \). Thus, the range of \( f(x) \) is: \[ \text{Range of } f(x): (-\infty, 0] \]
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For \( g(x) = -\sqrt[3]{x} \):
- The cube root decreases as \( x \) decreases from \( -\infty \) to \( \infty \), and thus \( -\sqrt[3]{x} \) will also take all real values from \( -\infty \) to \( \infty \). Hence, the range of \( g(x) \) is: \[ \text{Range of } g(x): (-\infty, \infty) \]
Monotonicity:
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For \( f(x) = -\sqrt{x} \):
- The function is decreasing as \( x \) increases because the square root function itself is increasing, and its negation is decreasing.
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For \( g(x) = -\sqrt[3]{x} \):
- The function is also decreasing; the cubic root function is increasing, and its negation is decreasing.
Summary of Features:
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Domains: The domains of \( f(x) \) and \( g(x) \) are different.
- \( f(x): [0, \infty) \)
- \( g(x): (-\infty, \infty) \)
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Ranges: The ranges of \( f(x) \) and \( g(x) \) are also different.
- \( f(x): (-\infty, 0] \)
- \( g(x): (-\infty, \infty) \)
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Monotonicity: Both functions are decreasing.
Conclusion:
- The true statements about the graphs of the functions are:
- They are both decreasing.
Thus, the correct response is: The graphs are both decreasing.