The parent fifth-root function \( f(x) = \sqrt[5]{x} \) has several key characteristics:
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Domain: The domain is all real numbers, \( (-\infty, \infty) \), because you can take the fifth root of any real number.
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Range: The range is also all real numbers, \( (-\infty, \infty) \), because the fifth root can result in any real output.
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Intervals of Increase/Decrease:
- The function is increasing on the entire domain \( (-\infty, \infty) \). This is because the derivative \( f'(x) = \frac{1}{5\sqrt[5]{x^4}} \) is positive for all \( x \neq 0 \).
- The function has no intervals where it is decreasing or constant.
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Intercepts:
- The x-intercept occurs at \( (0, 0) \), as \( f(0) = \sqrt[5]{0} = 0 \).
- The y-intercept also occurs at \( (0, 0) \).
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Odd, Even, or Neither:
- The function is odd. This can be confirmed because \( f(-x) = \sqrt[5]{-x} = -\sqrt[5]{x} = -f(x) \).
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Continuity: The function is continuous everywhere on its domain \( (-\infty, \infty) \). There are no breaks, holes, or jumps in the graph.
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Asymptotes: There are no asymptotes for the function \( f(x) = \sqrt[5]{x} \). The function does not approach infinity or negative infinity in a way that creates vertical or horizontal asymptotes.
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End Behavior:
- As \( x \to -\infty \), \( f(x) \to -\infty \).
- As \( x \to +\infty \), \( f(x) \to +\infty \).
In summary, the key characteristics of the function \( f(x) = \sqrt[5]{x} \) are:
- Domain: \( (-\infty, \infty) \)
- Range: \( (-\infty, \infty) \)
- Increasing: \( (-\infty, \infty) \)
- Intercepts: \( (0,0) \)
- Odd Function: Yes
- Continuous: Yes
- Asymptotes: None
- End Behavior: \( f(x) \to -\infty \) as \( x \to -\infty \) and \( f(x) \to +\infty \) as \( x \to +\infty \)