Identify the key characteristics of the parent fifth-root function . Include the following: domain, range, intervals where the function is increasing, decreasing, or constant, intercepts, whether the function is even, odd, or neither, whether the function is continuous or discontinuous, asymptotes, and end behavior

Answer 1

The parent fifth-root function can be represented as \(f(x) = \sqrt[5]{x}\).

Here are the key characteristics of the function:

1. Domain: The domain of the fifth-root function is all real numbers, since we can take the fifth root of any real number.
2. Range: The range of the function is also all real numbers, since the fifth root can produce both positive and negative values.
3. Increasing, decreasing, or constant intervals: The function is increasing for all positive numbers, decreasing for all negative numbers, and constant for \(f(0) = 0\).
4. Intercepts: The function has an x-intercept at the point (0, 0).
5. Even, odd, or neither: The function is neither even nor odd. \(f(-x) = -\sqrt[5]{x}\) is not equal to \(-f(x)\) and \(f(x)\) is not equal to \(f(-x)\).
6. Continuity: The fifth-root function is continuous for all real numbers. There are no discontinuities in the graph.
7. Asymptotes: The function does not have any asymptotes.
8. End behavior: As \(x\) approaches positive infinity, \(f(x)\) also approaches positive infinity. As \(x\) approaches negative infinity, \(f(x)\) approaches negative infinity.