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
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.