Question

Identify the key characteristics of the parent fifth-root function f left parenthesis x right parenthesis equals root index 5 start root x end root. 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 \( f(x) = \sqrt[5]{x} \) has several key characteristics:

1. **Domain**: The domain is all real numbers, \( (-\infty, \infty) \), because you can take the fifth root of any real number.

2. **Range**: The range is also all real numbers, \( (-\infty, \infty) \), because the fifth root can result in any real output.

3. **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.

4. **Intercepts**:
- The x-intercept occurs at \( (0, 0) \), as \( f(0) = \sqrt[5]{0} = 0 \).
- The y-intercept also occurs at \( (0, 0) \).

5. **Odd, Even, or Neither**:
- The function is odd. This can be confirmed because \( f(-x) = \sqrt[5]{-x} = -\sqrt[5]{x} = -f(x) \).

6. **Continuity**: The function is continuous everywhere on its domain \( (-\infty, \infty) \). There are no breaks, holes, or jumps in the graph.

7. **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.

8. **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 \( x \to +\infty \)