Decide whether each relation defines \( y \) as a function of \( x \). Give the domain and range. \[ y=\sqrt[8]{x} \]

To determine whether the relation \( y = \sqrt[8]{x} \) defines \( y \) as a function of \( x \), we need to check if for every value of \( x \) in the domain, there is exactly one corresponding value of \( y \). 1. **Function Definition**: A relation defines \( y \) as a function of \( x \) if each input \( x \) has exactly one output \( y \). 2. **Analyzing the Relation**: The expression \( y = \sqrt[8]{x} \) means we are taking the eighth root of \( x \). The eighth root is defined for all non-negative values of \( x \) (i.e., \( x \geq 0 \)). 3. **Determining the Domain**: - The domain of \( y = \sqrt[8]{x} \) is all non-negative real numbers: \[ \text{Domain} = [0, \infty) \] 4. **Determining the Range**: - The output \( y \) will also be non-negative since the eighth root of a non-negative number is non-negative. Therefore, the range is: \[ \text{Range} = [0, \infty) \] 5. **Conclusion**: Since for every \( x \) in the domain there is exactly one corresponding \( y \), the relation \( y = \sqrt[8]{x} \) does define \( y \) as a function of \( x \). ### Summary: - **Function**: Yes, \( y \) is a function of \( x \). - **Domain**: \( [0, \infty) \) - **Range**: \( [0, \infty) \)