To determine whether \( y \) is a function of \( x \) in the relation \( x = 17y^6 \), we need to analyze how \( y \) behaves in relation to \( x \).
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Rearranging the equation: We can rearrange \( x = 17y^6 \) to solve for \( y \):
\[ y^6 = \frac{x}{17} \]
\[ y = \left( \frac{x}{17} \right)^{1/6} \]
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Considering the sixth root: The expression \( y = \left( \frac{x}{17} \right)^{1/6} \) suggests that for each value of \( x \) (assuming \( x \geq 0 \)), there will be a single positive sixth root. However, the sixth root of a number can also be negative.
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Two possible values of \( y \): For any positive \( x \), the sixth root will yield:
- A positive value \( \left( \frac{x}{17} \right)^{1/6} \)
- A corresponding negative value \( -\left( \frac{x}{17} \right)^{1/6} \)
This means that for a given \( x \), there can be two values of \( y \) (one positive and one negative), making \( y \) not uniquely determined for every value of \( x \).
Conclusion
Since there can be multiple values of \( y \) for a single value of \( x \), we conclude that \( y \) is not a function of \( x \). Thus, the relation does not define \( y \) as a function of \( x \).