Question

Determine whether the given relation defines y as a function of x.
x equals 17 y Superscript 6
Question content area bottom
Part 1
Is y a function of​ x?

Answer 1

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 \( x \).

1. **Rearranging the equation:** We can rearrange \( x = 17y^6 \) to solve for \( y \):

\[
y^6 = \frac{x}{17}
\]

\[
y = \left( \frac{x}{17} \right)^{1/6}
\]

2. **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.

3. **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 \( x \).