Decide whether the given equation defines $$ y $$ as a function of $$ x $$. Then give the domain and range. $$ x=y^{12} $$

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To determine whether the equation $$ x = y^{12} $$ defines $$ y $$ as a function of $$ x $$, we need to check if for every value of $$ x $$, there is exactly one corresponding value of $$ y $$.

1. **Rearranging the equation**: We can express $$ y $$ in terms of $$ x $$:
   $$
   y = x^{1/12}
   $$
   This means that for each non-negative value of $$ x $$, there is a corresponding non-negative value of $$ y $$. However, for negative values of $$ x $$, there are no real values of $$ y $$ that satisfy the equation since $$ y^{12} $$ (being an even power) cannot be negative.

2. **Function definition**: Since for each non-negative $$ x $$ there is exactly one corresponding $$ y $$, we can conclude that $$ y $$ is a function of $$ x $$ for $$ x \geq 0 $$.

3. **Domain**: The domain of the function is the set of all possible $$ x $$ values for which the function is defined. Since $$ y^{12} $$ can only be non-negative, the domain is:
   $$
   \text{Domain: } [0, \infty)
   $$

4. **Range**: The range of the function is the set of all possible $$ y $$ values. Since $$ y = x^{1/12} $$ can take any non-negative value as $$ x $$ varies from $$ 0 $$ to $$ \infty $$, the range is:
   $$
   \text{Range: } [0, \infty)
   $$

In summary:
- The equation $$ x = y^{12} $$ defines $$ y $$ as a function of $$ x $$.
- Domain: $$ [0, \infty) $$
- Range: $$ [0, \infty) $$
$$ y $$ is a function of $$ x $$ for $$ x \geq 0 $$. The domain and range are both $$ [0, \infty) $$.

Decide whether the given equation defines \( y \) as a function of \( x \). Then give the domain and range. \( x=y^{12} \)

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