Let's analyze each equation to determine if it defines $y$ as a function of $x$. A function means that for each value of $x$, there is exactly one corresponding value of $y$.

1. $x^2 + 4y = 6$

   • Solving for $y$, we get $y = \frac{6 - x^2}{4}$. This equation defines $y$ uniquely in terms of $x$.
     Answer: Yes

2. $x^2y + y = 6$

   • Factoring the $y$, we get $y(x^2 + 1) = 6$, or $y = \frac{6}{x^2 + 1}$. Since this gives a unique value of $y$ for each $x$, this is a function.
     Answer: Yes

3. $2|x| + y = 10$

   • Solving for $y$, we get $y = 10 - 2|x|$. The absolute value introduces no ambiguity, as for each $x$, there is still only one value of $y$.
     Answer: Yes

4. $9 + x = y^3$

   • Solving for $y$, we get $y = \sqrt[3]{9 + x}$. The cube root function is well-defined and provides a unique value for $y$ for each $x$.
     Answer: Yes

In summary, all four equations define $y$ as a function of $x$.

Would you like further details on any of the steps? Here are 5 related questions to deepen your understanding:

1. How does solving for $y$ determine whether an equation defines a function?
2. What distinguishes a function from a relation in mathematics?
3. Can an equation involving absolute values still define a function?
4. How can you determine if a multivariable equation defines one variable as a function of the other?
5. What would happen if we had a square root instead of a cube root in equation 4?

Tip: When verifying if an equation defines a function, always check whether solving for $y$ results in a unique value for every $x$.