The question asks which equation best describes the inverse of the relationship between $x$ and $y$ shown in the table, with the domain restricted to $x \geq 0$.

To solve this:

1. Observe the given table:

   • For $x = 1$, $y = 0.5$
   • For $x = 2$, $y = 2$
   • For $x = 4$, $y = 8$
   • For $x = 7$, $y = 24.5$
   • For $x = 9$, $y = 40.5$

2. We are tasked with finding the inverse, so we need an equation where $y$ is expressed in terms of $x$.

3. Let's analyze the choices:

   • Option A: $x = \sqrt{2y}$ implies $y = \frac{x^2}{2}$, which seems to align with the pattern when looking at the values.
   • Option B: $y = \frac{1}{2}x^2$, also a quadratic relation, does not appear to match the inverse pattern.
   • Option C: $y = \sqrt{2x}$, involves a square root and doesn't fit the pattern well.
   • Option D: $x = \frac{1}{2}y^2$, suggests a quadratic relation in the wrong format for the inverse.

Given the table values, Option A $x = \sqrt{2y}$ best fits the inverse relationship between $x$ and $y$.

Would you like further details or clarifications on this? Here are some related questions:

1. How do you find the inverse of a function in general?
2. What are the conditions for a function to have an inverse?
3. How can you confirm that two functions are inverses of each other?
4. How do domain restrictions affect the inverse of a function?
5. What is the difference between a square root function and a quadratic function?

Tip: Always verify inverse relationships by checking whether applying one function to the result of the other returns the original value.