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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.

Which equation best describes the inverse of the relationship between x and y shown in the table? The domain of the function is restricted to x ≥ 0.

Discover how to identify the inverse relationship between two variables using a table of values. This problem demonstrates key concepts such as inverse functions, quadratic functions, and square root relationships. Suitable for students in Grades 9-12.

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