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The given equation is:

\[
y = \sqrt{x} + 2
\]

We need to determine whether this equation defines a function. To do this, we will consider whether for each input \(x\), there is exactly one output \(y\).

### Analysis:
1. The square root function, \( \sqrt{x} \), is defined only for non-negative values of \(x\) (i.e., \(x \geq 0\)), because the square root of a negative number is not defined in the real number system.

2. For each non-negative value of \(x\), \( \sqrt{x} \) gives exactly one non-negative output. Adding 2 to this result shifts the output upwards, but it does not affect the fact that for each valid \(x\), there is exactly one corresponding \(y\).

### Conclusion:
- The equation defines a function because, for every valid input \(x \geq 0\), there is exactly one output \(y\).
- The domain of the function is \(x \geq 0\), i.e., \([0, \infty)\), since the square root function is not defined for negative values of \(x\).

Would you like more details or have any questions?

### Related questions:
1. How does the domain of a function affect whether it is a valid function?
2. What would happen if the square root function allowed negative inputs in complex numbers?
3. How can we determine the range of this function \(y = \sqrt{x} + 2\)?
4. What is the graphical representation of \(y = \sqrt{x} + 2\)?
5. Can you provide another example of an equation that does not define a function?

### Tip:
Remember, a function must map each input to exactly one output—this is called the "vertical line test" when looking at graphs.

Can the following equations define functions? y = √x + 2. Justify your answer.

Explore whether y = √x + 2 defines a function, with a clear explanation of the function definition, domain restrictions, and analysis of the square root function. Suitable for students in Grades 9-12.

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