Let's analyze the function given:

Function:
$z = x + \sqrt{y}$

1. Domain:

   • The function involves a square root, $\sqrt{y}$. For $\sqrt{y}$ to be defined in the real number system, $y$ must be non-negative.
   • Therefore, the domain requires $y \geq 0$.
   • $x$ has no restrictions since it is added directly without any constraints.

   Domain:
   $x \in \mathbb{R}$ (all real numbers)
   $y \geq 0$

2. Range:

   • Since $z = x + \sqrt{y}$, the minimum value of $\sqrt{y}$ is 0 (when $y = 0$).
   • As $y$ increases, $\sqrt{y}$ can take any non-negative value.
   • Given that $x$ can take any real value, $z$ can also take any real value.

   Range:
   $z \in \mathbb{R}$ (all real numbers)

3. Evaluate at (5, 0):

   • Substitute $x = 5$ and $y = 0$: $z = 5 + \sqrt{0} = 5 + 0 = 5$

   Result:
   $z = 5$ at $(5,0)$

Would you like more details or have any questions?

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Follow-up Questions:

1. How would the domain change if the function were $z = x + \sqrt{y - 2}$?
2. What if the function were $z = x + \sqrt{|y|}$? How does that affect the domain and range?
3. If the function were $z = x + y^{1/3}$, how would the domain and range differ?
4. How does adding constraints on $x$, such as $x \geq 0$, affect the domain and range?
5. Can the domain or range of this function change in complex analysis?

Tip: When dealing with square roots, always check if the argument inside the root is non-negative to ensure it's within the real domain.