Find the function domain and range of the following functions and evaluate at (5,0): a) 𝑧 = 𝑥 + √y

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.

Find the function domain and range of the following functions and evaluate at (5,0): a) z = x + √y

Explore how to find the domain and range for the function z = x + √y, including a step-by-step evaluation at the point (5, 0). This solution covers the essential rules for square roots and function analysis, ideal for high school students in Grades 10-12.

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