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From the uploaded image, I can see that you are working on problems related to finding the range of functions over specific domains. Let me summarize and verify the solutions provided step-by-step. If you want detailed explanations or corrections for any of them, please let me know.

Here’s a quick analysis:

1. **Problem 1**:
   Function: \( f(x) = x^2 \). Domain: \(-2 \leq x \leq 3\).  
   **Range:** Since \(f(x)\) is quadratic, with its vertex as a minimum point at \(x = 0\), the range is \([0, 9]\).

2. **Problem 2**:
   Function: \( k(x) = x^2 - 5 \). Domain: All real \(x\).  
   **Range:** Shifted quadratic. Minimum value of \(x^2\) is 0, so range starts at \(-5\). Range is \([k(x), \infty)\) or \([-5, \infty)\).

3. **Problem 3**:
   Function: \( g(x) = (x - 1)^2 \). Domain: \(-2 \leq x \leq 4\).  
   **Range:** Completing the square and analyzing endpoints, range is \([0, 9]\).

4. **Problem 4**:
   Function: \( f(x) = 2x - x^2 \). Domain: \(0 \leq x \leq 2\).  
   **Range:** Finding critical points and analyzing vertex, range is \([0, 1]\).

5. **Problem 5**:
   Function: \( f(x) = x^2 - 4 \). Domain: \(-3 \leq x \leq 3\).  
   **Range:** Adding/subtracting \(-4\), the range is \([-4, 5]\).

6. **Problem 6**:
   Function: \( g(x) = x^2 - x + 1 \). Domain: \(1 \leq x \leq 3\).  
   **Range:** Analyzing \(g(x)\), range is \([3/4, 7]\).

7. **Problem 7**:
   Function: \( f(x) = x^2 - 3x + 5 \). Domain: \(0 \leq x \leq 4\).  
   **Range:** Completing the square and analyzing endpoints, range is \([11/4, 9]\).

Would you like detailed step-by-step solutions or clarifications for any specific part? Here are 5 follow-up questions to help you further:

1. Can I show you how to calculate the vertex for quadratic functions to find ranges systematically?  
2. Would you like a graph for any of the functions to visualize the domain and range relationships?  
3. Do you need help analyzing transformations like shifting or scaling for quadratic equations?  
4. Are there any questions about finding critical points or analyzing endpoints for these ranges?  
5. Would you like a refresher on completing the square for quadratic functions?

**Tip:** Always graph the function if possible; it provides a quick way to verify your calculated range visually!

The image contains a series of problems related to finding the range of functions, such as quadratic and transformed quadratic equations, over specific domains. Example problems include finding the range of f(x) = x^2, k(x) = x^2 - 5, and g(x) = (x - 1)^2, with given domains.

Learn how to determine the range of quadratic functions like f(x) = x^2 and transformed functions such as k(x) = x^2 - 5 over given domains. This guide provides detailed explanations, key formulas, and solutions, suitable for high school math students in Grades 9-11.

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