https://file.aiquickdraw.com/mathbot/file/17296980086277kjt29zm.jpg

The image shows a graph on a coordinate plane and asks to match it with one of the given functions based on its appearance. The options are:

1. \( f(x) = \sqrt{x + 2} + 1 \)
2. \( f(x) = \sqrt{x + 2} - 1 \)
3. \( f(x) = \sqrt{x - 2} + 1 \)
4. \( f(x) = \sqrt{x - 1} + 2 \)

### Let's analyze the graph and functions:

- The graph starts at \( x = -2 \), and increases. The point \( (-2, 1) \) is highlighted as the start of the curve, which is consistent with a square root function that has been shifted.

1. **Option 1: \( f(x) = \sqrt{x + 2} + 1 \)**
   - The domain is \( x \geq -2 \), because the expression inside the square root must be non-negative (\( x + 2 \geq 0 \)).
   - The function is shifted 2 units to the left and 1 unit up.
   - The starting point would be \( (-2, 1) \), which matches the given graph.

2. **Option 2: \( f(x) = \sqrt{x + 2} - 1 \)**
   - The domain is the same as Option 1 (\( x \geq -2 \)), but the graph would start at \( (-2, -1) \), which does not match the graph.

3. **Option 3: \( f(x) = \sqrt{x - 2} + 1 \)**
   - The domain is \( x \geq 2 \), so it does not match the graph which starts at \( x = -2 \).

4. **Option 4: \( f(x) = \sqrt{x - 1} + 2 \)**
   - The domain is \( x \geq 1 \), which does not match the graph either.

### Conclusion:
The graph matches **Option 1: \( f(x) = \sqrt{x + 2} + 1 \)**.

Would you like further details on any of the functions, or help with other related questions?

Here are some related questions:
1. How do we derive the domain of a square root function?
2. What happens to a graph when the function is shifted horizontally or vertically?
3. How can we determine the range of a square root function?
4. Why do square root functions have restricted domains?
5. How can we find the inverse of a square root function?

**Tip:** Always check the domain and range of a function when comparing it to a graph, as they are key to identifying the correct match.

The image shows a graph on a coordinate plane and asks to match it with one of the given functions based on its appearance. The options are: 
1. \( f(x) = \sqrt{x + 2} + 1 \)
2. \( f(x) = \sqrt{x + 2} - 1 \)
3. \( f(x) = \sqrt{x - 2} + 1 \)
4. \( f(x) = \sqrt{x - 1} + 2 \).

Learn how to identify the correct square root function from a graph. Analyze the function transformations, domain, and range to determine the best match. This problem explores graph shifts, domain restrictions, and square root functions, suitable for students in Grades 9-11.

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