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The problem in your image asks to find the inverse of the given function \( f(x) = \sqrt{3 + x}, \, x \geq -3 \).

### Step-by-step solution:

1. **Find the inverse function**:  
   Start by letting \( y = f(x) = \sqrt{3 + x} \), which can be rewritten as:
   \[
   y = \sqrt{3 + x}
   \]
   To find the inverse, solve for \( x \) in terms of \( y \):
   \[
   y^2 = 3 + x
   \]
   \[
   x = y^2 - 3
   \]
   Now, swap \( x \) and \( y \) to express the inverse function \( f^{-1}(x) \):
   \[
   f^{-1}(x) = x^2 - 3
   \]

2. **Domain and range**:
   - For the original function \( f(x) = \sqrt{3 + x} \), the domain is \( x \geq -3 \) since the expression inside the square root must be non-negative. The range is \( f(x) \geq 0 \), meaning \( y \geq 0 \).
   - For the inverse function \( f^{-1}(x) = x^2 - 3 \), the domain is \( x \geq 0 \) because the range of the original function becomes the domain of the inverse. The range of the inverse function is \( y \geq -3 \), as \( f^{-1}(x) = x^2 - 3 \).

### Answer:
Looking at the multiple-choice options, the correct choice is:

- **B. \( f^{-1}(x) = x^2 - 3, \, x \geq 0 \)**.

Would you like a detailed explanation of the inverse function process, or do you have any other questions?

### Related Questions:
1. How do you check if a function is one-to-one?
2. Can every function have an inverse?
3. What is the difference between a one-to-one function and a many-to-one function?
4. How can you graphically determine if a function has an inverse?
5. Why does the domain of the original function become the range of the inverse function?

### Tip:
Always remember that when finding the inverse, it's important to swap the domain and range of the original function for the inverse function.

For the following function defined that is one-to-one, write an equation for the inverse function in the form y = f^(-1)(x). Give the domain and range of f and f^(-1). If the function is not one-to-one, say so. f(x) = √(3 + x), x ≥ -3

Learn how to find the inverse of the function f(x) = √(3 + x), determine the domain and range, and understand when a function is one-to-one. This problem covers key concepts in algebra and inverse functions. Suitable for students in Grades 10-12.

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