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The given function is \( f(x) = \sqrt{x + 3} \).

### Step 1: Find the domain of \( f(x) \)

For the square root function to be defined, the expression under the square root must be non-negative:
\[
x + 3 \geq 0
\]
\[
x \geq -3
\]

Thus, the domain of \( f(x) \) is \( [-3, \infty) \).

### Step 2: Determine the inverse of \( f(x) \)

We will now solve for \( f^{-1}(x) \).

1. Start with \( y = \sqrt{x + 3} \).
2. Square both sides:
   \[
   y^2 = x + 3
   \]
3. Solve for \( x \):
   \[
   x = y^2 - 3
   \]

So, the inverse function is \( f^{-1}(x) = x^2 - 3 \).

### Step 3: Domain of the inverse function \( f^{-1}(x) \)

The domain of the inverse function is the range of the original function. The range of \( f(x) = \sqrt{x + 3} \) is \( [0, \infty) \) because the square root function only produces non-negative values.

Thus, the domain of \( f^{-1}(x) \) is \( [0, \infty) \).

### Conclusion

The domain of the inverse function \( f^{-1}(x) \) is \( [0, \infty) \).

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Do you have any questions or need further details?

Here are 5 related questions to deepen your understanding:
1. How would the domain change if the function was \( f(x) = \sqrt{x - 3} \)?
2. What happens to the domain if you work with a cubic root function instead of a square root?
3. How do you find the range of a function algebraically?
4. What is the graphical relationship between a function and its inverse?
5. Can a function have more than one inverse?

**Tip:** Always check the domain of the original function before determining the domain of its inverse.

Consider the function f(x) = sqrt(x + 3). What is the possible domain of the inverse function f^(-1)(x)?

Explore the process of determining the domain of the inverse function for f(x) = sqrt(x + 3). The solution involves understanding the behavior of square root functions and their inverses, leading to the determination of the inverse function as x^2 - 3 and its domain as [0, infinity).

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