To find the inverse of the function $f(x) = \sqrt{x + 2} - 7$, we need to follow these steps:

Step 1: Replace $f(x)$ with $y$

Let: $y = \sqrt{x + 2} - 7$

Step 2: Swap $x$ and $y$

To find the inverse, swap $x$ and $y$: $x = \sqrt{y + 2} - 7$

Step 3: Solve for $y$

Now, solve for $y$:

1. Add 7 to both sides: $x + 7 = \sqrt{y + 2}$

2. Square both sides to eliminate the square root: $(x + 7)^2 = y + 2$

3. Subtract 2 from both sides: $y = (x + 7)^2 - 2$

Step 4: Write the inverse function

Thus, the inverse function is: $f^{-1}(x) = (x + 7)^2 - 2$

Would you like a more detailed explanation or have any questions?

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Here are 5 related questions you might find useful:

1. How do you determine the domain and range of inverse functions?
2. What are the properties of square root functions and their inverses?
3. How do you verify that two functions are inverses of each other?
4. Can all functions have inverses? Why or why not?
5. How do restrictions on the domain affect the inverse of a function?

Tip: When finding the inverse, always ensure the original function is one-to-one, meaning it passes the horizontal line test.