https://file.aiquickdraw.com/mathbot/file/1726195272387fb5q3u8l.jpg

The function given is \( f(x) = \sqrt{x + 2} - 7 \), and we are asked to find which graph represents the inverse of this function, \( f^{-1}(x) \).

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

1. Start with the equation for \( f(x) \):
   \[
   y = \sqrt{x + 2} - 7
   \]

2. Swap \( x \) and \( y \) to begin solving for the inverse:
   \[
   x = \sqrt{y + 2} - 7
   \]

3. Solve for \( y \):
   - Add 7 to both sides:
     \[
     x + 7 = \sqrt{y + 2}
     \]
   - Square both sides to remove the square root:
     \[
     (x + 7)^2 = y + 2
     \]
   - Subtract 2 from both sides:
     \[
     y = (x + 7)^2 - 2
     \]

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

### Step 2: Analyze the inverse function.

The inverse function \( f^{-1}(x) = (x + 7)^2 - 2 \) is a parabola that is shifted 7 units to the left and 2 units down. The parabola opens upward because of the positive square term.

### Step 3: Compare the graphs.

- The correct graph should show a parabola, and the key features to look for are the shift and the direction of the parabola. It should have its vertex at \( (-7, -2) \) and open upwards.

### Conclusion:
Based on the graphs in the image, **Graph B** correctly represents the inverse function \( f^{-1}(x) = (x + 7)^2 - 2 \).

Would you like further details on how the transformation works? Here are five related questions for practice:

1. How would the graph of \( f^{-1}(x) \) change if \( f(x) = \sqrt{x + 3} - 7 \)?
2. What is the domain and range of the inverse function \( f^{-1}(x) \)?
3. How do you find the inverse of a function in general?
4. Can every function have an inverse? Why or why not?
5. How do the graphs of a function and its inverse relate to each other geometrically?

**Tip:** A function's graph and its inverse are reflections of each other across the line \( y = x \).

Which graph correctly shows the graph of f^(-1)(x), the INVERSE of f(x)?

This solution explains how to find the inverse of the function f(x) = sqrt(x + 2) - 7, step by step, and identifies the correct graph for the inverse function f^(-1)(x) = (x + 7)^2 - 2. Learn about function inverses and transformations, suitable for high school math students in Grades 10-12.

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