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 $y = x$.