Question:

Step 1: Understand what reflection in the line $y = x$ means. The reflection of the graph of $y = f(x)$ in the line $y = x$ gives us the inverse function $g(x) = f^{-1}(x)$. Therefore, we need to find the inverse of the given function. Step 2: Set up the equation to find the inverse function. Let $y = f(x)$, so: $$y = (x+2)^2 - 2$$ To find the inverse, we need to solve for $x$ in terms of $y$. Step 3: Isolate the squared term. Add 2 to both sides: $$y + 2 = (x+2)^2$$ Step 4: Take the square root of both sides. $$\sqrt{y+2} = |x+2|$$ Since we are given that $x \geq -2$, we have $x + 2 \geq 0$. Therefore, we take the positive square root: $$\sqrt{y+2} = x + 2$$ Step 5: Solve for $x$. Subtract 2 from both sides: $$x = \sqrt{y+2} - 2$$ Step 6: Write the inverse function by swapping variables. In the inverse function $g(x) = f^{-1}(x)$, we replace $y$ with $x$ and $x$ with $g(x)$: $$g(x) = \sqrt{x+2} - 2$$ Step 7: State the final answer. The function $g(x)$ whose graph is the reflection of $y = f(x)$ in the line $y = x$ is: $$g(x) = \sqrt{x+2} - 2$$ This matches **Option 3**.