The problem asks for the graph of the inverse function $f^{-1}(x)$, given the graph of the function $f(x)$. To find the graph of the inverse function, we need to reflect the graph of $f(x)$ across the line $y = x$. This reflection will swap the roles of the $x$- and $y$-coordinates, meaning every point $(x, y)$ on the original graph will map to the point $(y, x)$ on the graph of the inverse.

Let's break down the options based on the given graph in the problem:

• Original Graph Features (Top graph):
  • It has a vertical asymptote at $x = 0$ and a curve that is decreasing as it approaches the asymptote from both sides.

Now let's compare the reflected graphs for the inverse function:

1. Option A: The graph here has a vertical asymptote at $y = 0$, which is correct because reflecting the vertical asymptote in the original graph should give a horizontal asymptote for the inverse function. The shape of the curve also correctly reflects the behavior of the original function.

2. Option B: The curve and asymptote do not appear to reflect the original graph's behavior correctly, as the graph has a different curvature.

3. Option C: The reflection looks incorrect because the shape of the curve and the placement of the asymptote are not aligned with what the inverse should look like.

4. Option D: Similar to option B, the graph doesn't correctly reflect the behavior of the original function.

Correct Answer: Option A best represents the graph of the inverse function $f^{-1}(x)$ after reflecting the original graph across the line $y = x$.

Let me know if you want further details or have any questions!

Follow-up questions:

1. What are the key characteristics of inverse functions in graphs?
2. How does reflecting a function over the line $y = x$ affect its shape?
3. Can every function have an inverse? What conditions must be met?
4. What is the significance of the vertical and horizontal asymptotes in functions and their inverses?
5. How can you check algebraically if a function has an inverse?

Tip: To check if a function has an inverse, ensure it passes the "horizontal line test." If any horizontal line intersects the graph more than once, the function does not have an inverse.