To find the correct graph of the inverse function $f^{-1}$, we need to follow these steps:

1. Identify Points on the Original Graph $f$: Observe any specific points on the graph of $f$, as these will help us determine corresponding points on the inverse.

2. Swap Coordinates for the Inverse: For any point $(x, y)$ on $f$, the corresponding point on $f^{-1}$ will be $(y, x)$.

3. Check for Reflection Over $y = x$: The inverse function's graph should reflect the original graph across the line $y = x$.

Analysis of the Options

In the given options (A, B, C, D):

• Option A reflects the graph correctly across the line $y = x$ based on the points of the original graph, suggesting that it is the correct representation of the inverse function.

• Options B, C, and D do not match the reflection pattern of the inverse function, as their shapes or orientations do not correctly mirror the dashed original function across $y = x$.

Conclusion

The correct answer is Option A.

Would you like more details on this process or have any questions?

Here are some further questions you might explore:

1. What is the significance of the line $y = x$ in finding inverse functions?
2. How can you determine if two functions are inverses by looking at their graphs?
3. What other characteristics of functions help in identifying if they have inverses?
4. How does the domain and range of $f$ change when finding $f^{-1}$?
5. Can all functions have an inverse? Why or why not?

Tip: When looking for the inverse of a function graphically, remember to reflect across the line $y = x$ and look for the correct swapped points.