To graph the inverse $f^{-1}$ of the given function $f$, you can follow these steps:

1. Understand the nature of an inverse function: The graph of the inverse function $f^{-1}$ is a reflection of the graph of $f$ over the line $y = x$. This means that for every point $(x, y)$ on the graph of $f$, the corresponding point $(y, x)$ will be on the graph of $f^{-1}$.

2. Identify key points on the graph of $f$: From the image you uploaded, identify points such as $(0,0)$, $(2,4)$, $(3,8)$, etc., as examples of points on the graph of $f$. The inverse function will have the corresponding points $(0,0)$, $(4,2)$, $(8,3)$, etc.

3. Reflect the graph across the line $y = x$: Plot the reflected points from step 2 on the new graph and connect them accordingly to match the shape of the original function, but reflected.

Would you like further assistance with plotting this inverse graph or any details on how to proceed? Let me know!

Here are 5 related questions to explore this further:

1. How does the process of finding the inverse function differ algebraically from graphing it?
2. What is the relationship between the domain and range of $f$ and $f^{-1}$?
3. What are the conditions for a function to have an inverse (bijectivity)?
4. How can you verify if two functions are inverses of each other algebraically?
5. How does the derivative of a function relate to the derivative of its inverse?

Tip: Always ensure that the function you're dealing with is one-to-one before finding its inverse, which means each $y$-value is associated with exactly one $x$-value.