To find the inverse of the function $f(x) = \frac{x^5 - 3}{2}$, let's go step by step:

1. Start with the given function:

   $y = \frac{x^5 - 3}{2}$

2. Swap $x$ and $y$ to begin the process of finding the inverse:

   $x = \frac{y^5 - 3}{2}$

3. Solve for $y$ (which will be the inverse function):

   • Multiply both sides by 2 to get rid of the denominator: $2x = y^5 - 3$

   • Add 3 to both sides: $2x + 3 = y^5$

   • Take the fifth root of both sides: $y = \sqrt[5]{2x + 3}$

4. Thus, the inverse function is:

   $f^{-1}(x) = \sqrt[5]{2x + 3}$

Let me know if you'd like a deeper explanation or have further questions!

Related Questions:

1. How can we verify that this inverse function is correct?
2. What are the domain and range of the inverse function?
3. How does the graph of a function compare to its inverse?
4. What happens to the inverse if we change the original function slightly, such as $\frac{x^5 - 3}{3}$?
5. How can we use derivatives to analyze the behavior of the inverse function?

Tip: To find the inverse of a function, always remember to swap $x$ and $y$ and solve for $y$.