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 \( y \).

find inverse of the function x^5 - 3 over 2

Learn how to find the inverse of the function (x^5 - 3)/2 with a step-by-step breakdown. This problem covers core algebraic concepts, including solving for inverse functions and working with exponents. Suitable for high school students in Grades 10-12.

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