To find the inverse of the function $f(x) = x^5 - \frac{3}{2}$, we need to follow these steps:

1. Rewrite the function using $y$ instead of $f(x)$:

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

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

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

3. Solve for $y$ (which will give us $f^{-1}(x)$):

   • First, isolate the $y^5$ term by adding $\frac{3}{2}$ to both sides:

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

   • Now, solve for $y$ by taking the fifth root of both sides:

     $y = \sqrt[5]{x + \frac{3}{2}}$

Thus, the inverse function is:

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

Let me know if you'd like a more detailed explanation or if you have any questions!

5 Related Questions:

1. What is the domain and range of $f(x) = x^5 - \frac{3}{2}$?
2. How do we verify that two functions are inverses of each other?
3. What is the difference between an inverse function and a reciprocal function?
4. How do the graphs of $f(x)$ and $f^{-1}(x)$ compare?
5. Can all functions be inverted? If not, why?

Tip:

Always check that the inverse function returns the original value when plugged back into the original function to confirm its correctness.