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We are asked to find the inverse function \( f^{-1}(x) \) given that \( f(x) = \frac{x^3 - 7}{2} \).

### Steps to find the inverse:
1. Start by setting \( y = f(x) = \frac{x^3 - 7}{2} \).
   
   So, we have the equation:
   \[
   y = \frac{x^3 - 7}{2}
   \]

2. Solve for \( x \) in terms of \( y \).
   - Multiply both sides by 2:
     \[
     2y = x^3 - 7
     \]
   - Add 7 to both sides:
     \[
     x^3 = 2y + 7
     \]
   - Take the cube root of both sides:
     \[
     x = \sqrt[3]{2y + 7}
     \]

3. Now, replace \( y \) with \( x \) to express the inverse function:
   \[
   f^{-1}(x) = \sqrt[3]{2x + 7}
   \]

Thus, the inverse function is:
\[
f^{-1}(x) = \sqrt[3]{2x + 7}
\]

Would you like further clarification or more details on any step?

### Related Questions:
1. What is the domain of the inverse function \( f^{-1}(x) \)?
2. How can you verify that two functions are inverses of each other?
3. How does the process of finding an inverse change for different types of functions?
4. What are the characteristics of a function that guarantee the existence of an inverse?
5. What is the derivative of \( f^{-1}(x) \) using implicit differentiation?

### Tip:
When solving for the inverse of a function, always make sure the original function is one-to-one (i.e., passes the horizontal line test). This ensures that an inverse function exists.

Learn how to find the inverse of the function f(x) = (x^3 - 7)/2 through a detailed step-by-step process. This algebraic problem involves manipulating cubic functions and applying the inverse function theorem, suitable for students in Grades 11-12.

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