You have given the function \( f^{-1}(x) = \frac{1}{\sqrt{3 - 2x}} \), which represents the inverse of some original function \( f(x) \). Now, if we want to find the original function \( f(x) \), we need to follow these steps to reverse the inverse process.

### Steps to find \( f(x) \):
1. Start with the inverse function:
   \[
   y = \frac{1}{\sqrt{3 - 2x}}
   \]
   where \( y = f^{-1}(x) \).

2. Swap \( x \) and \( y \) to reverse the function:
   \[
   x = \frac{1}{\sqrt{3 - 2y}}
   \]

3. Solve for \( y \) (which will now represent \( f(x) \)):

   Multiply both sides by \( \sqrt{3 - 2y} \) to eliminate the denominator:
   \[
   x \sqrt{3 - 2y} = 1
   \]

4. Divide both sides by \( x \):
   \[
   \sqrt{3 - 2y} = \frac{1}{x}
   \]

5. Square both sides to remove the square root:
   \[
   3 - 2y = \frac{1}{x^2}
   \]

6. Isolate \( y \) by subtracting 3 from both sides:
   \[
   -2y = \frac{1}{x^2} - 3
   \]

7. Finally, divide by \( -2 \) to solve for \( y \):
   \[
   y = \frac{3 - \frac{1}{x^2}}{2}
   \]

Thus, the original function is:
\[
f(x) = \frac{3 - \frac{1}{x^2}}{2}
\]

Would you like further details on any part of this process, or do you have any questions?

Here are some related questions for practice:
1. What is the domain of the function \( f^{-1}(x) \)?
2. How would you verify that \( f(f^{-1}(x)) = x \)?
3. What are the restrictions on \( x \) for the original function \( f(x) \)?
4. How does the graph of \( f(x) \) compare to the graph of \( f^{-1}(x) \)?
5. Can \( f(x) \) or \( f^{-1}(x) \) have any horizontal asymptotes?

**Tip**: When dealing with inverse functions, always check the domain and range carefully since they are swapped between the function and its inverse.

Explore the detailed steps to find the original function from its inverse, given f^(-1)(x) = 1/√(3 - 2x). This solution covers key algebraic concepts, including inverse functions and radical equations, suitable for students in Grades 10-12.

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