find inverse of the function f(x)=x^5-3 over 2

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

### Step 1: Express the function in terms of \( y \)
Let \( y = f(x) = \frac{x^5 - 3}{2} \).

So, 
\[
y = \frac{x^5 - 3}{2}
\]

### Step 2: Solve for \( x \)
Now, we need to solve for \( x \) in terms of \( y \).

First, multiply both sides by 2 to eliminate the fraction:
\[
2y = x^5 - 3
\]

Next, add 3 to both sides:
\[
x^5 = 2y + 3
\]

Finally, take the 5th root of both sides:
\[
x = \sqrt[5]{2y + 3}
\]

### Step 3: Write the inverse function
The inverse of the function is:
\[
f^{-1}(y) = \sqrt[5]{2y + 3}
\]

### Step 4: Substitute \( y \) back to \( x \)
If you want to express the inverse as a function of \( x \), replace \( y \) with \( x \):
\[
f^{-1}(x) = \sqrt[5]{2x + 3}
\]

Thus, the inverse of \( f(x) = \frac{x^5 - 3}{2} \) is:
\[
f^{-1}(x) = \sqrt[5]{2x + 3}
\]

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How do you verify if a function has an inverse?
2. What are the domain and range of \( f^{-1}(x) \)?
3. How do you prove that \( f(f^{-1}(x)) = x \)?
4. What happens if you apply transformations to the function \( f(x) \)?
5. How do you find the derivative of \( f^{-1}(x) \)?

**Tip**: Inverse functions "undo" the operations of the original function.

Learn how to find the inverse of the function f(x) = (x^5 - 3) / 2 with a step-by-step solution. This problem covers important algebraic concepts like solving radical equations and understanding the inverse function theorem. Suitable for students in Grades 10-12.

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