To find the inverse of the function \( f(x) = -x^2 + 3x \), we first need to check if the function is invertible. Quadratic functions are generally not one-to-one, meaning they don't pass the horizontal line test and therefore don't have inverses over their entire domain. However, we can find the inverse over a restricted domain.

### Steps to find the inverse:

1. **Rewrite the function** in terms of \( y \):
   \[
   y = -x^2 + 3x
   \]

2. **Solve for \( x \) in terms of \( y \)**:
   The equation \( y = -x^2 + 3x \) is a quadratic equation. Rearrange it into standard quadratic form:
   \[
   -x^2 + 3x - y = 0
   \]
   Multiply the equation by \(-1\) to simplify:
   \[
   x^2 - 3x + y = 0
   \]

3. **Use the quadratic formula** to solve for \( x \). The quadratic formula is:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   In our case, \( a = 1 \), \( b = -3 \), and \( c = y \). Substituting these into the formula:
   \[
   x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(y)}}{2(1)}
   \]
   \[
   x = \frac{3 \pm \sqrt{9 - 4y}}{2}
   \]

4. **Simplify**:
   \[
   x = \frac{3 \pm \sqrt{9 - 4y}}{2}
   \]

Since this is a quadratic equation, there are two possible solutions for \( x \). To ensure the function is invertible, we must restrict the domain. Typically, the domain is restricted to \( x \geq \frac{3}{2} \) or \( x \leq \frac{3}{2} \). Let's choose the domain \( x \geq \frac{3}{2} \), which corresponds to using the positive square root. Therefore, the inverse function is:
\[
f^{-1}(y) = \frac{3 + \sqrt{9 - 4y}}{2}
\]

### Final Answer:
The inverse of \( f(x) = -x^2 + 3x \) over the restricted domain \( x \geq \frac{3}{2} \) is:
\[
f^{-1}(y) = \frac{3 + \sqrt{9 - 4y}}{2}
\]

Would you like more details on any of the steps, or do you have any questions?

Here are 5 related questions for further exploration:

1. How do you find the inverse of a function in general?
2. Why do we need to restrict the domain of quadratic functions to find their inverse?
3. What is the graphical interpretation of a function and its inverse?
4. How can you verify that two functions are inverses of each other?
5. How does the quadratic formula help solve for inverses?

**Tip**: Remember that for a function to have an inverse, it must be one-to-one (pass the horizontal line test), which sometimes requires restricting its domain.

Learn how to find the inverse of the quadratic function f(x) = -x^2 + 3x. This detailed solution includes the quadratic formula, explains the need for restricting the domain, and ensures the function is invertible. Suitable for high school students in Grades 10-12.

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