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.