The given function is:

$f(x) = \sqrt{x^2 + 9x}, \quad x > 0$

We are tasked with finding the inverse of this function, denoted as $f^{-1}(x)$.

Step-by-Step Process to Find the Inverse:

1. Start by replacing $f(x)$ with $y$:

$y = \sqrt{x^2 + 9x}$

2. Square both sides to eliminate the square root:

$y^2 = x^2 + 9x$

3. Rearrange the equation to form a quadratic in $x$:

$x^2 + 9x - y^2 = 0$

4. Solve the quadratic equation using the quadratic formula:

$x = \frac{-9 \pm \sqrt{9^2 - 4(1)(-y^2)}}{2(1)}$

Simplifying the discriminant:

$x = \frac{-9 \pm \sqrt{81 + 4y^2}}{2}$

$x = \frac{-9 \pm \sqrt{81 + 4y^2}}{2}$

5. Since $x > 0$, choose the positive root:

$x = \frac{-9 + \sqrt{81 + 4y^2}}{2}$

Thus, the inverse function is:

$f^{-1}(y) = \frac{-9 + \sqrt{81 + 4y^2}}{2}$

Alternatively, if we rewrite it in terms of $f^{-1}(x)$, we get:

$f^{-1}(x) = \frac{-9 + \sqrt{81 + 4x^2}}{2}$

This is the formula for the inverse of the given function.

Would you like more details on any step or have any questions?

Here are some related questions:

1. How do you verify if two functions are inverses of each other?
2. What conditions must be met for a function to have an inverse?
3. Can you describe the domain and range of the inverse function in this case?
4. How does the quadratic formula help in solving inverse functions?
5. How do the positive and negative roots in the quadratic formula affect the inverse?

Tip: Always remember that the inverse function swaps the roles of $x$ and $y$. This helps in verifying your results.