To find the inverse of the function $f(x) = 8\sqrt{x}$ for $x \geq 0$, let's proceed step by step.

Step 1: Set up the equation

The given function is: $f(x) = 8\sqrt{x}$ To find the inverse, we start by replacing $f(x)$ with $y$: $y = 8\sqrt{x}$

Step 2: Swap $x$ and $y$

To find the inverse function, we swap $x$ and $y$: $x = 8\sqrt{y}$

Step 3: Solve for $y$

Now, solve for $y$ in terms of $x$:

1. Divide both sides by 8: $\frac{x}{8} = \sqrt{y}$
2. Square both sides to eliminate the square root: $y = \left(\frac{x}{8}\right)^2$
3. Simplify: $y = \frac{x^2}{64}$

Thus, the inverse function is: $f^{-1}(x) = \frac{x^2}{64}$

Answer

This corresponds to option A: $f^{-1}(x) = \frac{1}{64} x^2, \text{ for } x \geq 0$

Would you like further details or have any questions?

Here are five related questions that might interest you:

1. How do we verify that two functions are inverses of each other?
2. Why does squaring eliminate the square root in this problem?
3. What are the conditions for a function to have an inverse?
4. How does restricting the domain affect the existence of an inverse?
5. How would the inverse function change if the original function was $f(x) = k\sqrt{x}$ for any constant $k$?

Tip: When solving for the inverse of a function, remember to swap $x$ and $y$ and solve for $y$ in terms of $x$ to isolate the new function.