To find the inverse function [tex]\( f^{-1}(x) \)[/tex] of the given function [tex]\( f(x) = 8 \sqrt{x} \)[/tex] for [tex]\( x \geq 0 \)[/tex], we need to follow these steps:
1. Express the function in terms of [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
y = 8 \sqrt{x}
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
- Start by isolating the square root:
[tex]\[
\sqrt{x} = \frac{y}{8}
\][/tex]
- Next, square both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \left( \frac{y}{8} \right)^2
\][/tex]
[tex]\[
x = \frac{y^2}{64}
\][/tex]
3. Express the inverse function:
- Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to write the inverse function:
[tex]\[
f^{-1}(x) = \frac{x^2}{64}
\][/tex]
4. Identify the correct option:
- Based on the obtained inverse function, we compare it with the given options:
[tex]\[
f^{-1}(x) = \frac{1}{64} x^2, \text{ for } x \geq 0
\][/tex]
This corresponds to Option C.
Therefore, the correct answer is:
[tex]\[
\boxed{\text{C: } f^{-1}(x) = \frac{1}{64} x^2, \text{ for } x \geq 0}
\][/tex]
