To find the inverse of the function [tex]\( f(x) = 8 \sqrt{x} \)[/tex] for [tex]\( x \geq 0 \)[/tex], we follow these steps:
1. Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = f(x) = 8 \sqrt{x}
\][/tex]
2. Isolate [tex]\( \sqrt{x} \)[/tex]:
[tex]\[
\sqrt{x} = \frac{y}{8}
\][/tex]
3. Square both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \left(\frac{y}{8}\right)^2
\][/tex]
4. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to denote the inverse function [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
f^{-1}(x) = \left(\frac{x}{8}\right)^2
\][/tex]
5. Simplify the expression:
[tex]\[
f^{-1}(x) = \frac{1}{64} x^2
\][/tex]
Given the above steps, the correct inverse function is:
[tex]\[
f^{-1}(x) = \frac{1}{64} x^2
\][/tex]
So the correct answer is:
[tex]\[
\boxed{\text{C. } f^{-1}(x) = \frac{1}{64} x^2, \text{ for } x \geq 0}
\][/tex]
