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Find the inverse of the following function.

[tex]\[ f(x) = 8 \sqrt{x}, \text{ for } x \geq 0 \][/tex]

A. [tex]\[ f^{-1}(x) = 8 x^2, \text{ for } x \geq 0 \][/tex]

B. [tex]\[ f^{-1}(x) = \frac{1}{8} x^2, \text{ for } x \geq 0 \][/tex]

C. [tex]\[ f^{-1}(x) = \frac{1}{64} x^2, \text{ for } x \geq 0 \][/tex]

D. [tex]\[ f^{-1}(x) = 64 x^2, \text{ for } x \geq 0 \][/tex]



Answer :

GinnyAnswer
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]

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