To determine the inverse of the function [tex]\(f(x) = \frac{\sqrt{x-2}}{6}\)[/tex], we follow these steps:
1. Start with the function [tex]\( f(x) = \frac{\sqrt{x-2}}{6} \)[/tex].
2. To find the inverse, we need to express [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex] where [tex]\(y = f(x)\)[/tex].
So, we set [tex]\( y = \frac{\sqrt{x-2}}{6} \)[/tex].
3. To isolate the square root, multiply both sides by 6:
[tex]\[ 6y = \sqrt{x-2} \][/tex]
4. Next, square both sides to remove the square root:
[tex]\[ (6y)^2 = x - 2 \][/tex]
[tex]\[ 36y^2 = x - 2 \][/tex]
5. Finally, solve for [tex]\(x\)[/tex] by adding 2 to both sides:
[tex]\[ 36y^2 + 2 = x \][/tex]
6. To express the inverse function [tex]\( f^{-1}(x) \)[/tex] in the standard form, replace [tex]\(y\)[/tex] with [tex]\(x\)[/tex]:
[tex]\[ f^{-1}(x) = 36x^2 + 2 \][/tex]
Therefore, the correct inverse function is:
[tex]\[ f^{-1}(x) = 36x^2 + 2 \][/tex]
The correct answer is:
A. [tex]\( f^{-1}(x) = 36x^2 + 2 \)[/tex], for [tex]\(x \geq 0\)[/tex]
