To find the inverse of the function [tex]\( f(x) = \frac{\sqrt{x-2}}{6} \)[/tex], let's follow the steps:
1. Set the function equal to [tex]\( y \)[/tex]:
[tex]\[ y = \frac{\sqrt{x-2}}{6} \][/tex]
2. Isolate the square root term:
Multiply both sides by 6 to get:
[tex]\[ 6y = \sqrt{x-2} \][/tex]
3. Remove the square root by squaring both sides:
Square both sides of the equation to get rid of the square root:
[tex]\[ (6y)^2 = x - 2 \][/tex]
4. Simplify the equation:
[tex]\[ 36y^2 = x - 2 \][/tex]
5. Solve for [tex]\( x \)[/tex]:
Add 2 to both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ x = 36y^2 + 2 \][/tex]
Therefore, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = 36x^2 + 2 \][/tex]
Comparing with the given options:
A. [tex]\( f^{-1}(x) = 36 x^2 + 2 \)[/tex], for [tex]\( x \geq 0 \)[/tex]
B. [tex]\( f^{-1}(x) = 6 x^2 + 2 \)[/tex], for [tex]\( x \geq 0 \)[/tex]
C. [tex]\( f^{-1}(x) = 36 x + 2 \)[/tex], for [tex]\( x \geq 0 \)[/tex]
D. [tex]\( f^{-1}(x) = 6 x^2 - 2 \)[/tex], for [tex]\( x \geq 0 \)[/tex]
The correct option is:
[tex]\[ \boxed{A} \][/tex]
