To find the inverse of the function [tex]\( f(x) = \frac{\sqrt{x-2}}{6} \)[/tex], we follow a step-by-step process:
1. First, express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = \frac{\sqrt{x-2}}{6} \][/tex]
2. Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to begin finding the inverse:
[tex]\[ x = \frac{\sqrt{y-2}}{6} \][/tex]
3. Clear the fraction by multiplying both sides by 6:
[tex]\[ 6x = \sqrt{y-2} \][/tex]
4. Square both sides to eliminate the square root:
[tex]\[ (6x)^2 = y - 2 \][/tex]
[tex]\[ 36x^2 = y - 2 \][/tex]
5. Solve for [tex]\( y \)[/tex] by adding 2 to both sides:
[tex]\[ y = 36x^2 + 2 \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = 36x^2 + 2 \][/tex]
Among the given choices:
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 answer matches our derived inverse function. Therefore, the correct answer is:
A. [tex]\( f^{-1}(x) = 36 x^2 + 2 \)[/tex], for [tex]\( x \geq 0 \)[/tex].
